1. Introduction
Lie’s theory of symmetry groups for differential equations was initiated and utilized for obtaining solutions or reductions of the differential equations [1]. Lie constructed a highly algorithmic technique for the solutions of differential equations. Lie established that several techniques to find the solutions of differential equations can be described and deduced by considering the Lie group analysis of differential equations [2]. Thus, Lie symmetry methodology has a great significance in the theory and applications of differential equations. It is broadly applied by several researchers to solve difficult nonlinear problems. Lie studied the groups of continuous transformations. These transformations (symmetries) can be described by the infinitesimal generators.
The symmetries, which are not continuous are called discrete symmetries. Discrete symmetries have several applications in differential equations, e.g., they are used to simplify the numerical scheme and to find the new exact solutions from the known solutions [3]. The nature of bifurcations in nonlinear dynamical systems are also obtained by using discrete symmetry groups [4]. The procedure to find the discrete symmetries of the differential equations is discussed in [5,6,7,8].
Nowadays, it is a challenging area of research to solve the dynamical equations, as various phenomena in nature are modeled in dynamical systems. Many researchers have considered dynamical systems, e.g., Martinez, Yu Zhang, and Timothy Gordon studied the uses of the control scheme in the classical dynamical systems theory to predict driver behavior and vehicle trajectories [9]. Martinez and Timothy also discussed the uses of machine learning for the systematic understanding of human control behaviors in driving [10]. Lie group theory has became a universal tool for the analysis of dynamical equations. Lie symmetry analysis provides an effective way to solve the partial differential equations.
But recently, interest is rising on the applications of Lie group analysis in the partial differential equations for their numerical solutions. Some work has been dedicated to building those numerical schemes that preserve the symmetries of the given differential equations. Invariantized finite difference schemes by using the idea of moving frame were constructed by Kim [11] and Olver [12]. The technique of discretization that preserves some continuous symmetries of the original differential equation was also studied in [13,14,15,16,17]. In [18], the exact solutions of Fisher’s type equation with the help of Lie symmetries, which are continuous symmetries in nature, are studied. Since most of the partial differential equations have some geometrical properties and some discrete symmetry groups correspond to these geometrical features of partial differential equations [8], the new invaiantized finite difference methods constructed by using these discrete symmetry groups may show better performances than the other finite difference methods.
In this article, first, it is shown that the Crank Nicolson scheme of a diffusion (heat) equation is invariant under a group of the discrete symmetry transformation. Here, a modification is proposed for the invariantization of the Crank Nicolson method given by Kim et al. [19]. This modification is proposed with the help of the composition of discrete and continuous symmetry transformations of the heat equation. It is also shown that the proposed invariantized scheme gives better results, as compared to the other classical finite difference methods.
2. Heat Equation
The homogeneous heat (diffusion) [20] equation
∂w∂t−D∂2w∂x2=0,
plays a vital role in the study of heat conduction and other diffusion processes, in which a thin metal rod of length L, whose sides are insulated, is considered. The temperature of the bar at the point x and at the time t is represented byw(x,t). The parameter D is called the thermal conductivity and it depends only upon the material from which the rod is made. For simplicity, we take the parameterD=1 then the Equation (1) becomes
∂w∂t−∂2w∂x2=0.
2.1. Continuous Symmetry Groups
Equation (2) is a linear and homogenous partial differential equation with one dependent variable w and two independent variables x and t. The point transformation of Equation (2) defines the local diffeomorphism
Γ:(x,t,w)↦(x^(x,t,w),t^(x,t,w),w^(x,t,w)).
This transformation maps any surfacew=f(x,t)to the following surface
x^=x^(x,t,f(x,t)),
t^=t^(x,t,f(x,t)),
w^=w^(x,t,f(x,t)).
The heat Equation (2) has infinite dimensional Lie algebra. The infinitesimal generators and the continuous symmetry groups of the Equation (2) [21] are presented in Table 1.
Due to the arbitrariness of the function appearing in the last generator of Table 1, Lie algebra of Equation (2) is infinite dimensional. Each of the groups given in Table 1 has the property of mapping solutions of heat Equation (2) to the other solutions. For example, consider the explicit form of the projection
(x,t,w)↦x1−4ϵt,t1−4ϵt,w1−4ϵtexp(−ϵx21−4ϵt),
whereϵis the group parameter. Computing the induced action on graphs of functions, we conclude that ifw=f(x,t) is any solution to heat Equation (2), so is
w=11−4ϵtexpϵx21−4ϵt)fx1−4ϵt,t1−4ϵt.
2.2. Discrete Symmetry Group
Hydon introduced a technique [7] by which all discrete point symmetries of the partial differential equations can be found on basis of the results [6] that each continuous symmetry generator of a Lie algebra ℓ of a differential equation brings an automorphism that preserve the following commutator relation
[Xl,Xk]=clkm Xm.
Hydon’s method categorizes and factor out all those automorphisms of a Lie algebra ℓ that are equivalent under the action of a Lie symmetry in a Lie group that is generated by the Lie algebra ℓ and provide the most general realization of these automorphisms as point transformations. Finally, by using these point transformations, an entire list of discrete point symmetries of a partial differential equation, is obtained.
The discrete symmetry group of Equation (2) has already been obtained in [8], which is isomorphic toZ4= {group of residues modulo 4} and is generated by
(x,t,w)→x2t,−14t,2ιtexpx24tw,
whereι=−1.
3. Finite Difference Schemes for the Heat Equation
Some finite difference schemes are available in the literature, which help us to find the numerical solutions of the partial differential equations. In this section, the backward difference scheme, forward difference scheme, and the Crank Nicolson method for the heat equation along their stability conditions, are given [22].
3.1. Forward Difference Scheme
The forward difference scheme for heat Equation (2) is given by
wn,j+1=αwn+1,j+(1−2α)wn,j+αwn−1,j,
whereα=kh2. Here h is thex-axis step size and k is the time step size for the grid points(xi,tj), wherexi=ih,tj=jkfor non-negative integersi,j. This scheme is Forward in Time and Centered in Space (FTCS). This method is explicit and converges to the solution for0<α≤12 , so is conditionally stable [22].
3.2. Backward Difference Scheme
The backward difference scheme for Equation (2) is given by
wn,j−1=−αwn+1,j+(1+2α)wn,j−αwn−1,j.
This scheme is Backward in Time and Centered in Space (BTCS). This is an implicit and unconditionally stable scheme [22].
3.3. Crank Nicolson Method
The Crank Nicolson method (CNM) for Equation (2) is
αwn−1,j+2(1−α)wn,j+αwn+1,j=−αwn−1,j+1+2(1+α)wn,j+1−αwn+1,j+1.
The Crank Nicolson method is also implicit and unconditionally stable [22]. It has significant advantages for the time-accurate solutions. The temporal truncation error of CNM isO(△t2), whereas the truncation error of FTCS and BTCS isO(△t).
3.4. Invariantization of the Crank Nicolson Method under the Discrete Symmetry Transformation
It is observed that among the mentioned finite difference schemes for the heat equation, the Crank Nicolson method gives the more accurate results [23], so we are interested in constructing an invariantization of the Crank Nicolson method. In the present subsection, we show that the Crank Nicolson method is invariant under the discrete symmetry transformation (3).
Letwn,j=w(n▵x,j▵t)be an approximation ofw(x,t)at the mesh point(xn,tj) . Now, by using the discrete symmetry group of heat equation given in (3), we have the following transformation
w=2itexpx24tw.
By using (5), the Crank Nicolson method transformed to
αwn−1,j2ij▵texp(n−1)2(▵x)24j▵t+2(1−α)wn,j2ij▵texp(n▵x)24j▵t+αwn+1,j2ij▵texp(n+1)2(▵x)24j▵t=−αwn−1,j+12i(j+1)▵texp(n−1)2(▵x)24(j+1)▵t+2(1+α)wn,j+12i(j+1)▵texp(n▵x)24(j+1)▵t−αwn+1,j+12i(j+1)▵texp(n+1)2(▵x)24(j+1)▵t.
By considering the following transformation
wN,J=JexpN24Jwn,j,
with
J=j▵tandN=n▵x,
(6) can be written as
αwN−1,J+2(1−α)wN,J+αwN+1,J=−αwN−1,J+1+2(1+α)wN,J+1−αwN+1,J+1.
The finite difference approximation to heat Equation (2) obtained in (7) is similar to the Crank Nicolson method for heat Equation (4). Thus, the Crank Nicolson method remains invariant under the discrete symmetry transformation (3). The consequence of the result obtained in Section 3 can be written in the form of the following theorem.
Theorem 1.
The Crank Nicolson method for heat equation given in (4) is invariant under the only discrete symmetry transformation (3) of the heat Equation (2).
4. Discrete Symmetry Numerical Scheme For the Heat Equation
Most of the finite difference methods including the Crank Nicolson method are invariant under time, space translations, and scale transformation [24]. It is also proved in the previous literature that the Crank Nicolson method for the heat Equation (2) is invariant under the transformation of the discrete symmetry. Notice that the Galilean boost and projection group are the transformations under which the Crank Nicolson method is not invariant. However, the Crank Nicolson method invariantized by Galilean boost, projection transformations or composition of these transformations, becomes unstable (i.e., does not converge to the exact solution). Nevertheless, by taking the composition of the discrete symmetry group and the projective symmetry group, which is a continuous symmetry group of the heat equation, invariantized Crank Nicolson method converges to the exact solution and gives the more adequate results as compare to other existing finite difference schemes of Equation (2).
In the present section, we construct an invariantization of the Crank Nicolson method for Equation (2) by using the composition of discrete and continuous symmetry groups. The construction of this method is based on the composition of the variable w of these two groups. We deal with the following transformation to construct the new scheme:
w=2ιtexpx24(1−4ϵt)1t−4ϵw,
whereι=−1 . By transforming the variable w in the Crank Nicolson method (4) with the above transformation for Equation (2), we get:
αtjwn−1,jexp(xn−1)24(1−4ϵtj)1tj−4ϵ
+(2−2α)tjwn,jexp(xn)24(1−4ϵtj)1tj−4ϵ
+αtjwn+1,jexp(xn+1)24(1−4ϵtj)1tj−4ϵ
=−αtj+1wn−1,j+1exp(xn−1)24(1−4ϵtj+1)1tj+1−4ϵ
+(2+2α)tj+1wn,j+1exp(xn)24(1−4ϵtj+1)1tj+1−4ϵ
−αtj+1wn+1,j+1exp(xn+1)24(1−4ϵtj+1)1tj+1−4ϵ,
which can be further simplified as
tj[αwn−1,jexp(xn−1)24(1−4ϵtj)1tj−4ϵ
+(2−2α)wn,jexp(xn)24(1−4ϵtj)1tj−4ϵ+
αwn+1,jexp(xn+1)24(1−4ϵtj)1tj−4ϵ]=
tj+1[−αwn−1,j+1exp(xn−1)24(1−4ϵtj+1)1tj+1−4ϵ
+(2+2α)wn,j+1exp(xn)24(1−4ϵtj+1)1tj+1−4ϵ
−αwn+1,j+1exp(xn+1)24(1−4ϵtj+1)1tj+1−4ϵ].
Sinceϵis a continuous parameter and we can choose the optimal value ofϵfor which the proposed method gives better performance than the Crank Nicolson method. We chooseϵ=ci, where0≤ci≤1and i is a positive integer, so the above equation takes the form:
tj[αwn−1,jexp(xn−1)24(1−4ϵtj)1tj−4ci+(2−2α)wn,jexp(xn)24(1−4ϵtj)1tj−4ci+
αwn+1,jexp(xn+1)24(1−4ϵtj)1tj−4ci]=
tj+1[−αwn−1,j+1exp(xn−1)24(1−4ϵtj+1)1tj+1−4ci
+(2+2α)wn,j+1exp(xn)24(1−4ϵtj+1)1tj+1−4ci
−αwn+1,j+1exp(xn+1)24(1−4ϵtj+1)(1tj+1−4ci].
The final form of the method is
Aαwn−1,j B1+(2−2α)wn,j B2+αwn+1,j B3=C−αwn−1,j+1 D1+(2+2α)wn,j+1 D2−αwn+1,j+1 D3,
where
A=tj,
B1=exp(xn−1)24(1−4ϵtj)1tj−4ci,
B2=exp(xn)24(1−4ϵtj)1tj−4ci,
B3=exp(xn+1)24(1−4ϵtj)1tj−4ci,
and
C=tj+1,
D1=exp(xn−1)24(1−4ϵtj+1)1tj+1−4ci,
D2=exp(xn)24(1−4ϵtj+1)1tj+1−4ci,
D3=exp(xn+1)24(1−4ϵtj+1)1tj+1−4ci.
Since the Crank Nicolson method for the heat equation is unconditionally stable [22] and the discritized Crank Nicolson method (DCNM) for heat Equation (2) provided in (8) is the invariantization of the Crank Nicolson method, so this method also preserves the unconditional stability condition and therefore converges to the exact solution without having any condition onα.
5. Solutions of the Heat Equation
In this section, we find the analytic solution of heat Equation (2) and compare it with the numerical solutions calculated by using the CNM and the proposed method DCNM.
5.1. Analytic Solution
We consider one dimensional homogeneous heat Equation (2) with the following boundary conditions
w(x,0)=g(x),0≤x≤L,w(0,t)=f1(t),0≤t≤T,w(1,t)=f2(t),
whereg(x),f1(t)andf2(t)are two times continuously differentiable functions onx∈[0,L], L is the length of the rod and T is the maximum time.
Heat Equation (2) is reduced to an ordinary differential equation by using the following similarity variable
ξ=t,V=wg(x),
whereg(x)is obtained from the initial condition(5.1)withg(x)≠0 . An exact solution of the system given by the Equations (2) and (9) for particular values of arbitrary functionsg(x),f1(t), andf2(t)is given in the following example.
Example 1.
Our aim is to find the solution of (2) with the following initial and boundary conditions
w(x,0)=sinπx,0≤x≤1,
w(0,t)=0,0≤t≤1,
w(1,t)=0.
Using the similarity transformation
ξ=t,V=wsinπx,
the above problem is reduced to the following ODE
Vξ+π2V=0,
with the solutionV=e−π2t.
Hencew(x,t)=e−π2tsin(πx)is the exact solution of the above boundary value problem.
5.2. Numerical Solutions Using CNM and the Proposed Method DCNM
In this subsection, the performance of the proposed method DCNM is investigated by applying it to Example 1. The efficiency of the present method DCNM is shown by calculating the absolute errors, root mean square errorsL2and maximum errorsL∞ . These errors are computed by the following formulas [18]:
Absoluteerror=|ei|,
L2=∑i=1nei21/2,
L∞=max1≤i≤n|ei|,
withei=(wi−Wi), wherewiare numerical andWiare the exact solutions.
To check the computational accuracy of the proposed method (DCNM), we reconsider Example 1 given in the previous subsection, which has the analytic solutionw(x,t)=exp(−π2t)sin(πx) [22]. We compare our results of the numerical solutions (DCNM) with the exact solutions and the solutions that are obtained by the Crank Nicolson method for the x-axis step sizeh=0.1,a time step sizek=0.01and for the timeT=0.5 . The comparisons of the numerical solutions obtained by DCNM with the solutions calculated by FTCS, CNM, and the exact solutions are presented in Table 2, where Table 3 is showing the absolute errors of the solutions calculated by FTCS, CNM and the DCNM given in (8) for the different x-axis step sizes. Table 4 shows the values ofw(x,t)for fixedh=0.1 and for different values of k, where Table 5 shows the values ofw(x,t)for fixedk=0.02and for different values of h.
Table 6 presents root mean square errorsL2and the maximum errorsL∞for DCNM and CNM of Example 1 for the different values of t.
Figure 1 shows the comparisons between the numerical solutions, obtained by the DCNM given in (8), the Crank Nicolson method, and the exact solutions of Example 1 in 2D for fixedT=0.5. Forh=0.1,k=0.05andT=1, the graphical representations of the space-time graph of the numerical solutions calculated by the DCNM and exact solutions for the above boundary value problem forx∈[0,1] are presented in Figure 2 and Figure 3, respectively, and it can be observed that both solutions are very similar.
6. Conclusions
This paper contains the application of the discrete symmetry transformation for the boundary value problem of the diffusion equation. An invariantized finite difference method to find the solution of the heat equation using the composition of discrete symmetry group and projection group of the heat equation is developed. Table 2, Table 3 and Table 4 show that the proposed invariantized method DCNM improves the efficiency and accuracy of the existing Crank Nicolson method.
Similarly, with the help of discrete symmetry groups of partial differential equations, different invariantized finite difference schemes can be constructed to improve the efficiency and performance of the existing finite difference methods.
Enlarge this image.Figure 1. Comparison of the exact solution with solutions obtained by CNM and DCNM.
| Generators | Symmetry Transformations | |
|---|---|---|
| 1 | ∂∂x | Space translation:(x,t,w)↦(x+ϵ,t,w) |
| 2 | ∂∂t | Time translation:(x,t,w)↦(x,t+ϵ,w) |
| 3 | w∂∂w | Scale Transformation:(x,t,w)↦(x,t,eϵw) |
| 4 | x∂∂x+2t∂∂t | Scale Transformation:(x,t,w)↦(eϵx,e2ϵt,w) |
| 5 | 2t∂∂x−xw∂∂w | Galilean boost:(x,t,w)↦(x+2ϵt,t,−xwϵ+w) |
| 6 | 4xt∂∂x+4t2∂∂t−(x2+2t)w∂∂w | Projection:(x,t,w)↦x1−4ϵt,t1−4ϵt,w1−4ϵtexp(−ϵx21−4ϵt) |
| 7 | q(x,t)∂∂w |
| xi | FTCS | CNM | DCNM | Exact Solutions |
|---|---|---|---|---|
| 0.0 | 0.000000 | 0.000000 | 0.0000000000 | 0.000000 |
| 0.2 | 0.000892 | 0.004517 | 0.0042956284 | 0.004227 |
| 0.4 | 0.001444 | 0.007309 | 0.00695047277 | 0.006840 |
| 0.6 | 0.001444 | 0.007309 | 0.00695047277 | 0.006840 |
| 0.8 | 0.000892 | 0.004517 | 0.0042956284 | 0.004227 |
| 1.0 | 0.000000 | 0.000000 | 0.0000000000 | 0.000000 |
| xi | FTCS | CNM | DCNM |
|---|---|---|---|
| 0.0 | 0.000000 | 0.000000 | 0.0000000000 |
| 0.2 | 0.003335 | 2.9×10−4 | 6.8345438×10−5 |
| 0.4 | 0.005396 | 4.69×10−4 | 1.10585242×10−4 |
| 0.6 | 0.005396 | 4.69×10−4 | 1.10585242×10−4 |
| 0.8 | 0.003335 | 2.9×10−4 | 6.8345438×10−5 |
| 1.0 | 0.000000 | 0.000000 | 0.000000000000 |
| xi | k=0.03 | k=0.02 | k=0.01 |
|---|---|---|---|
| 0.0 | 0.0000 | 0.0000 | 0.0000 |
| 0.2 | 0.0053 | 0.0045 | 0.0042 |
| 0.4 | 0.0085 | 0.0076 | 0.0069 |
| 0.6 | 0.0085 | 0.0076 | 0.0069 |
| 0.8 | 0.0053 | 0.0045 | 0.0042 |
| 1.0 | 0.0000 | 0.0000 | 0.0000 |
| xi | h=0.3 | h=0.2 | h=0.1 |
|---|---|---|---|
| 0.0 | 0.0000 | 0.0000 | 0.0000 |
| 0.2 | 0.0050 | 0.0047 | 0.0045 |
| 0.4 | 0.0081 | 0.0078 | 0.0076 |
| 0.6 | 0.0081 | 0.0078 | 0.0076 |
| 0.8 | 0.0050 | 0.0047 | 0.0045 |
| 1.0 | 0.0000 | 0.0000 | 0.0000 |
| t | L2DCNM | L∞DCNM | L2CNM | L∞CNM |
|---|---|---|---|---|
| 0.1 | 9.6×10−3 | 4.3×10−3 | 4.86×10−3 | 6.79×10−3 |
| 0.3 | 4.0×10−4 | 1.8×10−4 | 8.87×10−5 | 3.76×10−4 |
| 0.5 | 9.0830×10−4 | 4.0812×10−4 | 1.73×10−3 | 2.44×10−4 |
| 0.7 | 1.8024×10−4 | 1.0097×10−4 | 2.04×10−4 | 3.17×10−4 |
| 0.9 | 1.0224×10−4 | 6.1223×10−5 | 2.14×10−3 | 3.14×10−3 |
| 1 | 1.1556×10−4 | 5.0808×10−5 | 2.15×10−3 | 3.32×10−3 |
Author Contributions
All authors have equal contribution to this research and in preparation of the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding
This research was funded by NUST, Pakistan.
Acknowledgments
The author would like to thanks the editor and anonymous referees for their suggestions and valuable comments that improved the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
| FTCS | Forward in Time and Centered in Space |
| CNM | Crank Nicolson Method |
| DCNM | Discritized Crank Nicolson Method |
1. Lie, S. Theorie der Transjormations Gruppen; Chelsea: New York, NY, USA, 1970.
2. Yang, H.; Shi, Y.; Yin, B.; Dong, H. Discrete Symmetries Analysis and Exact Solutions of the Inviscid Burgers Equation. Discret. Nat. Soc. 2012, 56, 1-15.
3. Ibragimov, N.H. Elementary Lie Group Analysis and Ordinary Differential Equations; John Wiley & Sons: Chichester, UK, 1999.
4. Golubitsky, M.; Stewart, I.; Schaeffer, D.G. Singularities and Groups in Bifurcation Theory; Springer: New York, NY, USA, 1988.
5. Hydon, P.E. Discrete point symmetries of ordinary differential equations. R. Soc. Lond. Proc. A 1998, 454, 1961-1972.
6. Hydon, P.E. How to construct the discrete symmetries of partial differential equations. Eur. J. Appl. Math. 2000, 11, 515-527.
7. Hydon, P.E. Symmetry Methods for Differential Equations; Cambridge University Press: Cambridge, UK, 2007.
8. Laine, F.E.; Hydon, P.E. Classification of discrete symmetries of ordinary differential equations. Stud. Appl. Math. 2003, 111, 269-299.
9. Garcıa, M.M.; Zhang, Y.; Gordon, T. Modelling Lane Keeping by a Hybrid Open-Closed-Loop Pulse Control Scheme. IEEE Trans. Ind. Inform. 2016, 12, 2256-2265.
10. Garcıa, M.M.; Gordon, T. A New Model of Human Steering Using Far-Point Error Perception and Multiplicative Control. In Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics (SMC2018), Miyazaki, Japan, 7-10 October 2018.
11. Kim, P. Invariantization of numerical schemes using moving frames. Numer. Math. Springer 2006, 10, 142-149.
12. Olver, P.J. Geometric foundations of numerical algorithms and symmetry. Appl. Algebra Eng. Commun. Comput. 2001, 11, 417-436.
13. Budd, C.; Dorodnitsyn, V.A. Symmetry-adapted moving mesh schemes for the nonlinear Schrodinger equation. J. Phys. A Math. Gen. 2001, 34, 10387-10400.
14. Dorodnitsyn, V.A. Finite difference models entirely inheriting continuous symmetry of original differential equations. Int. J. Mod. Phys. Ser. C 1994, 5, 723-734.
15. Dorodnitsyn, V.A.; Kozlov, R.; Winternitz, P. Lie group classification of second order difference equations. J. Math. Phys. 2000, 41, 480-504.
16. Valiquette, F.; Winternitz, P. Discretization of partial differential equations preserving their physical symmetries. J. Phys. A Math. Gen. 2005, 38, 9765-9783.
17. Budd, C.J.; Iserles, A. Geometric integration: Numerical solution of differential equations on manifolds. Philos. Trans. R. Soc. Lond. A 1999, 357, 945-956.
18. Verma, R.; Jiwari, R.; Koksal, M.E. Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions. Adv. Differ. Equ. 2014, 10, 142-149.
19. Kim, P. Invariantization of the Crank-Nicolson method for Burgers' equation. Phys. D Nonlinear Phenom. 2008, 237, 243-254.
20. Stavroulakis, I.P.; Tersian, S.A. Partial Differential Equations: An Introduction with Mathematica and Maple; World Scientific Publishing Company Ltd.: Hackensack, NJ, USA, 2004.
21. Bluman, G.W.; Kumei, S. Symmetries and Differential Equations; Springer: New York, NY, USA, 1989.
22. Burden, R.L.; Faires, J.D. Numerical Analysis, 9th ed.; Brooks/Cole Cengage Learning: Boston, MA, USA, 2010.
23. Kharab, A.; Guenther, R.B. An Introduction to Numerical Methods, A Matlab Approach, 3rd ed.; CRC Press Taylor and Francis Group: New York, NY, USA, 2011.
24. Marx, C.; Aziz, H. Lie Symmetry preservation by Finite difference Schemes for the Burgers Equation. Symmetry 2010, 2, 868-883.
Khudija Bibi1,2,* and Tooba Feroze11Department of Mathematics, School of Natural Sciences, National University of Science and Technology Islamabad 44000, Pakistan
2Department of Mathematics, Faculty of Basic ans Applied Sciences, International Islamic University Islamabad 44000, Pakistan
*Author to whom correspondence should be addressed.
© 2020. This work is licensed under http://creativecommons.org/licenses/by/3.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.