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An Algebraic Hyperbolic Spline Quasi-Interpolation Scheme for Solving Burgers-Fisher Equations

Written By

Mohamed Jeyar, Abdellah Lamnii, Mohamed Yassir Nour, Fatima Oumellal and Ahmed Zidna

Reviewed: June 24th, 2021 Published: December 10th, 2021

DOI: 10.5772/intechopen.99033

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Simulation Modeling

Edited by Constantin Volosencu and Cheon Seoung Ryoo

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Abstract

In this work, the results on hyperbolic spline quasi-interpolation are recalled to establish the numerical scheme to obtain approximate solutions of the generalized Burgers-Fisher equation. After introducing the generalized Burgers-Fisher equation and the algebraic hyperbolic spline quasi-interpolation, the numerical scheme is presented. The stability of our scheme is well established and discussed. To verify the accuracy and reliability of the method presented in this work, we select two examples to conduct numerical experiments and compare them with the calculated results in the literature.

Keywords

  • Burgers-Fisher equation
  • Algebraic Hyperbolic Spline
  • Quasi-interpolation

1. Introduction

The utilization of quasi-interpolation methods has been advanced in several fields of numerical analysis. This method can yield directly to solutions and does not require the solution of any linear system. In general, quasi-interpolation methods have attracted much attention because of their potential for solving partial differential equations [1, 2, 3], curve and surface fitting, integration, differentiation, and so on. In [2], Foucher and Sablonnière developed some collocation methods based on quadratic spline quasi-interpolants for solving the elliptic boundary value problems. In [4], Bouhiri et al. have used the cubic spline collocation method to solve a two-dimensional convection-diffusion equation. Generally, the problems involving Burger’s equation arise in several important applications throughout science and engineering, including fluid motion, gas dynamics, [5] transfer and number theory [6].

In literature, recent developments in the resolution of the nonlinear Burger’s-Fisher equation have been achieved. In a recent study [7], Mohammadi developed a stable and accurate numerical method, based on the exponential spline and finite difference approximations, to solve the generalized Burgers’-Fisher equation. The main advantage of the last method is its simplicity. Kaya et al. [8] presented numerical simulation and explicit solutions of the generalized Burgers-Fisher. Ismail et al. [9] used the Adomian decomposition method for the solutions of Burger-Huxley and Burgers-Fisher equations. In [10] Mickens proposed a non-standard finite difference scheme for the Burgers-Fisher equation. A compact finite difference method for the generalized Burgers-Fisher equation was proposed by Sari et al. [11]. Khattak [12] presented a computational radial basis function method for the Burgers-Fisher equation and some various powerful mathematical methods such as factorization method [13], tanh function methods [6, 14], spectral collocation method [15, 16] and variational iteration method [17]. In [18], the fractional-order Burgers-Fisher and generalized Fisher’s equations have been solved by using the Haar wavelet method. Recently, in Ref. [19] discontinuous Legendre wavelet Galerkin method is presented for the numerical solution of the Burgers-Fisher and generalized Burgers-Fisher equations. It consists to combines both the discontinuous Galerkin and the Legendre wavelet Galerkin methods. In [20], Zhu and Kang presented a numerical scheme to solve the hyperbolic conservation laws equation based on cubic B-spline quasi-interpolation. Nonlinear partial differential equations are encountered in a variety of domains of science. Burgers- Fisher equation is a well nonlinear equation because it combines the reaction, convection and diffusion mechanisms. The sticky tag of this equation is called Burgers-Fisher because it gathers the properties of the convective phenomenon from the Burgers equation and the diffusion transport as well as the reaction mechanism from the Fisher equation. This equation shows an exemplary model to express the interaction between the reaction mechanisms, convection effect and diffusion transport. For current applications, Burgers-Fisher equation is much known in financial mathematics, physics, applied mathematics.

In this work, we consider the generalized Burger’s-Fisher equation ([9]) of the form:

ut+αuδux=2u2x+βu1uδ,xΩ=01,t0E1

with the initial condition

ux0=12+12tanhαδ2δ+1x1δ,xΩ=01E2

and the boundary conditions

u0t=12+12tanhαδ2δ+1αδ+1+βδ+1αt1δ,t0E3

and

u1t=12+12tanhαδ2δ+11αδ+1+βδ+1αt1δ,t0E4

The exact solution of Eq. (1) (presented in [9]) is given by:

uxt=12+12tanhαδ2δ+1xαδ+1+βδ+1αt1δ.E5

Our main purpose in this chapter is to use the univariate quasi-interpolant associated with the algebraic hyperbolic B-spline of order 4for solving the Burgers-Fisher Eqs. (10). Firstly, we approximate first and second-order partial derivatives by those of the algebraic hyperbolic spline Q4uxitnquasi-interpolant. Then, we use this derivatives to approximate uxinand 2u2xin. The resulting system can be solved using MATLAB’s ode solver. More precisely, we provide a powerful numerical scheme applying a hyperbolic quasi interplant used in [21] to solve Burger’s Fisher equation. This method produces better results compared to the results obtained by all the schemes in the literature, for example, those studied in [22, 23].

The chapter is organized as follows. Section 2 is dedicated to the description of the quasi-interpolation of the algebraic hyperbolic splines. Afterward, Section 3 is devoted to the presentation of numerical techniques to solve the Burger’s-Fisher equation. The stability of the scheme has been studied in Section 4. In Section 5, two examples of the Burger’s-Fisher equation are illustrated and compared to those obtained with some previous results. Finally, our conclusion is presented in Section 6.

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2. Algebraic hyperbolic spline quasi-interpolation of order 4

In this section, we recall the results on hyperbolic spline quasi-interpolation that we will use to establish the numerical method (see [21]). Let T=xi=ihi=+(0<h<π) be a set of knots which partition the parameter axis x uniformly.

For k3, the B-spline family that generates the space Γk=sinhxcoshx1xxk3is called algebraic hyperbolic B-spline (for more details see [24]), which can be defined as for k=2:

N0,2x=hsinhx2coshh1,0x<h,hsinh2hx2coshh1,hx<2h,0,otherwise,E6
Ni,2x=N0,2hxih,i=0±1±2E7

and for k3,

Ni,kx=1hxhxNi,k1hsds,i=0±1±2.E8

We apply the recursion formula (8) to get the algebraic hyperbolic B-spline of order 4, which is defined in Γ4as follows:

Ni,4x=xxi+sinhxxi2h1coshh,xix<xi+1,xxi+22hcoshh+2xxicoshh+2sinhxi+1x+sinhxi+2x2hcoshh1,xi+1x<xi+2,xi+2x+6hcoshh2xxicoshhsinhxi+2x2sinhxi+3x2hcoshh1,xi+2x<xi+3,xxi+4+sinhxi+4x2hcoshh1,xi+3x<xi+4,0,otherwise.E9

According to [21], the univariate Quasi-Interpolant associated to the algebraic hyperbolic B-spline of order 4, can be expressed as operators of the form

Q41fx=i=3n1ν¯h1fi+2+νh1fi+1+fi+3Ni,4xE10

where νh1=14cschh22hcschh1,ν¯h1=12νh1and fi=fxi.

The error associated with the quadrature formula based on Q41fis of order 5as the following theorem describes.

Theorem 1 There exists a constant C2such that for all fL14aband for all partitions τhof ab,

fQ41fC2h5L4f,E11

with L4is an operator defined by: L4D2D21and L4f=0for all fΓ4.

ProofThe proof is almost the same as that of Theorem 15 in [21].

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3. Numerical scheme using hyperbolic spline quasi-interpolation

For approximate derivatives of fby derivatives of Q41fup to the order h4, we can evaluate the value of fat xiby

Q41f=j=3n1ν¯h1fj+2+νh1fj+1+fj+3Nj,4E12

and

(Q41f)=j=3n1ν¯h1fj+2+νh1fj+1+fj+3Nj,4.E13

The values of Nj,4and Nj,4using the formula (9) are

Ni,4x=1coshxxi2h1coshh,xix<xi+1,1+2coshh2coshxi+1xcoshxi+2x2hcoshh1,xi+1x<xi+2,2coshxi+3x+coshxi+2x2coshh12hcoshh1,xi+2x<xi+3,1coshxi+4x2hcoshh1,xi+3x<xi+4,0,otherwise,E14

and

Ni,4x=sinhxxi2h1coshh,xix<xi+1,2sinhxi+1x+sinhxi+2x2hcoshh1,xi+1x<xi+2,sinhxi+2x2sinhxi+3x2hcoshh1,xi+2x<xi+3,sinhxi+4x2hcoshh1,xi+3x<xi+4,0,otherwise,E15

By using Eq. (10), the first derivative of algebraic hyperbolic spline quasi-interpolation at xifor all i2n2is

Q41fxi=j=3n1ν¯h1fj+2+νh1fj+1+fj+3Nj,4xi=ν¯h1fi+νh1fi1+fi+1Ni2,4'xi+ν¯h1fi+1+νh1fi+fi+2Ni1,4xi=νh1fj2ν¯h1fi1+ν¯h1fi+1+νh1fi+2E16

That is to say

Q41fxi=νh1fj2ν¯h1fi1+ν¯h1fi+1+νh1fi+2E17

and the second derivative of algebraic hyperbolic spline quasi-interpolation at xifor all i2n2is

Q41fxi=j=3n1ν¯h1fj+2+νh1fj+1+fj+3Nj,4xi=ν¯h1fi+νh1fi1+fi+1Ni2,4xi+ν¯h1fi+1+νh1fi+fi+2Ni1,4xiE18

That is to say

Q41fxi=ahνh1fi2+2νh1+ν¯h1fi1+2νh1ν¯h1fi+2νh1+ν¯h1fi+1+νh1fi+2E19

with ah=sinhh2hcoshh1.

Discretizing (1) in time we get

uin+1uin+αuδinτuxin=τ2u2xin+βτuin1uδinE20

where uinis the approximation of the value uxtat xitn, tn=and τis the time step with 0iMand 0nN. Then, we use the derivatives of the algebraic hyperbolic spline Q4uxitnquasi-interpolant to approximate uxinand 2u2xin.

Assume that Un=u0nu1nuMnis known for the non-negative integer n. We set unknown vectors as

Uxn=ux0nux1nuxMn2U2xn=2u2x0n2u2x1n2u2xMnE21

From the initial conditions ((3), (4)) and boundary conditions (2), we can compute the numerical solution of (1) step by step using the scheme (20) and formulas ((17), (19)).

According to (17), (19) and (21). the scheme (20) can be rewritten as

uin+1=uin+αuδinτ2hνh1ui2nν¯h1ui1n+ν¯h1ui+1n+νh1ui+2n+τ2hah(νh1ui2n+2νh1+ν¯h1ui1n+2νh1ν¯h1uin+2νh1+ν¯h1ui+1n+νh1ui+2n)+βτuin1uδin,foralli2M2E22

with ah=sinhh2hcoshh1.

This scheme is called the algebraic hyperbolic quasi-interpolation (AHQI) scheme.

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4. Stability analysis

Sharma and Singh provided a method to study the stability of the nonlinear partial equation in [25], which we used in this section to study the stability of our scheme.

If we set r=τ2h,Ain=αuδin,Bin=βuδin, then the scheme (22) becomes

uin+1=Ainrνh1+rahνh1ui2n+Ainrν¯h1+rahν¯h12νh1ui1n+(1+βττBin+2rahνh1ν¯h1)uin+Ainrν¯h1+rahν¯h12νh1ui+1n+Ainrνh1+rahνh1ui+2n.E23

If we move to the L-infinity norm then we obtain

Un+1LSupiAinrνh1+rahνh1UnL+SupiAinrν¯h1+rahν¯h12νh1UnL+Supi1+βττBin+2rahνh1ν¯h1UnL+SupiAinrν¯h1+rahν¯h12νh1UnL+SupiAinrνh1+rahνh1UnL.E24

If we set M1n=SupiahAin,M2n=Supiah+Ain,M3n=Supi1+βττBin, then the Eq. (24) becomes

Un+1L(M3n+rν¯h16ah+M1n+M2n+νh12ah+M1n+M2nUnL.E25

It implies that the scheme is stable if

M3n+rν¯h16ah+M1n+M2n+νh12ah+M1n+M2nC,E26

with Cis a finite positive constant.

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5. Numerical results

In this section, the proposed quasi-interpolation splines collocation methods are tested for their validity for solving the generalized Burgers-Fisher equation with the initial condition (2) and the boundary conditions (3). Two different examples for the Burgers-Fisher equation are solved and the obtained results are compared with those presented in [22, 25]. To verify the accuracy and reliability of the present method in this article, we select two examples to conduct numerical experiments and compare them with the calculated results in the existing literature. That’s why we divided this section into two subsections, in each subsection we compared our scheme (AHQI scheme) to each example by comparing their maximum error which is defined by

e=Max1iMuiexactuiapproach.E27

5.1 First example: MCN scheme

In the first example, we compared the maximum error of AHQI scheme with MCN scheme proposed in [22]. In Table 1 we showed the maximum error of each scheme with different values of Nwith α=β=δ=1and we remarked that our method is better than that presented in [22], also we illustrated the numerical results of Eq. (1) by our method in Figure 1(a) and (b) for different values in space and time.

xN = 10N = 100N = 1000
AHQIMCNAHQIMCNAHQIMCN
1×1051×1041 ×1061×1051×1091×108
0.10.04420.09870.02480.08650.01280.2880
0.20.09230.12690.07600.11530.03840.2834
0.30.13990.13520.12250.12320.07270.2060
0.40.18690.13760.16620.12500.12930.1419
0.50.23290.13830.20640.12530.21800.1158
0.60.27780.13790.23940.12510.34390.1315
0.70.32130.13590.25950.12350.49290.1836
0.80.36330.12870.24580.11620.57400.2489
0.90.40370.24890.12250.08820.32410.2516

Table 1.

Values of errors by AHQI and MCN.

Figure 1.

The behavior of numerical results of equation1by AHQI forτ=0.001.

5.2 Second example: BSQI scheme

For the second example, we compared our scheme to BSQI scheme proposed in [25] for different values of α,βand τ: in the Table 2α=β=0.001,τ=0.0001and in the Table 3α=β=1,τ=0.0001with M=16. In each table we calculate the maximum error for different values of δand we remarked that for δ=1and δ=2our method is better than the other scheme but is close to it for δ=3. We also illustrated the maximum error for α=β=0.001,δ=1,τ=0.0001and α=β=1,δ=1,τ=0.00001in t=1and in different values of space as the Figures 1 and 2 respectively show. The results of our method for three different space size steps (δ=1,2,4) and five different time size steps tare shown in Tables 2 and 3. It is very clear that a good agreement between the analytical solution and the present numerical results with a minimum error is obtained, and the error becomes clear when using a large size step for time and space.

Figure 2.

The absolute errors forα=β=0.001,τ=0.0001,δ=1,t=1.

tδ=1δ=2δ=4
AHQIBSQIAHQIBSQIAHQIBSQI
0.0012.22044×10164.44089×10162.88657×10159.76996×10152.37587×10143.37508×1014
0.0053.88578×10161.55431×10151.26565×10144.17888×10131.16573×10131.66422×1013
0.0106.66133×10161.83187×10152.45359×10149.18154×10142.32702×10133.25628×1013
0.5001.19904×10143.81917×10142.12041×10131.07303×10122.12041×10123.88234×1012
1.0005.10702×10153.04201×10142.17048×10131.08047×10122.12352×10123.91087×1012

Table 2.

Error for various values of δand xwith α=β=0.001, τ=0.0001.

tδ=1δ=2δ=4
AHQIBSQIAHQIBSQIAHQIBSQI
0.23.99146×1085.55746×1071.26606×1072.56108×1061.76174×1071.76161×106
0.47.94950×1089.05507×1071.74941×1074.24308×1061.82135×1064.17351×107
0.61.76244×1072.18808×1062.71508×1073.56848×1061.32793×1062.42401×106
0.82.34913×1072.93314×1063.79941×1071.46518×1067.62594×1072.35757×106
12.47511×1073.01455×1063.98493×1075.54230×1063.79941×1071.44350×106

Table 3.

Error for various values of δand xwith α=β=1, τ=0.0001.

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6. Conclusion

In this work, a numerical scheme to solve the nonlinear Burgers -Fisher equation has been proposed using algebraic hyperbolic spline quasi-interpolation. The numerical scheme stability was well established. The scheme efficiency, as well as its accuracy, are justified by treating well-known examples in the literature, for each case the error is reported. We conclude that the scheme with algebraic hyperbolic spline quasi-interpolation can solve Burgers-Fisher equations since it produces reasonably good results, with high convergence with very small errors.

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Written By

Mohamed Jeyar, Abdellah Lamnii, Mohamed Yassir Nour, Fatima Oumellal and Ahmed Zidna

Reviewed: June 24th, 2021 Published: December 10th, 2021