A New Shifted Chebyshev Galerkin Operational Matrix of Derivatives: Highly Accurate Method for a Nonlinear Singularly Perturbed Problem with an Integral Boundary Condition

Abstract

The main goal of this paper is to come up with a new numerical algorithm for solving a first-order nonlinear singularly perturbed differential equation (SPDE) with an integral boundary condition (IBC). This paper builds a modified shifted Chebyshev polynomials of the first kind (CPFK) basis function that meets a homogeneous IBC, named IMSCPFK. It has established an operational matrix (OM) for derivatives of IMSCPFK. The numerical solutions are spectral, obtained by applying the spectral collocation method (SCM). This approach transforms the problem by their IBC into a set of algebraic equations, which may be resolved using any suitable numerical solver. Ultimately, we substantiate the proposed theoretical analysis by providing an example to verify the correctness, efficiency, and applicability of the developed method. We compare the acquired numerical findings with those derived from other methodologies. Tables and figures display the method’s highly accurate agreement with the residual error.

Introduction

Differential equations (DEs) with IBCs model many challenges in applied sciences, including heat conduction and chemical engineering. Researchers have explored various numerical approaches for SPDEs that include IBCs. The authors of [1] made a big contribution when they came up with a finite difference scheme that used a Shishkin mesh to solve problems (1.1). They showed that their method almost reached first-order uniform convergence. Also, Cakir [2] described a finite difference method that works on a Shishkin-type mesh to solve a second-order semi-linear SPDE with an IBC. This method almost proved first-order uniform convergence. Kudu [3] looked at a parameterized singularly perturbed problem with an IBC. He came up with a difference method that worked on a Bakhvalov mesh and led to first-order uniform convergence. It is important to note that these methods fall under the category of layer-adapted grid approaches. A new method called an adaptive grid method based on a difference technique was proposed in [4] to solve (1.1), and it also showed first-order uniform convergence.

The current paper discusses a new numerical algorithm for solving the SPDE:

1.1

where is the perturbation parameter, and d are given constants. The two functions and are assumed to be sufficiently continuously differentiable functions in and respectively, and for three constants and , we have

1.2

where and are defined as follows:Under these hypotheses, the problem (1.1) has a unique solution which contains a boundary layer near for (see [1, 4]).

The three prevalent spectral approaches are the collocation, tau, and Galerkin methods. The choice of the most appropriate form of these procedures is contingent upon the characteristics of the analyzed DE and the kind of boundary conditions (BCs) related to it. These methodologies use OMs (for instance, [5, 6, 7, 8, 9, 10, 11, 12, 13–14]) to formulate efficient algorithms that provide accurate numerical solutions for diverse DEs while reducing computational efforts. Additionally, the methodologies included in the papers [15, 16, 17–18] for solving DEs demonstrate the robustness and versatility of spectral methods in various applications.

To the best of our knowledge, the literature has not yet documented or traced a Galerkin OM using any basis function that satisfies the homogeneity IBC. This largely explains our interest in such an OM. Another motive is that the use of this form of OM leads to high accuracy for the numerical solution of BVP ((1.1)).Here is a summary of this paper’s main objectives:

1. Constructing a new class of basis polynomials, IMCPFK, that satisfy the homogeneous IBC.

2. The establishment of an OM for derivatives of IMCPFK is being discussed.

3. Using the collocation technique along with the suggested OM of derivatives to come up with a numerical way to solve the BVP (1.1).

The paper follows a structured format. In Section 2, the CPFK and their shifted ones are discussed. Section 3 is confined to the construction of IMCPFK, which satisfies the homogeneous IBC in (1.1). Section 4 is limited to developing a new OM of IMCPFK’ derivtives to address BVP (1.1). The application of SCM for the numerical resolution of (1.1) will be described in Section 5. The convergence and error analysis are presented in Section 6. Section 7 presents a numerical example, along with comparisons to many other methodologies from the literature. Finally, Section 8 outline findings and future work.

An Overview on CPFK and their Shifted Ones

the CPFK are represented by (see, [19])

2.1

and they fulfillThese polynomials can be built by using the following recurrence relationwith The polynomials are special ones of the Jacobi polynomials, . More definitely, we have

2.2

The shifted CPFK is defined by , and they fulfilland have the analytical form

2.3

and .

I-Modified CPFK

Here, we construct a new type of polynomial, which will be called IMCPFK and will be symbolized by , in order to satisfy IBCs:

3.1

Regarding this, we suggest IMCPFK with the form

3.2

where is a unique constant such that satisfies the condition (3.1). Substitution of into (3.1) and using and leads towhere .The proposed IMCPFK have the special values

3.3

where

Remark 3.1

In particular, and for the special case, and ,

3.4

and this aligns with the initial condition .

Operational Matrix of Derivatives of IMCPFK

In this section, we will construct the OM of derivatives of IMCPFK . To do that, we need to express in terms of themselves. First, we must note that

, .

.

We will introduce a novel Galerkin OM of derivatives by stating and proving the following theorem.

Theorem 4.1

For all , we have

4.1

where the expansion coefficients , staisfy the system

4.2

And the coefficients have the form:

4.3

Proof

It is easy to see that takes the form (4.3). So the expansion (4.1) leads to

4.4

Using the Maclaurin series for two polynomials and , Eq.(4.4) takes the form

4.5

This leads to the following triangle system in ,

4.6

which can be written as (4.2) and this finishes the proof.

Now, we have attained the primary objective, which is the OM of derivtives of

4.7

as a result of Theorem 4.1.

Corollary 4.1

4.8

where ,

A Collocation Algorithm for Handling (1.1)

Here, we utilize the OM obtained in Corollary 4.1 to get approximate solutions for (1.1).

Homogeneous Boundary Conditions

Suppose that the given BC in (1.1) is homogeneous, that is, . We can consider an approximate solution to in the form

5.1

where

In view of Corollary 4.1, can be approximated in the form:

5.2

This approach use the approximations (5.1) and (5.2) to express the residual of the ODE given in (1.1) as

5.3

To obtain , a spectral approach is suggested in the current section: the integrated shifted Chebyshev first-kind collocation operational matrix method, ISC1COMM, in which we apply SCM using the given OM for derivatives of IMCPFK. The collocation points aresuch that

5.4

Then can be obtained by solving the algebraic system (5.4) using Newton’s iterative method.

Nonhomogeneous Boundary Conditions

A crucial phase in developing the proposed technique involves transforming the BVP (1.1) with non-homogeneous BCs into its equivalent homogeneous form. To do that, we use the following transformation:

5.5

wheresuch that . So that,

5.6

Then

5.7

Convergence and Error Estimates For ISC1COMM

Here, we examine the convergence and error estimates of ISC1COMM. Consider the space defined byAdditionally, the error between and its approximation is defined by

6.1

The paper analyzes the error of ISC1COMM using the following two norms:

6.2

and

6.3

The following theorem confirms and tend to zero as increases.

Theorem 6.1

Assume that with . Suppose that is given by (5.1) and represents the best possible approximation for out of . Then, we get:

6.4

6.5

Proof

The function may be expressed as

6.6

where . So,

6.7

Then the inequality (6.7) enables us to obtain

6.8

and

6.9

Since the approximate solution represents the best possible approximation to , consequently, we have

6.10

and

6.11

Employing in particular in (6.10) and (6.11) produces (6.4) and (6.5).

Corollary 6.1 explains that exhibits a quick rate of convergence.

Corollary 6.1

6.12

and

6.13

Proof

Using

6.14

and making use of some computational manipulations, the two estimates  (6.4) and (6.5) lead to (6.12) and (6.13).

The following theorem further emphasizes error stability by estimating error propagation.

Theorem 6.2

For any two successive approximations of for the problem (1.1), we get:

6.15

where means that a generic constant d exists such that .

Proof

We haveBy considering Theorem 6.1, the successive approximations for the problem (1.1) satisfy (6.15).

Numerical Simulations

In order to verify the validity of the theoretical results obtained, we consider the following nonlinear SPDE with an IBC in the form [20, 21–22]:

7.1

The exact solution to the problem is not available. Table 1, shows the residual error obtained and the convergence rate using the application of ISC1COMM for for different values of , where the approximate solutions take the formwheresuch thatFor , we get:thanks to Mathematica. In particular, for , and , we get

7.2

The residual error, , is defined as follows:then we have

Note 7.1

It is worthwhile to note thatfor all values of and demonstrated in Table 1 which emphasize that the verification of the presented study of convergence in Section 6.

Furthermore, Fig.1a and 1b show the approximate solution and the residual error obtained. Additionally, Table 2 shows that ISC1COMM gives more accurate results than other methods in [20, 21–22].

Fig. 1 [Images not available. See PDF.]

Figures of obtained approximate and error using and

Table 1. Maximum absolute errors using ISC1COMM

	CPU Time
3	0.22s	6.5E-02	1.3E-02	1.2E-02	2.6E-02	6.6E-02	1.6E-02	4.1E-02	1.4E-02	2.5E-02
7	0.34s	4.1E-04	2.7E-04	3.3E-04	1.5E-04	3.5E-04	1.7E-04	2.1E-04	4.4E-04	1.3E-04
10	0.49s	5.1E-07	1.1E-07	2.8E-07	6.1E-07	1.7E-07	4.4E-07	1.1E-07	2.7E-07	6.9E-07
13	0.55s	2.1E-10	1.5E-10	3.2E-10	3.2E-10	2.2E-10	1.2E-10	2.0E-10	3.3E-10	1.3E-10

Table 2. Comparison between ISC1COMM and other different methods

13	ISC1COMM	2.1E-10	3.2E-10	2.2E-08	2.0E-10	3.3E-10	3.5E-10	1.4E-10
2048	[20]	2.33E-06	2.25E-06	2.45E-06	2.27E-06	2.27E-06	2.27E-06	2.27E-06
2048	[21]	2.54E-07	4.51E-07	3.30E-07	6.61E-07	6.43E-07	4.80E-07
1024	[22]	5.68E-04	6.90E-04	6.41E-04	6.68E-04	6.79E-04

Remark 7.1

When making numerical comparisons, only results from three previous papers [20, 21–22] that presented only the same example were used. Up to the best of our knowledge, these three papers are the only ones in which the BVP (1.1) was discussed.

Remark 7.2

The computations were performed using Mathematica 13.3 on a computer system equipped with an Intel(R) Core(TM) i9-10850 CPU operating at 3.60GHz, featuring 10 cores and 20 logical processors. The algorithmic steps for solving the SPDE using ISC1COMM are expressed as follows:

Remark 7.3

For interested readers, we utilized several built-in functions in Mathematica for our numerical implementation of the provided Algorithm. Below is a summary of the tools used, along with minimal information about each function:

• Array: For creating and manipulating arrays, which are used to hold coefficients and operational matrices throughout the computations.

• NSolve: For finding numerical solutions to nonlinear algebraic equations, it is utilized to compute the zeros of .

• FindRoot: For solving equations by finding roots, it is essential for handling the nonlinear aspects of our system, using a zero initial approximation.

• ChebyshevT: For generating IMCPFK, which serve as basis functions that serve as the foundation for approximating the solution in our collocation method.

• D: To compute ordinary derivatives to determine the defined residuals.

• Table: For generating lists and arrays of values based on specified formulas, particularly for collocation points and other parameterized data.

Conclusion

Here, we have established a modified SCPFK system that satisfies homogeneous IBC. When one uses these polynomials with the SCM, he can get a close approximation of the given first-order DE. TWe tested the proposed method ISC1COMM and found that it provides a high-accuracy and efficient solution. We believe that the algorithm presented in this study holds significant potential for further development and application to a broader range of ordinary and fractional differential equations with different types of IBCs.

Author Contributions

Not Applicable.

Funding

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). No funding was received to assist with the preparation of this manuscript.

Data Availability

No data is associated with this research.

Declarations

Competing interests

The author declares no competing interests.

Ethical Approval

Not Applicable.

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