**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 *<sup>r</sup>* <sup>¼</sup> *<sup>τ</sup>* <sup>2</sup>*<sup>h</sup>* , <sup>A</sup>*<sup>n</sup> <sup>i</sup>* <sup>¼</sup> *<sup>α</sup> <sup>u</sup><sup>δ</sup>* ð Þ*<sup>n</sup> <sup>i</sup>* , <sup>B</sup>*<sup>n</sup> <sup>i</sup>* <sup>¼</sup> *<sup>β</sup> <sup>u</sup><sup>δ</sup>* ð Þ*<sup>n</sup> <sup>i</sup>* , then the scheme (22) becomes

$$\begin{aligned} u\_i^{n+1} &= \left(-\mathcal{A}\_i^n r \nu\_h^1 + r a\_h \nu\_h^1\right) u\_{i-2}^n + \left(-\mathcal{A}\_i^n r \overline{\nu}\_h^1 + r a\_h \left(\overline{\nu}\_h^1 - 2\nu\_h^1\right)\right) u\_{i-1}^n + \left(1 + \beta \tau - \tau \mathcal{B}\_i^n\right) u\_i^n \\ &+ 2r a\_h \left(\nu\_h^1 - \overline{\nu}\_h^1\right)) u\_i^n + \left(\mathcal{A}\_i^n r \overline{\nu}\_h^1 + r a\_h \left(\overline{\nu}\_h^1 - 2\nu\_h^1\right)\right) u\_{i+1}^n + \left(\mathcal{A}\_i^n r \nu\_h^1 + r a\_h \nu\_h^1\right) u\_{i+2}^n. \end{aligned} \tag{23}$$

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

$$\begin{aligned} \left\lVert \left. U^{n+1} \right\rVert\_{L} \overset{\infty}{\leq} & \sup\_{i} \left| -\mathcal{A}\_{i}^{n} r \nu\_{h}^{1} + r a\_{h} \nu\_{h}^{1} \right| \left\lVert \left. U^{n} \right\rVert\_{L} \overset{\infty}{+} + \sup\_{i} \left| -\mathcal{A}\_{i}^{n} r \overline{\nu}\_{h}^{1} + r a\_{h} \left( \overline{\nu}\_{h}^{1} - 2 \nu\_{h}^{1} \right) \right| \left\lVert \left. U^{n} \right\rVert\_{L} \right| \\ & + \sup\_{i} \left| 1 + \beta \tau - \tau \mathcal{B}\_{i}^{n} + 2 r a\_{h} \left( \nu\_{h}^{1} - \overline{\nu}\_{h}^{1} \right) \right| \left\lVert \left. U^{n} \right\rVert\_{L} \right| \\ & + \sup\_{i} \left| \mathcal{A}\_{i}^{n} r \overline{\nu}\_{h}^{1} + r a\_{h} \left( \overline{\nu}\_{h}^{1} - 2 \nu\_{h}^{1} \right) \right| \left\lVert \left. U^{n} \right\rVert \right\rVert\_{L}^{\infty} + \sup\_{i} \left| \mathcal{A}\_{i}^{n} r \nu\_{h}^{1} + r a\_{h} \nu\_{h}^{1} \right| \left\lVert \left. U^{n} \right\rVert \right\rVert\_{L}^{\infty} . \end{aligned} \tag{24}$$

If we set M*<sup>n</sup>* <sup>1</sup> <sup>¼</sup> *Supi* <sup>∣</sup>*ah* � *<sup>A</sup><sup>n</sup> <sup>i</sup>* ∣,M*<sup>n</sup>* <sup>2</sup> <sup>¼</sup> *Supi* <sup>∣</sup>*ah* <sup>þ</sup> *<sup>A</sup><sup>n</sup> <sup>i</sup>* ∣,M*<sup>n</sup>* <sup>3</sup> <sup>¼</sup> *Supi* <sup>∣</sup><sup>1</sup> <sup>þ</sup> *βτ* � *<sup>τ</sup>*B*<sup>n</sup> i* ∣, then the Eq. (24) becomes

$$\left\|\left|\boldsymbol{U}^{n+1}\right\|\right\|\_{L}^{\infty} \leq \left(\mathcal{M}\_{3}^{n} + |\boldsymbol{r}|\left(|\overline{\nu}\_{h}^{1}|\left(\boldsymbol{\Theta}|a\_{h}| + \left(\mathcal{M}\_{1}^{n} + \mathcal{M}\_{2}^{n}\right)\right) + |\nu\_{h}^{1}|\left(2|a\_{h}| + \left(\mathcal{M}\_{1}^{n} + \mathcal{M}\_{2}^{n}\right)\right)\right) \left\|\boldsymbol{U}^{n}\right\|\_{L}^{\infty}.\tag{25}$$

It implies that the scheme is stable if

$$|\mathcal{M}\_3^n + |r| \left( |\overline{\nu}\_h^1| \left( \left( \Theta |a\_h| + \left( \mathcal{M}\_1^n + \mathcal{M}\_2^n \right) \right) + |\nu\_h^1| \left( 2|a\_h| + \left( \mathcal{M}\_1^n + \mathcal{M}\_2^n \right) \right) \right) \le \mathcal{C}, \tag{26}$$

with C is a finite positive constant.

#### **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

*An Algebraic Hyperbolic Spline Quasi-Interpolation Scheme for Solving Burgers-Fisher… DOI: http://dx.doi.org/10.5772/intechopen.99033*

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

$$e = \max\_{1 \le i \le M} |u\_i^{\text{exact}} - u\_i^{\text{append}}|. \tag{27}$$

#### **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 *N* with *α* ¼ *β* ¼ *δ* ¼ 1 and we remarked that our method is better than that presented in [22], also we illustrated the


#### **Table 1.**

*Values of errors by AHQI and MCN.*

**Figure 1.** *The behavior of numerical results of equation* ð Þ 1 *by AHQI for τ* ¼ 0*:*001*.*



*An Algebraic Hyperbolic Spline Quasi-Interpolation Scheme for Solving Burgers-Fisher… DOI: http://dx.doi.org/10.5772/intechopen.99033*

illustrated the maximum error for *α* ¼ *β* ¼ 0*:*001, *δ* ¼ 1, *τ* ¼ 0*:*0001 and *α* ¼ *β* ¼ 1, *δ* ¼ 1, *τ* ¼ 0*:*00001 in *t* ¼ 1 and 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* ¼ 1, 2, 4) and five different time size steps *t* are 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.

## **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.

## **Author details**

Mohamed Jeyar1#, Abdellah Lamnii2 \*#, Mohamed Yassir Nour2,3#, Fatima Oumellal2# and Ahmed Zidna3

1 Faculty of Sciences, First Mohammed University, Oujda, Morocco

2 Hassan First University of Settat, Faculté des Sciences et Technique, LaboratoryMISI, Morocco

3 LGIPM, Université Lorraine, Metz, France

\*Address all correspondence to: a\_lamnii@yahoo.fr
