Computational methods and dynamical analysis for studying (1+1) dimensional functional equations of mixed integro-differential type

Abstract

In the present paper, the Fibonacci collocation method is implemented to solve $(1+1)$ dimensional difference equations of mixed integro-differential type. First, using the quadratic numerical technique, the mixed functional integro-differential equations are reduced to a system of Fredholm functional integro-differential equations in one dimension. Then, the Fibonacci collocation method is applied to transform the Fredholm functional integro-differential equations into a system of linear algebraic equations. The convergence analysis of the functional integro-differential equations is discussed. The method’s error analysis is presented. Several examples are provided to demonstrate the use of the Fibonacci collocation approach. Maple 18 is used to carry out all of the numerical calculations. In addition, the corresponding errors of the numerical results are computed. Novelty: the numerical simulation demonstrates the reliability and efficiency of the Fibonacci collocation method. The proposed method is very effective, simple, and suitable for solving functional integro-differential equations. Motivation for using the Fibonacci collocation method is that it gives highly accurate solutions.

Introduction

Many scientists and engineers deal with the integro-differential equations (IDEs) through their research in electric circuit problems, neutron diffusion, heat and mass diffusion processes, and electromagnetic theory. Because it is challenging to locate an analytical solution to equations of this type, there are a wide variety of various numerical techniques that may be applied to solve IDEs. Kumar and Gupta in [1] presented Lagrange polynomials for solving Volterra–Fredholm IDEs, also Lagrange polynomials were applied for solving Riccati differential equations [2]. Jafarzadeh and Keramati [3] evaluated the solution of system of IDEs via using Taylor polynomial. Dhundel and Waghmare in [4] presented the Laplace transform method for the purpose of getting the answer to the problem of incomplete IDEs. In order to find a solution to fractional IDEs, Abdou et al. [5] presented the Adomian decomposition approach. Elzaki and Ahmed [6] presented Adomian decomposition, Sumudu transform, homotopy perturbation, and variational iteration techniques for solving IDEs. Al-Hamami et al. [7] designed an analysis to determine the best strategy for the axillary function to get the solution of IDEs. Pandey in [8] applied a finite difference approach to get to the bottom of the problem of IDEs. Al-Bugami applied Toeplitz matrix algorithm and the product Nystrom procedure to solve IDEs in his work [9]. The Haar wavelet procedure was used for the research that Maleknejad and Babaaghaie [10] conducted on the solution of IDEs.

In recent years, functional integro-differential equations (FIDEs) garnered the attention of a great number of authors such as Yavuz and Özkol [11] who applied differential transform approach to get to the resolution of FIDEs. Bernoulli polynomial technique with residual correction was first presented by Sezer et al. in [12] to obtain a solution to FIDEs. During [13], Sezer et al. researched a solution to FIDEs using the Taylor expansion method. Additionally, Sezer and Mollaoglu [14] utilized Gegenbauer polynomials to resolve FIDEs. Lastly, to solve FIDEs, Sezer et al. [15] provided Lucas polynomials and Chebyshev collocation points.

In this paper, the solution of $(1+1)$ dimensional mixed functional integro-differential equations type (MFIDEs) in $(1+1)$ dimension in displacement and time in the space $L^2[a,b]\times C[0,T],T<1$, is investigated by using the quadratic numerical technique (QNT) and Fibonacci collocation method (FCM). Abdou et al. [16] presented the solution of MFIDEs by using the separation of variables and Bernoulli polynomial technique. Araghi and colleagues [17] investigated the application of FCM to find a solution to Cauchy integral equations. Bahşi and Yalçinbaş in [18] applied FCM for solving fractional diffusion equations. Also in [19], Bahşi and Yalçinbaş used FCM with a residual error function to solve IDEs. Nadir [20] applied FCM to get done the solution of one-dimensional Volterra and Fredlom IEs. Mirzaee and Hoseini, in their paper [21], investigated the use of FCM in the solution of Volterra–Fredholm IEs. Abo-Eldahab et al. [22] presented a spectral collocation algorithm for solving IDEs by FCM. Mohamed [23] used FCM to solve fractional diffusion equations. In [24], Lotfi et al. applied the FCM approach to first-order fuzzy IDEs. Mokhtar et al. in [25] introduced FCM to resolve Volterra–Fredholm integral equations. Ebadi and Haghkhah [26] investigated some properties of Fibonacci numbers. Darabi and Parandin [27] studied the solution of fuzzy Volterra integral equations via FCM. In [28], Haq and Ali studied the response of the two-dimensional Sobolev problem using FCM. In the research carried out by Kumar et al. [29], the authors developed an operational matrix technique that made use of fractional Lagrange polynomials to deal with fractional order that is brought up in the hypothesis of chemical reactors with the nonlocal BVPs. Mahdy and colleagues in [30] presented an approach of computing that is used for solving mixed integral equations of two kinds that have singular kernels. Using Alhazmi and his colleagues’ work, [31] discusses computational ways for solving combined IEs concerning symmetric highly singular kernels with $(1+1)$ dimension. A computer technique was developed by Mahdy et al. in [32] for the purpose of solving three-dimensional mixed Volterra–Fredholm IEs. Approximate calculation solving the Cauchy IEs employing Lucas polynomials was presented by Mahdy and Mohamed in [33]. There are collocation methods based on polynomials for solving IDEs such as Taylor Chebyshev in [34], Laguerre polynomials in [35], Pell–Lucas polynomials in [36], Lucas polynomials in [37], shifted Chebyshev polynomials in [38], Bernoulli polynomials in [39], Chebyshev polynomials of sixth kind in [40, 41], Chebyshev polynomials of eighth kind in [42], the least square method in [43]. Existence and uniqueness of solutions for fractional IDEs were studied in [44] by Schaefer’s fixed point theorem. Approximations for partial integro differential equations (PIDEs) on an adaptive mesh were presented in [45]. Homotopy analysis was presented in [46] for solving Volterra–Fredholm IDEs. Rostami in [47] used a new technique for solving system of PIDEs. Rostami and Maleknejad in [48] applied hybrid functions to obtain the solution of PIDEs. In [49], Rostami presented a new wavelet method for solving nonlinear PIDEs. In [50], Rostami and Maleknejad used hybrid functions for solving nonlinear mixed PIDEs. Also Rostami and Maleknejad in [51] presented hybrid modified block-pulse functions to obtain the solution of a system of PIDEs. Rostami in [52] applied Hermite wavelet Galerkin for solving Volterra PIDEs.

The main advantage of Fibonacci polynomials is that the coefficients of individual terms in Fibonacci polynomials are smaller than the coefficients of individual terms in the classical orthogonal polynomials such as Chebyshev, shifted Chebyshev, and Legendre polynomials. These properties encourage us to implement these polynomials as functional expansion. Also Fibonacci numbers and their related polynomials are of fundamental importance in different disciplines such as numerical analysis, algebra, geometry, combinatorics, approximation theory, statistics, and number theory.

The rest of the work is organized as described below. The discussion of convergence analysis is presented in Sect. 2. The error analysis is discussed in Sect. 3. In Sect. 4, quadratic numerical technique is applied to transform the MFIDEs into a system of Fredholm FIDEs. Section 5 is devoted to studying the numerical solution of Fredholm FIDEs by FCM. Some numerical applications and error in each example are presented in Sect. 6. The final section has been committed to the discussion of the conclusion.

Take into consideration the nth FIDEs with variable coefficients in the space $L^2[a,b]\times C[0,T],T<1$, which are as follows:

1.1

$\begin{array}{rl}L[\mathrm{\Phi }(x,t)]=& \sum _{r=0}^n{A}_r(x)\frac{{\partial }^r\mathrm{\Phi }(x,t)}{\partial x^r}+\sum _{r=0}^k{B}_r(x)\frac{{\partial }^r\mathrm{\Phi }(x+\alpha ,t)}{\partial x^r}\\ & -{\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w)\mathrm{\Phi }(w-\gamma ,\tau )dwd\tau =H(x,t),n\ge k,\end{array}$ assuming the $(m-1)$ conditions:

1.2

$\begin{array}{c}\sum _{r=0}^{m-1}[{\alpha }_{jk}\frac{{\partial }^r\mathrm{\Phi }(a,t)}{\partial x^r}+{\beta }_{jr}\frac{{\partial }^r\mathrm{\Phi }(b,t)}{\partial x^r}+{\gamma }_{jr}\frac{{\partial }^r\mathrm{\Phi }(c,t)}{\partial x^r}]\hfill \\ ={\eta }_j(t),a\le c\le b,\mathrm{\forall }t\in [0,T],T<1,{\eta }_j(0)\ne 0,j=1,2,\dots ,m-1,\hfill \end{array}$ where $A_r(x),B_r(x),0\le r\le n$, $x\in [a,b]$, are known and continuous functions. In the integral term, the kernel of position is denoted by $k(x,w)\in L^2([a,b]\times [a,b])$, while $F(t,\tau )\in C([0,T]\times [0,T])$ is denoted as the kernel of time. The free term $H(x,t)$ has a known function and is determined by the space $L^2[a,b]\times C[0,T],T<1$, the function ${\eta }_j(t)$ is given and denoted in the class $C([0,T]),0\le t\le T<1$, the coefficients denoted by ${\alpha }_{jr}$, ${\beta }_{jr}$, ${\gamma }_{jr}$ are constants, whereas $\mathrm{\Phi }(x,t)$ represents an unknown function.

If you need a solution that is not duplicated somewhere else, you will need to impose some restrictions on a free function; as a result, the unidentified function will act in the same approach as the defined function works. Therefore, when specific requirements are imposed on the function that is available, those conditions will also be applied to the function that is unknown. In that case, it will be the only solution that utilizes and is compatible with the conditions that have been established.

Convergence analysis of the functional integro-differential equations

Consider the continuity of the following functions:

(i) For $0\le k\le n$, the two continuous functions $A_k(x)$, $B_k(x)$ satisfy $\mid A_k(x)\mid \le A_k$, $\mid B_k(x)\mid \le B_k$.

(ii) Both the time kernel, denoted by $g(t,\tau )$, and the position kernel, denoted by $k(x,y)$, are satisfied in their respective contexts $\mid g(t,\tau )\mid \le g$,

$\mid k(x,y)\mid \le c$.

(iii) The function $H(x,t)$ that is being presented along with its derivatives time in terms of n− with regard to x belong to the space $L^2[a,b]\times C[0,T]$, and the norm is $\parallel H(x,t)\parallel =\underset{0\le t\le T}{max}{\int }_0^t(\sqrt{{\int }_a^b{H}^2(x,\tau )}dx)d\tau =H.$

Lemma 2.1

In addition to requirements (i) through (iii), it is true that the infinite series${\sum }_{j=0}^{\mathrm{\infty }}{\mathrm{\Psi }}_j(x,t)$has uniform convergence to a continuous response function$\mathrm{\Phi }(x,t)$.

Proof

Consider the two solutions ${\mathrm{\Phi }}_{\ell -1}(x,t),{\mathrm{\Phi }}_{\ell }(x,t)\in {\{{\mathrm{\Phi }}_j(x,t)\}}_{j=0}^{\mathrm{\infty }}$. Hence, we have

2.1

$\begin{array}{c}\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_{\ell }(x,t)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_{\ell }(x+\alpha ,t)}{\partial x^k}\hfill \\ -{\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w){\mathrm{\Phi }}_{\ell -1}(w-\gamma ,\tau )dwd\tau =H(x,t),n\ge m,\hfill \end{array}$ and

2.2

$\begin{array}{c}\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_{\ell -1}(x,t)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_{\ell -1}(x+\alpha ,t)}{\partial x^k}\hfill \\ -{\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w){\mathrm{\Phi }}_{\ell -2}(w-\gamma ,\tau )dwd\tau =H(x,t),n\ge m,\hfill \end{array}$ under the following conditions:

2.3

$\begin{array}{c}\sum _{k=0}^{n-1}[{\alpha }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_{\ell }(a,t)}{\partial x^k}+{\beta }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_{\ell }(b,t)}{\partial x^k}+{\gamma }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_{\ell }(c,t)}{\partial x^k}]={\eta }_i(t),\hfill \\ \sum _{k=0}^{n-1}[{\alpha }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_{\ell -1}(a,t)}{\partial x^k}+{\beta }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_{\ell -1}(b,t)}{\partial x^k}+{\gamma }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_{\ell -1}(c,t)}{\partial x^k}]={\eta }_i(t).\hfill \end{array}$ We can conclude the following from (2.1) and (2.2):

2.4

$\begin{array}{c}\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k}{\partial x^k}[{\mathrm{\Phi }}_{\ell }(x,t)-{\mathrm{\Phi }}_{\ell -1}(x,t)]+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k}{\partial x^k}[{\mathrm{\Phi }}_{\ell }(x+\alpha ,t)-{\mathrm{\Phi }}_{\ell -1}(x+\alpha ,t)]\hfill \\ ={\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w)[{\mathrm{\Phi }}_{\ell -1}(w-\gamma ,\tau )-{\mathrm{\Phi }}_{\ell -2}(w-\gamma ,\tau )]dwd\tau .\hfill \end{array}$ Also, conditions (2.3) yield

2.5

$\begin{array}{c}\sum _{k=0}^{n-1}\{{\alpha }_{ik}\frac{{\partial }^k}{\partial x^k}[{\mathrm{\Phi }}_{\ell }(a,t)-{\mathrm{\Phi }}_{\ell -1}(a,t)]+{\beta }_{ik}\frac{{\partial }^k}{\partial x^k}[{\mathrm{\Phi }}_{\ell }(b,t)-{\mathrm{\Phi }}_{\ell -1}(b,t)]\hfill \\ +{\gamma }_{ik}\frac{{\partial }^k}{\partial x^k}[{\mathrm{\Phi }}_{\ell }(c,t)-{\mathrm{\Phi }}_{\ell -1}(c,t)]\}=0.\hfill \end{array}$ Introduce

2.6

${\mathrm{\Psi }}_{\ell }(x,t)={\mathrm{\Phi }}_{\ell }(x,t)-{\mathrm{\Phi }}_{\ell -1}(x,t),$ where

2.7

${\mathrm{\Phi }}_{\ell }(x,t)=\sum _{j=0}^{\ell }{\mathrm{\Psi }}_j(x,t),\ell =1,2,\dots .$ Equation (2.4) shall be adapted as follows:

2.8

$\begin{array}{c}\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(x,t)+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(x+\alpha ,t)\hfill \\ ={\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w){\mathrm{\Psi }}_{\ell -1}(w-\gamma ,\tau )dwd\tau ,\hfill \end{array}$ assuming the following conditions:

2.9

$\sum _{k=0}^{n-1}\{{\alpha }_{ik}\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(a,t)+{\beta }_{ik}\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(b,t)+{\gamma }_{ik}\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(c,t)\}=0,\mathrm{\forall }t\in [0,T].$ According to the successive approximation method, there are ${\mathrm{\Phi }}_0(x,t)={\mathrm{\Psi }}_0(x,t)$ and ${\mathrm{\Psi }}_0(x,t)$ satisfying the following equation:

2.10

$\{\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_0(x,t)+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_0(x+\alpha ,t)=H(x,t)\}.$ Applying the Taylor expansion for the unknown function ${\mathrm{\Psi }}_{\ell }(x\pm \sigma ,t)$ and neglecting the second order, we have

2.11

$\underset{k}{max}[\sum _{k=0}^n\{A_k(x)+B_k(x)\}\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(x,t)]\le {\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w){\mathrm{\Psi }}_{\ell -1}(w,\tau )dwd\tau .$ In addition,

2.12

$\underset{k}{max}[\sum _{k=0}^n\{A_k(x)+B_k(x)\}\frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_0(x,t)]\le H(x,t).$ By making use of the characteristics of the modulus, along with the assistance conditions (i), (ii), we have found that $\underset{k}{max}\sum _{k=0}^n\parallel \frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(x,t)\parallel \le \rho \parallel {\mathrm{\Psi }}_{\ell -1}(x,t)\parallel ,(\rho =\frac{gc(a-b)T}{{max}_k{\sum }_{k=0}^n\{A_k(x)+B_k(x)\}}).$ By induction and with the aid of inequality (2.12) and condition (iii), we get $\underset{k}{max}\sum _{k=0}^n\parallel \frac{{\partial }^k}{\partial x^k}{\mathrm{\Psi }}_{\ell }(x,t)\parallel \le {\rho }^{\ell }H.$ This bound can make the sequence $\{{\mathrm{\Psi }}_{\ell }(x,t)\}$ converge; after that, the sequence $\{{\mathrm{\Phi }}_{\ell }(x,t)\}$ converges. Therefore, the infinite series $\mathrm{\Phi }(x,t)={\sum }_{j=0}^{\mathrm{\infty }}{\mathrm{\Psi }}_j(x,t)$ is uniformly convergent with the terms ${\mathrm{\Psi }}_j(x,t)$ and can be controlled through ${\rho }^j,\mathrm{\forall }t\in [0,T]$. □

Error analysis

In this section, we shall use the technique of residual correction to provide an error analysis of MFIDEs (1.1) when operating under conditions (1.2). In their work [19], Yalcinbas and Bahsi presented a residual error function to solve one-dimensional IDEs. The residual function is defined in the following manner:

3.1

$\begin{array}{c}{R}_N(x,t)=\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_N(x,t)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_N(x+\alpha ,t)}{\partial x^k}\hfill \\ -{\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w){\mathrm{\Phi }}_N(w-\gamma ,\tau )dwd\tau -H(x,t),\hfill \end{array}$ where ${\mathrm{\Phi }}_N(x,t)$ is the approximate solution of (1.1). Moreover, ${\mathrm{\Phi }}_N(x,t)$ satisfies the following:

3.2

$\begin{array}{c}\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_N(x,t)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_N(x+\alpha ,t)}{\partial x^k}\hfill \\ -{\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w){\mathrm{\Phi }}_N(w-\gamma ,\tau )dwd\tau =R_N(x,t)+H(x,t),\hfill \end{array}$ given the conditions

3.3

$\sum _{k=0}^{n-1}[{\alpha }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_N(a,t)}{\partial x^k}+{\beta }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_N(b,t)}{\partial x^k}+{\gamma }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_N(c,t)}{\partial x^k}]={\eta }_i(t).$ Subtracting (3.2) from (1.1), also (3.3) from (1.2), we obtain

3.4

$\begin{array}{c}\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k{E}_N(x,t)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k{E}_N(x+\alpha ,t)}{\partial x^k}\hfill \\ ={\int }_0^t{\int }_a^{b}F(t,\tau )k(x,w)E_N(w-\gamma ,\tau )dwd\tau -R_N(x,t),\hfill \end{array}$ given the conditions

3.5

$\sum _{k=0}^{n-1}[{\alpha }_{ik}\frac{{\partial }^k{E}_N(a,t)}{\partial x^k}+{\beta }_{ik}\frac{{\partial }^k{E}_N(b,t)}{\partial x^k}+{\gamma }_{ik}\frac{{\partial }^k{E}_N(c,t)}{\partial x^k}]=0.$ Using conditions (i)–(iii), after applying the Taylor expansion and neglecting the second derivatives, we get $\underset{k}{max}\mid A_k+B_k\mid \parallel \frac{{\partial }^k{E}_N(x,t)}{\partial x^k}\parallel \le gc(a-b)T\parallel E_N(x,t)\parallel +\parallel R_N(x,t)\parallel .$ The error estimate at $k=0$ is $\parallel E_N(x,t)\parallel \le \frac{\parallel R_N(x,t)\parallel }{\{\mid A_0+B_0\mid -gc(a-b)T\}},(gc(a-b)T<\mid A_0+B_0\mid ).$ While for $k>0$, we have $\parallel \frac{{\partial }^k{E}_N(x,t)}{\partial x^k}\parallel \le \frac{\parallel R_N(x,t)\parallel }{{max}_k\mid A_k+B_k\mid }.$

Quadratic numerical technique

In this part, a quadratic numerical approach has been applied to MFIDEs (1.1) to get a system of Fredholm FIDEs in displacement. For this purpose, we split the interval $[0,T],0\le \tau \le t\le T<1$, as $0=t_0<t_1<\cdots <t_j<\cdots <t_N=T$, where $t=t_j$, $j=0,1,2,\dots ,N$. Therefore, MFIDEs (1.1) can be expressed as follows:

4.1

$\begin{array}{c}L[\mathrm{\Phi }(x,t_j)]=\sum _{k=0}^n{A}_k(x)\frac{{\partial }^k\mathrm{\Phi }(x,t_j)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k\mathrm{\Phi }(x+\alpha ,t_j)}{\partial x^k}\hfill \\ -{\int }_0^{t_j}{\int }_a^{b}g(t_j,\tau )k(x,w)\mathrm{\Phi }(w-\gamma ,\tau )dwd\tau =H(x,t_j),\hfill \end{array}$ given the conditions

4.2

$\sum _{k=0}^{n-1}[{\alpha }_{ik}\frac{{\partial }^k\mathrm{\Phi }(a,t_j)}{\partial x^k}+{\beta }_{ik}\frac{{\partial }^k\mathrm{\Phi }(b,t_j)}{\partial x^k}+{\gamma }_{ik}\frac{{\partial }^k\mathrm{\Phi }(c,t_j)}{\partial x^k}]={\eta }_i(t_j),i=1,2,\dots ,n-1.$ The expression for equation (4.1) can also be written as

4.3

$\begin{array}{rcl}L[\mathrm{\Phi }(x,t_j)]& =& \sum _{k=0}^n{A}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_j(x)}{\partial x^k}+\sum _{k=0}^m{B}_k(x)\frac{{\partial }^k{\mathrm{\Phi }}_j(x+\alpha )}{\partial x^k}\\ & & -\sum _{\xi =0}^j{\omega }_{\xi }{g}_{j,\xi }{\int }_a^{b}k(x,w){\mathrm{\Phi }}_{\xi }(w-\beta )dw=H_j(x),\end{array}$ given the conditions from (4.2)

4.4

$\sum _{k=0}^{n-1}[{\alpha }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_j(a)}{\partial x^k}+{\beta }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_j(b)}{\partial x^k}+{\gamma }_{ik}\frac{{\partial }^k{\mathrm{\Phi }}_j(c)}{\partial x^k}]={\eta }_{i,j},i=1,2,\dots ,n-1,$ where ${\omega }_{\xi }=\{\begin{array}{c}\frac{h}{2},\xi =0,\\ h,0<\xi <T,\\ \frac{h}{2},\xi =T,h=\frac{T}{N}.\end{array}$

Fibonacci collocation procedure

In this part, we shall use FCM to get the solution of the system of Fredholm FIDEs (4.3) under conditions (4.4).

Definition 5.1

([53])

Fibonacci polynomials (FPs) are defined from the recursion relation:

5.1

$F_{\ell +2}(x)=x{F}_{\ell +1}(x)+F_{\ell }(x),\ell \ge 0,F_0(x)=0,F_1(x)=1.$ Moreover FPs are given from the explicit form $F_{\ell }(x)=\frac{{(x+\sqrt{x^2+4})}^{\ell }-{(x-\sqrt{x^2+4})}^{\ell }}{2^{\ell }\sqrt{x^2+4}}$ or the following expansion

5.2

$F_{\ell }(x)=\sum _{ȷ=0}^{\lfloor \frac{\ell -1}{2}\rfloor }\left(\begin{array}{c}\ell -ȷ-1\\ ȷ\end{array}\right)x^{\ell -2ȷ-1}.$

Properties of Fibonacci polynomials

1) The first derivatives of FPs are presented by [53] in the following form: $\frac{d{F}_{j+1}(x)}{dx}=\sum _{m=0}^{\lfloor \frac{j-1}{2}\rfloor }{(-1)}^{m+1}(2m-j)F_{j-2m}(x),j\ge 1,$ where $\lfloor \frac{\ell }{2}\rfloor$ is the largest integer in $\frac{\ell }{2}$.

2) The inverse formula of FPs is given by [54] $x^m=\sum _{j=0}^{\lfloor \frac{m}{2}\rfloor }{(-1)}^{j+1}[\left(\begin{array}{c}m\\ j-1\end{array}\right)-\left(\begin{array}{c}m\\ j\end{array}\right)]F_{m-2j+1}(x),m\ge 0.$ 3) The FPs have a generating function $G(\xi ,t)=\frac{t}{1-t^2-t\xi }=\sum _{n=0}^{\mathrm{\infty }}{F}_n(\xi )t^2=t+\xi t^2+({\xi }^2+1)t^3+({\xi }^3+2\xi )t^4+\cdots .$ 4) The FPs are normalized so that $F_n(1)=F_n,$ where the $F_n$ are Fibonacci numbers.

Function approximation of solution

The FCM provides the following truncated form of the approximate solution to equation (4.3):

5.3

$\{\mathrm{\Phi }(x,t_j)={\mathrm{\Phi }}_j(x)\simeq \sum _{\ell =0}^{M+1}{c}_{\ell ,j}{F}_{\ell }(x)\},$ where $c_{\ell ,j},\ell =0,1,2,\dots ,M+1;j=0,1,\dots ,M$ have the Fibonacci coefficients unknown and $F_{\ell }(x)$ have FCM, the definition of which may be found in (5.2).

In a similar manner, the definitions of ${\mathrm{\Phi }}_j(x+\alpha )$ and ${\mathrm{\Phi }}_j^{(k)}(x+\alpha )$ are as follows:

5.4

$\mathrm{\Phi }(x+\alpha ,t_j)={\mathrm{\Phi }}_j(x+\alpha )\simeq \sum _{\ell =0}^{M+1}{c}_{\ell ,j}{F}_{\ell }(x+\alpha ),$

5.5

${\mathrm{\Phi }}_j^{(k)}(x+\alpha )\simeq \sum _{\ell =0}^{M+1}{c}_{\ell ,j}{F}_{\ell }^{(k)}(x+\alpha ).$ Substituting from (5.3)–(5.5) into (4.3), we obtain

5.6

$\begin{array}{c}\sum _{\ell =0}^{M+1}{c}_{\ell ,j}\{\sum _{k=0}^n{A}_k(x)F_{\ell }^{(k)}(x)+\sum _{k=0}^m{B}_k(x)F_{\ell }^{(k)}(x+\alpha )\hfill \\ -\sum _{\xi =0}^j{\omega }_{\xi }{g}_{j,\xi }{\int }_a^{b}k(x,u)F_{\ell }(u-\beta )du\}=H_j(x)\hfill \end{array}$ through the utilization of the following collocation points into (5.6):

5.7

$x_r=a+\frac{b-a}{M}(r-1),r=1,2,\dots ,M+1.$ Therefore, we obtain the following system of $(M+1)\times (N+1)$ linear algebraic equations with $(M+1)\times (N+1)$ unknowns:

5.8

$\begin{array}{c}\sum _{\ell =0}^{M+1}{c}_{\ell ,j}\{\sum _{k=0}^n{A}_k(x_r)F_{\ell }^{(k)}(x_r)+\sum _{k=0}^m{B}_k(x_r)F_{\ell }^{(k)}(x_r+\alpha )\hfill \\ -\sum _{\xi =0}^j{\omega }_{\xi }{g}_{j,\xi }{\int }_a^{b}k(x_r,u)F_{\ell }(u-\beta )du\}=H_j(x_r),\hfill \end{array}$ with the conditions

5.9

$\begin{array}{c}\sum _{k=0}^{n-1}\sum _{\ell =0}^{M+1}{c}_{\ell ,j}[{\alpha }_{ik}{F}_{\ell }^{(k)}(a)+{\beta }_{ik}{F}_{\ell }^{(k)}(b)+{\gamma }_{ik}{F}_{\ell }^{(k)}(c)]\hfill \\ ={\eta }_{i,j},i=1,2,\dots ,n-1,j=0,1,\dots ,N.\hfill \end{array}$ The following theorem discusses the convergence analysis of the Fibonacci expansion.

Theorem 5.1

([55])

If$\mathrm{\Phi }(x)$is defined on [0, 1] with$|{\mathrm{\Phi }}^{(k)}(0)|\le M^k$, $k\ge 0$, M is a positive constant, and$\mathrm{\Phi }(x)=\sum _{i=0}^{\mathrm{\infty }}{c}_i{F}_i(x),$then we have:

(1) $|c_i|\le \frac{M^{i-1}coshM}{2^{i-1}(i-1)!}$;

(2) The series converges absolutely.

Applications

In this section, we will examine the applicability of the provided approaches by analyzing four numerical instances to see whether or not they are accurate. The Maple 18 software was utilized throughout the entire process of arriving at the numerical results.

Example 1

Study the following the third-order MFIDEs:

6.1

$\begin{array}{c}\frac{{\partial }^3}{\partial x^3}(\mathrm{\Phi }(x,t))-x\frac{{\partial }^2}{\partial x^2}(\mathrm{\Phi }(x,t))+x^2\frac{\partial }{\partial x}(\mathrm{\Phi }(x,t))+\frac{\partial }{\partial x}(\mathrm{\Phi }(x+1,t))-2x(\mathrm{\Phi }(x+1,t))\hfill \\ -{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{3}xu\mathrm{\Phi }(u-1,\tau )dud\tau =H(x,t),\hfill \end{array}$ given the conditions $\begin{array}{c}\mathrm{\Phi }(-1,t)+16\mathrm{\Phi }(0,t)-\mathrm{\Phi }(1,t)=0,\frac{\partial }{\partial x}\mathrm{\Phi }(-1,t)+8\frac{\partial }{\partial x}\mathrm{\Phi }(0,t)-\frac{\partial }{\partial x}\mathrm{\Phi }(1,t)=0,\hfill \\ \frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(-1,t)+4\frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(0,t)-\frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(1,t)=0,\hfill \end{array}$ where $H(x,t)=(2x^5-4x^4-44x^3-60x^2+28x+56)(0.01+t).$ In the following illustration, we will apply QNT and FCM to equation (6.1) for $N=M=5$, $T=[0,0.9]$, and using the collocation points (5.7), we obtain system of $6\times 6$ equations with the same number of the unknown coefficients. Solving this system, we get the unknown coefficient of Fibonacci polynomials in the form: $\begin{array}{c}{c}_{1,0}=-0.02999999860,c_{2,0}=-0.04000000829,c_{3,0}=0.03000000267,\hfill \\ c_{4,0}=0.04000000018,\hfill \\ c_{5,0}=0.009999999802,c_{6,0}=6.98\times {10}^{-11},c_{1,1}=-0.6300001793,\hfill \\ c_{2,1}=-0.8399989057,\hfill \\ c_{3,1}=0.6299996492,c_{4,1}=0.8399999747,c_{5,1}=0.2100000281,\hfill \\ c_{6,1}=-8.96\times {10}^{-9},\hfill \\ c_{1,2}=-1.229999809,c_{2,2}=-1.640001454,c_{3,2}=1.230000499,\hfill \\ c_{4,2}=1.640000024,\hfill \\ c_{5,2}=0.4099999617,c_{6,2}=1.297\times {10}^{-8},c_{1,3}=-1.829999610,\hfill \\ c_{2,3}=-2.440002761,\hfill \\ c_{3,3}=1.830000938,c_{4,3}=2.440000044,c_{5,3}=0.6099999282,c_{6,3}=2.470\times {10}^{-8},\hfill \\ c_{1,4}=-2.429999369,c_{2,4}=-3.240003932,c_{3,4}=2.430001289,\hfill \\ c_{4,4}=3.240000075,\hfill \\ c_{5,4}=0.809999902,c_{6,4}=3.346\times {10}^{-8},c_{1,5}=-2.729999698,\hfill \\ c_{2,5}=-3.640001653,\hfill \\ c_{3,5}=2.730000523,c_{4,5}=3.640000036,c_{5,5}=0.9099999600,c_{6,5}=1.36\times {10}^{-8}.\hfill \end{array}$

Substituting from the above coefficients into equation (5.3), we acquire the approximate solution in the following forms having varying perceptions of the passage of time in the interval [0,0.9]: $\begin{array}{c}{\mathrm{\Phi }}_0(x)=0.01000000387+0.03999999228x+0.06000000208x^2+0.04000000046x^3\hfill \\ \phantom{{\mathrm{\Phi }}_0(x)=}+0.009999999802x^4\hfill \\ \phantom{{\mathrm{\Phi }}_0(x)=}+6.98\times {10}^{-11}{x}^5,t=0.\hfill \\ {\mathrm{\Phi }}_1(x)=0.2099994980+0.8400010164x+1.259999734x^2+0.8399999389x^3\hfill \\ \phantom{{\mathrm{\Phi }}_1(x)=}+0.2100000281x^4-8.96\times {10}^{-9}{x}^5,t=0.2.\hfill \\ {\mathrm{\Phi }}_2(x)=0.4100006517+1.639998633x+2.460000384x^2+1.640000076x^3\hfill \\ \phantom{{\mathrm{\Phi }}_2(x)=}+0.4099999617x^4+1.297\times {10}^{-8}{x}^5,t=0.4.\hfill \\ {\mathrm{\Phi }}_3(x)=0.6100012562+2.439997401x+3.660000723x^2+2.440000143x^3\hfill \\ \phantom{{\mathrm{\Phi }}_3(x)=}+0.6099999282x^4+2.47\times {10}^{-8}{x}^5,t=0.6.\hfill \\ {\mathrm{\Phi }}_4(x)=0.810001822+3.239996318x+4.860000995x^2+3.240000209x^3\hfill \\ \phantom{{\mathrm{\Phi }}_4(x)=}+0.809999902x^4+3.346\times {10}^{-8}{x}^5,t=0.8.\hfill \\ {\mathrm{\Phi }}_5(x)=0.9100007850+3.639998460x+5.460000403x^2+3.640000090x^3\hfill \\ \phantom{{\mathrm{\Phi }}_5(x)=}+0.90999996x^4+1.36\times {10}^{-8}{x}^5,t=0.9.\hfill \end{array}$ The absolute errors committed in Example 1 are compared with various values of time and position, and the results of this comparison are provided in Tables 1 and 2, as well as in Fig. 1.

[See PDF for image]

Figure 1

The absolute errors of Example 1

Table 1. Numerical outcomes (absolute error) of Example 1

x	T = 0	T = 0.2	T = 0.4
−1	6.98 × 10−11	8.96 × 10−9	1.297 × 10−8
−0.8	2.2872064 × 10−11	2.9360128 × 10−9	4.25 × 10−9
−0.6	5.427648 × 10−12	6.967296 × 10−10	1.0085472 × 10−9
−0.4	7.14752 × 10−13	9.17504 × 10−11	1.328128 × 10−10
−0.2	2.2336 × 10−14	2.8672 × 10−12	4.1504 × 10−12
0	0.00000000	0.00000000	0.00000000
0.2	2.2336 × 10−14	2.8672 × 10−12	4.1504 × 10−12
0.4	7.14752 × 10−13	9.17504 × 10−11	1.328128 × 10−10
0.6	5.427648 × 10−12	6.967296 × 10−10	1.0085472 × 10−9
0.8	2.2872064 × 10−11	2.9360128 × 10−9	4.2500096 × 10−9
1	6.98 × 10−11	8.96 × 10−8	1.297 × 10−9

Table 2. Numerical results (absolute error) of Example 1

x	T = 0.6	T = 0.8	T = 0.9
−1	2.47 × 10−8	3.346 × 10−8	1.36 × 10−8
−0.8	8.093696 × 10−9	1.09641728 × 10−8	4.456448 × 10−9
−0.6	1.920672 × 10−9	2.6018496 × 10−9	1.057536 × 10−9
−0.4	2.52928 × 10−10	3.426304 × 10−10	1.39264 × 10−10
−0.2	7.904 × 10−12	1.07072 × 10−11	4.352 × 10−12
0	0.00000000	0.00000000	0.00000000
0.2	7.904 × 10−12	1.07072 × 10−11	4.352 × 10−12
0.4	2.52928 × 10−10	3.426304 × 10−10	1.39264 × 10−10
0.6	1.920672 × 10−9	2.6018496 × 10−9	1.057536 × 10−9
0.8	8.09369 × 10−9	1.09641728 × 10−8	4.456448 × 10−9
1	2.47 × 10−8	3.346 × 10−8	1.360 × 10−9

Example 2

Take into consideration the MFIDEs of the second order:

6.2

$\begin{array}{c}\frac{{\partial }^2}{\partial x^2}(\mathrm{\Phi }(x,t))+x^2\frac{\partial }{\partial x}(\mathrm{\Phi }(x,t))+(\mathrm{\Phi }(x,t))+\frac{{\partial }^2}{\partial x^2}(\mathrm{\Phi }(x+1,t))\hfill \\ -2x\frac{\partial }{\partial x}(\mathrm{\Phi }(x+1,t))+(\mathrm{\Phi }(x+1,t))\hfill \\ -{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{2}x{u}^2\mathrm{\Phi }(u-1,\tau )dud\tau =e^t(3x^4+2x^3-12x^2+3x+27),\hfill \end{array}$ given the conditions $\mathrm{\Phi }(-1,t)+8\mathrm{\Phi }(0,t)-\mathrm{\Phi }(1,t)=0,\frac{\partial }{\partial x}\mathrm{\Phi }(-1,t)+4\frac{\partial }{\partial x}\mathrm{\Phi }(0,t)-\frac{\partial }{\partial x}\mathrm{\Phi }(1,t)=0.$ Similar as in Example 1, applying QNT and FCM for equation (6.2) for $N=M=5$, $T=[0,0.9]$, the following are the approximate solutions that we get from our calculations: $\begin{array}{c}{\mathrm{\Phi }}_0(x)=1.000000119+2.999999948x+2.999999971x^2+0.9999999998x^3\hfill \\ \phantom{{\mathrm{\Phi }}_0(x)=}-4\times {10}^{-9}{x}^4+6\times {10}^{-10}{x}^5,t=0.\hfill \\ {\mathrm{\Phi }}_1(x)=1.221401851+3.664208739x+3.664208472x^2+1.221402768x^3\hfill \\ \phantom{{\mathrm{\Phi }}_1(x)=}+2.9\times {10}^{-8}{x}^4-4\times {10}^{-9}{x}^5,t=0.2.\hfill \\ {\mathrm{\Phi }}_2(x)=1.491824219+4.475474332x+4.475474192x^2+1.491824704x^3\hfill \\ \phantom{{\mathrm{\Phi }}_2(x)=}+1.99\times {10}^{-8}{x}^4-3.2\times {10}^{-9}{x}^5,t=0.4.\hfill \\ {\mathrm{\Phi }}_3(x)=1.822118962+5.466356198x+5.466356382x^2+1.822118776x^3\hfill \\ \phantom{{\mathrm{\Phi }}_3(x)=}+9.2\times {10}^{-9}{x}^4+3\times {10}^{-9}{x}^5,t=0.6.\hfill \\ {\mathrm{\Phi }}_4(x)=2.225541126+6.676622617x+6.676622742x^2+2.225540916x^3\hfill \\ \phantom{{\mathrm{\Phi }}_4(x)=}+6.5\times {10}^{-9}{x}^4,t=0.8.\hfill \\ {\mathrm{\Phi }}_5(x)=2.459604708+7.378808429x+7.378809005x^2+2.459603058x^3\hfill \\ \phantom{{\mathrm{\Phi }}_4(x)=}-4.01\times {10}^{-8}{x}^4+1.16\times {10}^{-8}{x}^5,t=0,9.\hfill \end{array}$ Example 2’s absolute errors are displayed in Tables 3, 4 and Fig. 2, respectively.

[See PDF for image]

Figure 2

The absolute errors of Example 2

Table 3. The absolute errors in Example 2

x	T = 0	T = 0.2	T = 0.4
−1	4.6 × 10−9	3.3 × 10−8	2.31 × 10−8
−0.8	1.835008 × 10−9	1.318912 × 10−8	9.199616 × 10−9
−0.6	5.65056 × 10−10	4.06944 × 10−9	2.82787 × 10−9
−0.4	1.08544 × 10−10	7.8336 × 10−10	5.42208 × 10−10
−0.2	6.592 × 10−12	4.768 × 10−11	3.2864 × 10−11
0	0.00000000	0.00000000	0.00000000
0.2	6.208 × 10−12	4.512 × 10−11	3.0816 × 10−11
0.4	9.6256 × 10−11	7.0144 × 10−10	4.76672 × 10−10
0.6	4.71744 × 10−10	3.44736 × 10−9	2.330208 × 10−9
0.8	1.441792 × 10−9	1.056768 × 10−8	7.102464 × 10−9
1	3.4 × 10−9	2.5 × 10−8	1.67 × 10−8

Table 4. The absolute errors in Example 2

x	T = 0.6	T = 0.8	T = 0.9
−1	6.2 × 10−9	6.5 × 10−9	5.17 × 10−8
−0.8	2.78528 × 10−9	2.6624 × 10−9	2.0226048 × 10−8
−0.6	9.5904 × 10−10	8.424 × 10−10	6.098976 × 10−9
−0.4	2.048 × 10−10	1.664 × 10−10	1.145344 × 10−9
−0.2	1.376 × 10−11	1.04 × 10−11	6.7872 × 10−11
0	0.00000000	0.00000000	0.00000000
0.2	1.568 × 10−11	1.04 × 10−11	6.0448 × 10−11
0.4	2.6624 × 10−10	1.664 × 10−10	9.07776 × 10−10
0.6	1.4256 × 10−9	8.424 × 10−10	4.294944 × 10−9
0.8	4.75136 × 10−9	2.6624 × 10−9	1.2623872 × 10−8
1	1.22 × 10−8	6.5 × 10−9	2.85 × 10−8

Example 3

Let the third-order MFIDEs:

6.3

$\begin{array}{c}\frac{{\partial }^3}{\partial x^3}(\mathrm{\Phi }(x,t))-x\frac{{\partial }^2}{\partial x^2}(\mathrm{\Phi }(x,t))+x^2\frac{\partial }{\partial x}(\mathrm{\Phi }(x,t))+(\mathrm{\Phi }(x,t))+\frac{\partial }{\partial x}(\mathrm{\Phi }(x+1,t))\hfill \\ +2x(\mathrm{\Phi }(x+1,t))\hfill \\ -{\int }_0^t{\int }_{-1}^1{(t-\tau )}^{3}xu\mathrm{\Phi }(u-1,\tau )dud\tau =H(x,t),t\in [0,\pi ],\hfill \end{array}$ given the following conditions: $\begin{array}{c}\mathrm{\Phi }(-1,t)+16\mathrm{\Phi }(0,t)-\mathrm{\Phi }(1,t)=0,\hfill \\ \frac{\partial }{\partial x}\mathrm{\Phi }(-1,t)+8\frac{\partial }{\partial x}\mathrm{\Phi }(0,t)-\frac{\partial }{\partial x}\mathrm{\Phi }(1,t)=0,\hfill \\ \frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(-1,t)+4\frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(0,t)-\frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(1,t)=0,\hfill \end{array}$ at $\begin{array}{rcl}H(x,t)& =& 24(x+1))cos(t)-12x{(x+1)}^{2}cos(t)\\ & & +4x^2{(x+1)}^{3}cos(t)+{(x+1)}^{4}cos(t)\\ & & +4{(x+2)}^{3}cos(t)+2x{(x+2)}^{4}cos(t).\end{array}$ Similar as in the previous examples, the approximate solutions of equation (6.3) by using QNT and FCM at $N=M=5$, $T=[0,0.9\pi ]$ are given in the forms: $\begin{array}{c}{\mathrm{\Phi }}_0(x)=0.999999472+4.000000141x+6.000000123x^2+4.000000017x^3\hfill \\ \phantom{{\mathrm{\Phi }}_0(x)=}+1.000000014x^4-1.4\times {10}^{-8}{x}^5,t=0.\hfill \\ {\mathrm{\Phi }}_1(x)=0.8090151066+3.236068550x+4.854102440x^2+3.236068026x^3\hfill \\ \phantom{{\mathrm{\Phi }}_1(x)=}+0.8090170396x^4-5.13\times {10}^{-8}{x}^5,t=\frac{\pi }{5}.\hfill \\ {\mathrm{\Phi }}_2(x)=0.3090179606+1.236067696x+1.854101699x^2+1.236067957x^3\hfill \\ \phantom{{\mathrm{\Phi }}_2(x)=}+0.3090169586x^4+2.9747\times {10}^{-8}{x}^5,t=\frac{2\pi }{5}.\hfill \\ {\mathrm{\Phi }}_3(x)=-0.3090164645-1.236068086x-1.854102048x^2-1.236068013x^3\hfill \\ \phantom{{\mathrm{\Phi }}_3(x)=}-0.3090169891x^4+7.\times {10}^{-9}{x}^5,t=\frac{3\pi }{5}.\hfill \\ {\mathrm{\Phi }}_4(x)=-0.8090180864-3.236067752x-4.854101862x^2-3.236067902x^3\hfill \\ \phantom{{\mathrm{\Phi }}_4(x)=}-0.8090169724x^4-2.22\times {10}^{-8}{x}^5,t=\frac{4\pi }{5}.\hfill \\ {\mathrm{\Phi }}_5(x)=-0.9510598207-3.804224973x-5.706338667x^2-3.804225889x^3\hfill \\ \phantom{{\mathrm{\Phi }}_5(x)=}-0.9510564877x^4-6.82\times {10}^{-8}{x}^5,t=\frac{9\pi }{10}.\hfill \end{array}$ A comparison of Example 3’s absolute errors are given in Tables 5, 6 and Fig. 3.

[See PDF for image]

Figure 3

Errors of Example 3

Table 5. The absolute errors in Example 3

x	T = 0	$T=\frac{\pi }{5}$	$T=\frac{2\pi }{5}$
−1	1.4 × 10−8	5.13 × 10−8	2.9747 × 10−8
−0.8	4.58752 × 10−9	1.6809984 × 10−8	9.74749696 × 10−9
−0.6	1.08864 × 10−9	3.989088 × 10−9	2.31312672 × 10−9
−0.4	1.4336 × 10−10	5.25312 × 10−10	3.0460928 × 10−10
−0.2	4.48 × 10−12	1.6416 × 10−11	9.51904 × 10−12
0	0.00000000	0.00000000	0.00000000
0.2	4.48 × 10−12	1.6416 × 10−11	9.51904 × 10−12
0.4	1.4336 × 10−10	5.25312 × 10−10	3.0460928 × 10−10
0.6	1.08864 × 10−9	3.989088 × 10−9	2.31312672 × 10−9
0.8	4.58752 × 10−9	1.6809984 × 10−8	9.74749696 × 10−9
1	1.4 × 10−8	5.13 × 10−8	2.9747 × 10−8

Table 6. The absolute errors of Example 3

x	$T=\frac{3\pi }{5}$	$T=\frac{4\pi }{5}$	$T=\frac{9\pi }{10}$
−1	7.8 × 10−9	2.22 × 10−8	6.82 × 10−8
−0.8	2.555904 × 10−9	7.274496 × 10−9	2.23477760 × 10−8
−0.6	6.06528 × 10−10	1.726272 × 10−9	5.303232 × 10−9
−0.4	7.9872 × 10−11	2.27328 × 10−10	6.98368 × 10−10
−0.2	2.496 × 10−12	7.104 × 10−12	2.1824 × 10−11
0	0.00000000	0.00000000	0.00000000
0.2	2.496 × 10−12	7.104 × 10−12	2.1824 × 10−11
0.4	7.9872 × 10−11	2.27328 × 10−10	6.98368 × 10−10
0.6	6.06528 × 10−10	1.726272 × 10−9	5.303232 × 10−9
0.8	2.555904 × 10−9	7.274496 × 10−9	2.2347776 × 10−8
1	7.8 × 10−9	2.22 × 10−8	6.82 × 10−8

Example 4

Suppose the third-order MFIDEs [16]

6.4

$\begin{array}{c}\frac{{\partial }^3}{\partial x^3}(\mathrm{\Phi }(x,t))-x\frac{{\partial }^2}{\partial x^2}(\mathrm{\Phi }(x,t))+x^2\frac{\partial }{\partial x}(\mathrm{\Phi }(x,t))+\frac{\partial }{\partial x}(\mathrm{\Phi }(x+1,t))-2x(\mathrm{\Phi }(x+1,t))\hfill \\ -{\int }_0^t{\int }_{-1}^1{e}^{t-\tau }xu\mathrm{\Phi }(u-1,\tau )dud\tau =H(x,t),\hfill \end{array}$ given the following cases: $\begin{array}{c}\mathrm{\Phi }(-1,t)+16\mathrm{\Phi }(0,t)-\mathrm{\Phi }(1,t)=0,\hfill \\ \frac{\partial }{\partial x}\mathrm{\Phi }(-1,t)+8\frac{\partial }{\partial x}\mathrm{\Phi }(0,t)-\frac{\partial }{\partial x}\mathrm{\Phi }(1,t)=0,\hfill \\ \frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(-1,t)+4\frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(0,t)-\frac{{\partial }^2}{\partial x^2}\mathrm{\Phi }(1,t)=0,\hfill \end{array}$ where $H(x,t)=(2x^5-4x^4-44x^3-60x^2+28x+56)(0.01+t).$ Using QNT and FCM in a manner analogous to that of the other examples at $N=M=5$, comparison between FCM and Bernoulli’s polynomial method (BPM) [16] has been presented in Table 7 and Figs. 4, 5.

[See PDF for image]

Figure 4

Errors of Example 4 by using FCM

[See PDF for image]

Figure 5

Errors of Example 4 by using BPM

Table 7. Comparison between absolute errors of Example 4, N = 5 by BPM [16], and FCM

x	T = 0.2		$T=0.6$
	BCM, [16]	FCM	BCM,[16]	FCM
−1	1.38 × 10−9	8.92 × 10−11	7.37 × 10−10	1.29 × 10−11
−0.8	4.53 × 10−10	2.92 × 10−11	3.05 × 10−9	1.29 × 10−8
−0.6	1.07 × 10−10	6.94 × 10−12	1.0004 × 10−9	4.24 × 10−9
−0.4	1.41 × 10−11	9.13 × 10−13	3.12 × 10−11	1.32 × 10−10
−0.2	4.42 × 10−13	42.85 × 10−14	9.77 × 10−13	4.14 × 10−12
0	0.0000000	0.0000000	0.0000000	0.0000000
0.2	4.42 × 10−13	2.85 × 10−14	9.77 × 10−13	4.14 × 10−12
0.4	1.41 × 10−11	9.13 × 10−13	3.12 × 10−11	1.32 × 10−10
0.6	1.07 × 10−10	6.94 × 10−12	2.37 × 10−10	1.006 × 10−9
0.8	4.53 × 10−10	2.92 × 10−11	1.0004 × 10−9	4.24 × 10−9
1	1.38 × 10−9	8.92 × 10−11	3.05 × 10−9	1.29 × 10−8

It is very evident from the above results that the error grows when the time increases. Also from Table 7, when $T=0.2$, the results of FCM are better than the results of BPM [16]. But when $T=0.6$, the results of BPM [16] are better than the results of FCM.

Discussion of the results

(1) The absolute error approaches zero when the time is zero, but the absolute error starts to increase with increasing time.

(2) Figs. 1, 2, and 3 demonstrate a comparison of the absolute errors that were found in the data for various values of time. For example, in Fig. 1, it is very evident that the absolute error at $T=0.0$ is better than the absolute error at $T=0.2$, also the absolute error at $T=0.2$ is better than the absolute error at $T=0.4$, and so on.

(3) Comparison between the absolute errors is presented in Tables 1, 2, 3, 4, 5, and 6 for many standards of value of x, T. In Example 1, Table 1, at the point $x=-1$, $T=0$ the absolute error is $6.98\times {10}^{-11}$ but at the same point with different time $T=0.2$ the absolute error becomes $8.96\times {10}^{-9}$, the error increases by 10⁻², while absolute error becomes $1.297\times {10}^{-8}$ at $T=0.4$. Also, in Table 2, at the same point of x absolute errors are $2.47\times {10}^{-8}$, $3.346\times {10}^{-8}$, $1.36\times {10}^{-8}$ at the points $T=0.6,0.8$, and 0.9, respectively, it is clear that the absolute error at the beginning and end of time in the interval [0,1] is 10⁻³.

(4) The results of Example 4, comparing the error between two different methods at successive time periods, reveal an important fact: the advantage of using the numerical method depends on the starting point and the time required for a limited period. Then, at another successive time, it may be better to move to another method to obtain error stability.

Conclusion

In the paper [16], the authors discussed the solution of $(1+1)$ dimensional difference equations of mixed integro-differential type using the approach of separation of variables for functions of position and time. In this paper, the authors discussed the same equation but using a different technique with time and position. This method involves dividing the time interval by the integral period and then turning the formula in time and position into an algebraic system of functional integro-differential equations in position. The outcomes of this approach are the desired solution. Here the authors used the Fibonacci collocation method to get the approximate solution.

Future works: The Fibonacci collocation approach may be developed to solve mixed-difference integro-differential equations in (2 + 1) dimensions.

Acknowledgements

We thank the anonymous reviewers for their valuable comments and suggestions during the three rounds of review, which improved this work.

Author contributions

A.M.S.M., M.A.A. and D.S.M. wrote the main manuscript text and all authors prepared Figs. 1-5. All authors reviewed the manuscript.

Funding information

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Data availability

No datasets were generated or analysed during the current study.

Declarations

Credit author statement

Each author was responsible for conceiving the research question, developing the methodology, conducting the analysis, and writing the paper. The final paper was read and approved by every one of the authors.

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Kumar, S.; Gupta, V. An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations. J. Appl. Math. Comput.; 2023; 69, pp. 251-272.4541499

2. Kumar, S.; Gupta, V.; Kumar, A.; Gomez-Aguila, J.F. An accurate operational matrix method based on Lagrange polynomials for solving fractional-order pantograph delay and Riccati differential equations. Phys. Scr.; 2023; 98, [DOI: https://dx.doi.org/10.1088/1402-4896/acc0a8] 4.

3. Jafarzadeh, Y.; Keramati, B. Numerical method for a system of integro-differential equations and convergence analysis by Taylor collocation. Ain Shams Eng. J.; 2018; 9, pp. 1433-1438.

4. Dhundel, R.R.; Waghmare, G.L. Solving partial integro differential equations using double Laplace transform method. Am. J. Comput. Appl. Math.; 2015; 5, 1 pp. 7-10.

5. El-Borai, M.M.; Abdou, M.A.; Youssef, M.I.M. On Adomian’s decomposition method for solving nonlocal perturbed stochastic fractional integro-differential equations. Life Sci. J.; 2013; 10, 4 pp. 550-555.

6. Ahmed, S.A.; Elzaki, T.M. On the comparative study integro-differential equations using difference numerical methods. J. King Saud Univ., Sci.; 2020; 32, pp. 84-89.

7. Zada, L.; Al-Hamami, M.; Nawaz, R.; Jehanzeb, S.; Morsy, A.; Abdel-Aty, A.; Nisar, K.S. A new approach for solving Fredholm integro-differential equations. Inf. Sci. Lett.; 2021; 10, 3 pp. 407-415.

8. Pandey, P.K. Numerical solution of linear Fredholm integro-differential equations by non-standard finite difference method. Appl. Appl. Math.; 2015; 10, 2 pp. 1019-1026.3447629

9. Al-Bugami, A.M.: Two-dimensional Fredholm integro-differential equation with singular kernel and its numerical solutions. Adv. Math. Phys., 1–8 (2022)

10. Babaaghaie, A.; Maleknejad, K. Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet. J. Comput. Appl. Math.; 2017; 317, pp. 643-651.3606101

11. Yavuz, M.T.; Özkol, I. Solutions of integro-differential difference equations via differential transform method. J. Sci. Eng.; 2021; 18, 1 pp. 1-14.

12. Erdem, K.; Yalçinbaş, S.; Sezer, M. A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations. J. Differ. Equ. Appl.; 2013; 19, 10 pp. 1619-1631.3173507

13. Aslan, B.; Gurbuz, B.; Sezer, M. A new collocation method for solution of mixed linear integro-differential difference equations. New Trends Math. Sci.; 2015; 2, pp. 133-146.3358767

14. Mollaoglu, T.; Sezer, M. A numerical approach with residual error estimation for solution of high-order linear differential-difference equations by using Gegenbauer polynomials. CBU J. Sci.; 2017; 13, 1 pp. 39-49.

15. Gümgüm, S.; Savaşaneril, N.; Kürkçü, O.; Sezer, M. A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays. Sakarya Univ. J. Sci.; 2018; 22, 6 pp. 1659-1668.

16. Mahdy, A.M.S., Abdou, M.A., Mohamed, D.S.: Computational methods for solving higher-order $(1+1)$ dimensional mixed-difference integro-differential equations with variable coefficients. Mathematics, 1–25 (2023)

17. Ali, M.; Araghi, F.; Noeiaghdam, F. Fibonacci-regularization method for solving Cauchy integral equations of the first kind. Ain Shams Eng. J.; 2017; 8, pp. 363-369.

18. Bahşi, A.K.; Yalçinbaş, S. Numerical solutions and error estimations for the space fractional diffusion equation with variable coefficients via Fibonacci collocation method. SpringerPlus; 2016; 5, 1375 pp. 1-14.

19. Bahşi, A.K.; Yalçinbaş, S. Fibonacci collocation method with a residual error function to solve linear Volterra integro differential equations. New Trends Math. Sci.; 2016; 4, 1 pp. 1-14.3455647

20. Nadir, M. Solving linear integral equations with Fibonacci polynomials. Malaya J. Mat.; 2018; 6, 4 pp. 711-715.3891642

21. Mirzaee, F.; Hoseini, S.F. Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations. Appl. Math. Comput.; 2016; 273, pp. 637-644.3427783

22. Abo-Eldahab, E.M.; Mohamed, A.S.; Ali, S.M. Spectral collocation algorithm for solving fractional Volterra- Fredholm integro-differential equations via generalized Fibonacci polynomials. Contemp. Math.; 2022; 3, 3 pp. 1-18.

23. Mohamed, A.S. Fibonacci collocation pseudo-spectral method of variable-order space-fractional diffusion equations with error analysis. AIMS Math.; 2022; 7, 8 pp. 14323-14337.4431899

24. Seifi, A.; Lotfi, T.; Allahviranloo, T. A new efficient method using Fibonacci polynomials for solving of first-order fuzzy Fredholm-Volterra integro-differential equations. Soft Comput.; 2019; 23, pp. 9777-9791.

25. Mokhtar, M.M.; El Dewaik, M.H.; Mohamed, A.S. Semi-analytic Fibonacci polynomial solution for Volterra-Fredholm integral equation with error analysis. Fractals; 2022; 30, 8 pp. 1-10.

26. Ebadi, M.; Haghkhah, S. Investigation of the properties of a new class of interpolation polynomials based on Fibonacci numbers. Sahand Commun. Math. Anal.; 2023; 20, 2 pp. 133-146.

27. Darabi, M.; Parandin, N. A novel method to solve fuzzy Volterra integral equations using collocation method. Approx. Theory Appl.; 2020; 14, 2 pp. 115-130.

28. Haq, S., Ali, I.: Approximate solution of two-dimensional Sobolev equation using a mixed Lucas and Fibonacci polynomials. Eng. Comput., 1–11 (2021)

29. Kumar, S.; Gupta, V.; Aguilar, J.F.G. An efficient operational matrix technique to solve the fractional order non-local boundary value problems. J. Math. Chem.; 2022; 60, 1 pp. 1-17.4366373

30. Mahdy, A.M.S.; Abdou, M.A.; Mohamed, D.S. A computational technique for computing second-type mixed integral equations with singular kernels. J. Math. Comput. Sci.; 2024; 32, 2 pp. 137-151.

31. Alhazmi, S.E.; Mahdy, A.M.S.; Abdou, M.A.; Mohamed, D.S. Computational techniques for solving mixed $(1+1)$ dimensional integral equations with strongly symmetric singular kernel. Symmetry; 2023; 15, 6 pp. 1-27.

32. Mahdy, A.M.S.; Nagdy, A.S.; Hashem, K.M.; Mohamed, D.S. A computational technique for solving three-dimensional mixed Volterra Fredholm integral equations. Fractal Fract.; 2023; 7, 2 196.

33. Mahdy, A.M.S.; Mohamed, D.S. Approximate solution of Cauchy integral equations by using Lucas polynomials. Comput. Appl. Math.; 2022; 41, 8 pp. 1-20.4512254

34. Izadi, M.; Yüzbaşi, Ş.; Baleanu, D. A Taylor–Chebyshev approximation technique to solve the 1D and 2D nonlinear Burgers equations. Math. Sci.; 2022; 16, pp. 459-471.4502367

35. Daşcioğlu, A.; Salinan, S. Comparison of the orthogonal polynomial solutions for fractional integral equations. Mathematics; 2019; 7, 59 pp. 1-10.

36. Yüzbaşi, Ş.; Yildirim, G. Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integro-differential equations and residual correction. Turk. J. Math.; 2020; 44, pp. 1065-1091.4122891

37. Elmaci, D.; Savşaneril, N.B. Solutions of high-order linear Volterra integro-differential equations via Lucas polynomials. Montes Taurus J. Pure Appl. Math.; 2023; 5, 1 pp. 22-33.

38. Yüzbaşi, Ş. Shifted Legendre method with residual error estimation for delay linear Fredholm integro-differential equations. J. Taibah Univ. Sci.; 2017; 11, pp. 344-352.

39. Mohamed, D.S.; Abdou, M.A.; Mahdy, A.M.S. Dynamical investigation and numerical modeling of a fractional mixed nonlinear partial integro-differential problem in time and space. J. Appl. Anal. Comput.; 2024; 14, 6 pp. 3458-3479.4780292

40. Al-Bugami, A.M., Abdou, M.A., Mahdy, A.M.S.: Sixth kind Chebyshev and Bernoulli polynomial numerical methods for solving nonlinear mixed partial integro-differential equations with continuous kernels. J. Funct. Spaces, 1–14 (2023)

41. Mahdy, A.M.S., Abdou, M.A., Mohamed, D.S.: Numerical solution and dynamical studies for (2 + 1) dimensional Volterra-Fredholm integral equations with a discontinuous kernel. J. Appl. Math. Comput., 1–27 (2025)

42. Mahdy, A.M.S.; Abdou, M.A.; Mohamed, D.S. Numerical solution and dynamical studies for solving system of quadratic integral equations. Partial Differ. Equ. Appl. Math.; 2025; 13, 101070.

43. Das, P.; Rana, S. Theoretical prospects of fractional order weakly singular Volterra integro differential equations and their approximations with convergence analysis. Math. Methods Appl. Sci.; 2021; 44, pp. 9419-9440.4279858

44. Srivastava, H.M., Nain, A.K., Vats, R.K., Das, P.: A theoretical study of the fractional-order p-Laplacian nonlinear Hadamard type turbulent flow models having the Ulam-Hyers stability. Rev. R. Acad. Cienc. Exactas 117(160) (2023)

45. Santra, S.; Mohapatra, J.; Das, P.; Choudhuri, D. Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis. Comput. Math. Appl.; 2023; 150, pp. 87-101.4646748

46. Das, P.; Rana, S.; Ramos, H. A perturbation based approach for solving fractional order Volterra-Fredholm integro-differential equations and its convergence analysis. Int. J. Comput. Math.; 2019; 97, 10 pp. 1-18.4145344

47. Rostami, Y. New technique for solving system of variable-order fractional partial integro differential equations. Comput. Math. Math. Phys.; 2025; 65, 2 pp. 270-289.4883716

48. Rostami, Y.; Maleknejad, K. Approximate solution to solve singular variable-order fractional Volterra-Fredholm integral partial differential equations type defined using hybrid functions. Int. J. Comput. Math.; 2024; 101, pp. 668-693.4772013

49. Rostami, Y. A new wavelet method for solving a class of nonlinear partial integro-differential equations with weakly singular kernels. Math. Sci.; 2022; 16, pp. 225-235.4474040

50. Rostami, Y.; Maleknejad, K. Comparison of two hybrid functions for numerical solution of nonlinear mixed partial integro-differential equations. Iran. J. Sci.; 2022; 46, pp. 645-658.4410094

51. Rostami, Y.; Maleknejad, K. A novel approach to solving system of integral partial differential equations based on hybrid modified block-pulse functions. Math. Methods Appl. Sci.; 2024; 47, pp. 5798-5818.4730563

52. Rostami, Y. An effective computational approach based on Hermite wavelet Galerkin for solving parabolic Volterra partial integro differential equations and its convergence analysis. Math. Model. Anal.; 2023; 28, 1 pp. 163-179.4539606

53. Koshy, T. Fibonacci and Lucas Numbers with Applications; 2001; Hoboken, Wiley:

54. Abd-Elhameed, W.M.; Youssri, Y.H. Spectral tau algorithm for certain coupled system of fractional differential equations via generalized Fibonacci polynomial sequence. Iran. J. Sci. Technol., Trans. A, Sci.; 2019; 43, pp. 543-544.3922564

55. Abd-Elhameed, W.M.; Youssri, Y.H. A novel operational matrix of Caputo fractional derivatives of Fibonacci polynomials: spectral solutions of fractional differential equations. Entropy; 2016; 18, 345 pp. 1-15.3592289