# Comparative Study of Some Numerical Methods for the Burgers–Huxley Equation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Burgers–Huxley Equation

## 3. Numerical Experiment

- (1)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.
- (2)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\beta >\alpha )\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =2.0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.
- (3)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\beta >>\alpha )\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$. (singularly perturbed case)
- (4)
- $\alpha >\beta \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\alpha =2.0),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.
- (5)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.
- (6)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta >\alpha \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\beta =2.0),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.
- (7)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\beta >>\alpha )\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$. (singularly perturbed case)
- (8)
- $\alpha >\beta \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(\alpha =2.0),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.
- (9)
- (10)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$ and $k=0.1$ (FIEFDM)
- (11)
- $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$ and $k=0.1$ (FIEFDM)

## 4. Nonstandard Finite Difference Scheme

- Non-local representation of linear and non-linear terms on the computational grid; E.g., ${u}_{n}\approx 2{U}_{n}-{U}_{n+1}$, ${u}_{n}^{2}\approx \left({\displaystyle \frac{{U}_{n+1}+2{U}_{n}+{U}_{n-1}}{4}}\right){U}_{n}$, ${u}_{n}^{3}\approx 2{U}_{n}^{3}-{U}_{n}^{2}{U}_{n+1}$ etc.
- Use of numerator and denominator functions $\psi (h)$ and $\varphi (k)$, respectively with the property$$\underset{h\to 0}{lim}\psi (h)=h,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\underset{k\to 0}{lim}\varphi (k)=k,$$
- The difference equation should have the same order as the original differential equation. In general, when the order of the difference equation is larger than the order of the differential equation, spurious solutions will appear [58].
- The discrete approximation should preserve some important properties of the corresponding differential equation. Properties such as boundedness and positivity should be preserved.

#### 4.1. NSFD1 Scheme

**Theorem**

**1**

**.**If $1-2R\ge 0$, the numerical solution from NSFD1 satisfies

**Proof.**

- (a)
- $k\le 5.251\times {10}^{-3}$ for $\beta =0.5$.
- (b)
- $k\le 5.515\times {10}^{-3}$ for $\beta =1.0$.
- (c)
- $k\le 6.090\times {10}^{-3}$ for $\beta =2.0$.
- (d)
- $k\le 1.377\times {10}^{-2}$ for $\beta =10.0$.

#### 4.2. NSFD2 Scheme

**Theorem**

**2**

**.**If $1-2R\ge 0$, the numerical solution from NSFD2 satisfies

**Proof.**

## 5. Exponential Finite Difference Methods

#### 5.1. Explicit Exponential Finite Difference Method (EEFDM)

#### 5.2. Fully Implicit Exponential Finite Difference Method (FIEFDM)

- Set ${\mathbf{V}}^{(0)}$, an initial guess.
- For $m=0,1,2,\dots $ until convergence do:

## 6. Numerical Results

- In Cases 10 and 11, we compare the accuracy of the fully implicit exponential scheme by chosen parameters as $\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =0.5\phantom{\rule{3.33333pt}{0ex}}(\beta =\alpha ),$ $\gamma =0.001$ and $k=0.1$ and $\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =10.0\phantom{\rule{3.33333pt}{0ex}}(\beta >>\alpha ),$ $\gamma =0.5$ and $k=0.1$, respectively. Table 35, Table 36, Table 37 and Table 38 display the results for Cases 10 and 11 at $t=1.0$ and $t=10.0$. We note that, for Cases 1–9, $h=0.1$ and $k=0.005$.

**Case 1:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =\beta =0.5$ and $\gamma =0.001.$

**Case 2:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =2.0$ $(\beta >\alpha )$ and $\gamma =0.001$

**Case 3:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =10.0\phantom{\rule{3.33333pt}{0ex}}(\beta >>\alpha ),$ and $\gamma =0.001.$ (singularly perturbed)

**Case 4:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =2.0\phantom{\rule{3.33333pt}{0ex}}(\alpha >\beta ),$ $\beta =0.5$ and $\gamma =0.001.$

**Case 5:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =\beta =0.5$ and $\gamma =0.5$

**Case 6:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =2.0$ $(\beta >\alpha )$ and $\gamma =0.5$.

**Case 7:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =10.0\phantom{\rule{3.33333pt}{0ex}}(\beta >>\alpha ),$ and $\gamma =0.5.$ (singularly perturbed)

**Case 8:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =2.0\phantom{\rule{3.33333pt}{0ex}}(\alpha >\beta ),$ $\beta =0.5$ and $\gamma =0.5$

**Case 9:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =1.0\phantom{\rule{3.33333pt}{0ex}}(\alpha =\beta ),$ $\beta =1.0$ and $\gamma =0.001$

**Case 10:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =\beta =0.5\phantom{\rule{3.33333pt}{0ex}}(\beta =\alpha ),$ $\gamma =0.001$ and $k=0.1$

**Case 11:**$\phantom{\rule{3.33333pt}{0ex}}\alpha =0.5,$ $\beta =10.0\phantom{\rule{3.33333pt}{0ex}}(\beta >>\alpha ),$ $\gamma =0.5$ and $k=0.1$

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NSFD | Nonstandard Finite Difference |

EEFDM | Explicit Exponential Finite Difference Method |

FIEFDM | Fully Implicit Exponential Finite Difference Method |

MATLAB | Matrix Laboratory |

## References

- Albowitz, M.J.; Clarkson, P.A. Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform; University Press: Cambridge, UK, 1990. [Google Scholar]
- Lawrence, C.E. Partial Differential Equation; Graduate Studies in Mathematics; American Mathematical Society: Berkeley, CA, USA, 1997; pp. 1–662. [Google Scholar]
- Estevez, P.G. Non-classical symmetries and the singular manifold method: The Burgers and the Burgers–Huxley equations. J. Phys. A Math. Gen.
**1994**, 27, 2113–2127. [Google Scholar] [CrossRef] - Estevez, P.G.; Gordoa, P.R. Nonclassical symmetries and the singular manifold method: Theory and six examples. Stud. Appl. Math.
**1995**, 95, 73–113. [Google Scholar] [CrossRef] - Chou, M.H.; Lin, Y.T. Exotic dynamic behavior of the forced FitzHugh-Nagumo equations. Comput. Math. Appl.
**1996**, 32, 109–124. [Google Scholar] [CrossRef] [Green Version] - Olver, P. Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics; Springer: New York, NY, USA, 1994; Volume 107. [Google Scholar]
- Zhao, X.; Tang, D.; Wang, L. New soliton-like solutions for KdV equation with variable coefficient. Phys. Lett. A
**2005**, 346, 228–291. [Google Scholar] [CrossRef] - Yomba, E. The extended Fan’s sub-equation method and its applications to KdV-MkdV, BKK and variant Boussinesq equations. Phys. Lett. A
**2005**, 336, 463–476. [Google Scholar] [CrossRef] - Wazwaz, A.M. Solutions and periodic solutions for the fifth-order KdV equation. Appl. Math. Lett.
**2006**, 19, 1162–1167. [Google Scholar] [CrossRef] - Wang, M. Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A
**1996**, 213, 79–287. [Google Scholar] [CrossRef] - Biswas, A. Anjan S1-soliton solution of the K (m, n) equation with generalized evolution. Phys. Lett. A
**2008**, 25, 4601–4602. [Google Scholar] [CrossRef] - Yu, X.; Gao, Y.T.; Sun, Z.Y.; Liu, Y. N-soliton solutions, Backlund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg–de Vries equation. Phys. Scr.
**2010**, 81, 045402. [Google Scholar] [CrossRef] - Ismail, H.N.A.; Raslan, K.; Abd Rabboh, A.A. Adomian decomposition method for burgers—Huxley and burgers—Fisher equations. Appl. Math. Comput.
**2004**, 1, 291–301. [Google Scholar] - Batiha, M.S.M.; Noorani, B.; Hashim, I. Application of variational iteration method to the generalized burgers–huxley equation. Chaos Solitons Fractals
**2008**, 36, 660–663. [Google Scholar] [CrossRef] - Biazar, J.; Mohammadi, F. Application of differential transform method to the generalized burgers–huxley equation. Appl. Appl. Math.
**2010**, 5, 1726–1750. [Google Scholar] - Yaghouti, M.R.; Zabihi, A. Application of Laplace decomposition method for burgers–huxley and burgers–fisher equations. J. Math. Model.
**2013**, 1, 41–67. [Google Scholar] - Molabahramia, A.; Khan, F. The homotopy analysis method to solve the burgers–huxley equation. Nonlinear Anal. Real World Appl.
**2009**, 10, 589–600. [Google Scholar] [CrossRef] - Mittal, R.C.; Jiwari, R. A numerical study of burger–huxley equation by differential quadrature method. J. Appl. Math. Mech.
**2009**, 5, 1–9. [Google Scholar] - Javidi, M. A numerical solution of the generalized burgers–huxley equation by spectral collocation method. Appl. Math. Comput.
**2006**, 178, 338–344. [Google Scholar] - Tomasiello, S. Numerical solutions of the burgers–huxley equation by the IDQ method. Int. J. Comput. Math.
**2010**, 87, 129–140. [Google Scholar] [CrossRef] - Ray, S.S.; Gupta, A. Comparative analysis of variational iteration method and Haar wavelet method for the numerical solutions of Burgers–Huxley and Huxley equations. J. Math. Chem.
**2014**, 52, 1060–1080. [Google Scholar] - Ray, S.S.; Gupta, A. Application of novel schemes based on Haar wavelet collocation method for numerical simulations of Burger and Boussinesq–Burger equations. Appl. Math. Inf. Sci.
**2016**, 10, 1513–1524. [Google Scholar] [CrossRef] - Chen, Z.; Gumel, A.B.; Micken, R.E. Nonstandard discretizations of the generalized nagumo reaction-diffusion equation. Numer. Methods Partial Differ. Equ.
**2003**, 19, 363–379. [Google Scholar] [CrossRef] - Kyrychko, Y.N.; Bartuccelli, M.; Blysuss, K.B. Persistence of travelling wave solutions of a fourth order diffusion system. J. Comput. Appl. Math.
**2005**, 176, 433–443. [Google Scholar] [CrossRef] - Anguelov, R.; Lubuma, J.M.S. Contributions to the Mathematics of the Nonstandard Finite Difference Method and Applications. Numer. Methods Partial Differ. Equ.
**2001**, 17, 518–543. [Google Scholar] [CrossRef] - Mickens, R.E. Application of Nonstandard Finite Difference Scheme; World Scientific: Singapore, 2000; pp. 1–261. [Google Scholar]
- Mickens, R.E.; Gumel, A.B. Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation. J. Sound Vib.
**2002**, 4, 791–797. [Google Scholar] [CrossRef] - Zhang, L.; Wang, L.; Xiaohu, D. Exact finite difference scheme and nonstandard finite difference scheme for burgers and burgers-fisher equations. J. Appl. Math.
**2014**, 597926. [Google Scholar] [CrossRef] - Agbavon, K.M.; Appadu, A.R. Construction and analysis of some nonstandard finite difference methods for the Fitzhugh-Nagumo equation. Numer. Methods Partial Differ. Equ. In Review.
- Appadu, A.R.; Lubuma, J.M.S.; Mphephu, N. Computational study of three numerical methods for some linear and nonlinear advection-diffusion-reaction problems. Prog. Comput. Fluid Dyn.
**2017**, 17, 114–129. [Google Scholar] [CrossRef] - Appadu, A.R. Numerical solution of the 1D advection-diffusion equation using standard and nonstandard finite difference schemes. J. Appl. Math.
**2013**, 2013, 734374. [Google Scholar] [CrossRef] - Agabvon, K.M.; Appadu, A.R.; Khumalo, M. On the numerical solution of fishers equation with coefficient of diffusion term much smaller than coefficient of reaction term. Adv. Differ. Equ.
**2019**, 146, 33. [Google Scholar] [CrossRef] - Chapwanya, M.; Appadu, A.R.; Jejeniwa, M.O.; Lubuma, J.M.S. An explicit nonstandard finite difference scheme for the Fitzhugh-Nagumo equations. Int. J. Comput. Math.
**2019**, 96, 1993–2009. [Google Scholar] [CrossRef] - Mickens, R.E. Nonstandard finite difference schemes for reaction-diffusion equations. Numer. Methods Partial Differ. Equ.
**1999**, 15, 201–215. [Google Scholar] [CrossRef] - Jordan, P.M. A nonstandard finite difference scheme for nonlinear heat transfer in a thin finite rod. J. Differ. Equ. Appl.
**2003**, 9, 1015–1021. [Google Scholar] [CrossRef] - Aderogba, A.A.; Chapwanya, M. Positive and bounded nonstandard finite difference scheme for the hodgkin–huxley equations. Jpn. J. Ind. Appl. Math.
**2018**, 35, 773–785. [Google Scholar] [CrossRef] - Zibaei, M.; Zeinadini, S.; Namjoo, M. Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes. J. Differ. Equ. Appl.
**2016**, 22, 1098–1113. [Google Scholar] [CrossRef] - Oluwaseye, A.; Talitha, W. Nonstandard finite difference scheme for a Tacoma Narrows Bridge model. Appl. Math. Model.
**2018**, 62, 223–236. [Google Scholar] - Dai, W. Nonstandard finite difference schemes for solving nonlinear micro heat transport equations in double-layered metal thin films exposed to ultrashort pulsed lasers. In Advances in the Applications of Nonstandard Finite Difference Schemes; Mickens, R.E., Ed.; World Scientific: Hackensack, NJ, USA, 2005; pp. 191–248. [Google Scholar]
- Díaz-Rodríguez, M.; González-Parra, G.; Arenas, A.J. Nonstandard numerical schemes for modeling a 2-DOF serial robot with rotational spring-damper-actuators. Int. J. Numer. Methods Biomed. Eng.
**2011**, 27, 1211–1224. [Google Scholar] [CrossRef] - Inan, B.; Bahadir, A.R. A numerical solution of the Burgers equation using a Crank–Nicolson exponential finite difference method. J. Math. Comput. Sci.
**2014**, 4, 849–860. [Google Scholar] - Macías-Díaz, J.E. A modified exponential method that preserves structural properties of the solutions of the Burgers-Huxley equation. Int. J. Comput. Math.
**2017**. [Google Scholar] [CrossRef] - Bhattacharya, M.C. An explicit conditionally stable finite difference equation for heat conduction problems. Int. J. Numer. Method Eng.
**1985**, 21, 239–265. [Google Scholar] [CrossRef] - Inan, B.; Bahadir, A.R. Numerical solution of the one-dimensional burgers equation: Implicit and fully implicit exponential finite difference methods. Pramana J. Phys.
**2018**, 81, 547–556. [Google Scholar] [CrossRef] - Bahadir, A.R. Exponential finite-difference method applied to korteweg-de vries equation for small times. Appl. Math. Comput.
**2005**, 160, 675–682. [Google Scholar] - İnan, B. Finite difference methods for the generalized huxley and burgers-huxley equations. Kuwait J. Sci.
**2017**, 44, 20–27. [Google Scholar] - İnan, B. An exponential finite difference method based on Padé approximation. Celal Bayar Univ. Sci.
**2017**, 13, 71–80. [Google Scholar] [CrossRef] - Macías-Díaz, J.E.; İnan, B. Numerical efficiency of some exponential methods for an advection-diffusion equation. Int. J. Comput. Math.
**2019**, 96, 1005–1029. [Google Scholar] [CrossRef] - Appadu, A.R.; İnan, B.; Tijani, Y.O. Comparison of some numerical methods for the Burgers-Huxley equation. To appear in AIP Conference Proceedings.
- Griffiths, W.G.; Schiesser, E.W. Traveling Wave Analysis of Partial Differential Equation: Numerical and Analytical Methods with MATLAB and Maple; Academic Press: Cambridge, MA, USA, 2011; p. 461. ISBN 9780123846525. [Google Scholar]
- Deng, X. Travelling wave solutions for the generalized Burgers–Huxley equation. Appl. Math. Comput.
**2008**, 204, 733–737. [Google Scholar] [CrossRef] - Burgers, J.M. A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics; Academic Press: New York, NY, USA, 1948; Volume 1, pp. 171–199. [Google Scholar]
- Nourazar, S.S.; Soori, M.; Nazari-Golshan, A. On the Exact Solution of Newell-Whitehead-Segel Equation Using the Homotopy Perturbation Method. Aust. J. Basic Appl. Sci.
**2011**, 5, 1400–1411. [Google Scholar] - Wang, Z.S.; Zhuo, X.Y.; Lu, Y.K. Solitary wave solutions of the generalised Burgers-Huxley equation. J. Phys. A Math. Gen.
**1990**, 23, 271–274. [Google Scholar] [CrossRef] - Mickens, R.E. Pitfalls in the numerical integration of differential equations. In Analytical Techniques for Material Characterization; Collins, W.E., Chowdari, B.V.R., Radhakrishna, S., Eds.; World Scientific: Singapore, 1987; Volume 17, pp. 123–143. [Google Scholar]
- Mickens, R.E.; Smith, A. Finite-Difference Model of Ordinary Differential Equations: Influence of Denominator Function. J. Franklin Inst.
**1990**, 327, 143–149. [Google Scholar] [CrossRef] - Mickens, R.E. Lie methods in mathematical modeling: Difference equation models of differential equations. Math. Comput. Model.
**1988**, 11, 528–530. [Google Scholar] [CrossRef] - Hilderband, F.B. Finite-Difference Equations and Simulations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1968; p. 338. [Google Scholar]
- Taha, T.; Ablowitz, M. Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical Korteweg-de Vries equation. J. Comput. Phys.
**1984**, 2, 231–235. [Google Scholar] [CrossRef]

**Figure 1.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.

**Figure 2.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =2.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.

**Figure 3.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.

**Figure 4.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =2.0,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.

**Figure 5.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.

**Figure 6.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =2.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.

**Figure 7.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.

**Figure 8.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =2.0,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$.

**Figure 9.**Plot of $|\xi |$ vs. k vs. $\omega $ for $\alpha =1.0,\phantom{\rule{3.33333pt}{0ex}}\beta =1.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$.

Cases | Parameter Values | Condition for Stability |
---|---|---|

1 | $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$ | $k\le 0.005$ |

2 | $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =2.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$ | $k\le 0.005$ |

3 | $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$ | $k\le 0.005$ |

4 | $\alpha =2.0,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$ | $k\le 0.005$ |

5 | $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$ | $k\le 0.005$ |

6 | $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =2.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$ | $k\le 0.005$ |

7 | $\alpha =0.5,\phantom{\rule{3.33333pt}{0ex}}\beta =10.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$ | $k\le 0.005$ |

8 | $\alpha =2.0,\phantom{\rule{3.33333pt}{0ex}}\beta =0.5,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.5$ | $k\le 0.005$ |

9 | $\alpha =1.0,\phantom{\rule{3.33333pt}{0ex}}\beta =1.0,\phantom{\rule{3.33333pt}{0ex}}\gamma =0.001$ | $k\le 0.005$ |

t | x | Exact | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|---|

1 | $0.1$ | $5.000858\times {10}^{-4}$ | $5.000764\times {10}^{-4}$ | $5.000764\times {10}^{-4}$ | $5.000768\times {10}^{-4}$ | $5.000769\times {10}^{-4}$ |

$0.5$ | $5.001249\times {10}^{-4}$ | $5.000985\times {10}^{-4}$ | $5.000985\times {10}^{-4}$ | $5.000998\times {10}^{-4}$ | $5.000998\times {10}^{-4}$ | |

$0.9$ | $5.001640\times {10}^{-4}$ | $5.001545\times {10}^{-4}$ | $5.001545\times {10}^{-4}$ | $5.001549\times {10}^{-4}$ | $5.001549\times {10}^{-4}$ | |

10 | $0.1$ | $5.007711\times {10}^{-4}$ | $5.007616\times {10}^{-4}$ | $5.007616\times {10}^{-4}$ | $5.007621\times {10}^{-4}$ | $5.007621\times {10}^{-4}$ |

$0.5$ | $5.008102\times {10}^{-4}$ | $5.007838\times {10}^{-4}$ | $5.007838\times {10}^{-4}$ | $5.007851\times {10}^{-4}$ | $5.007851\times {10}^{-4}$ | |

$0.9$ | $5.008492\times {10}^{-4}$ | $5.008397\times {10}^{-4}$ | $5.008397\times {10}^{-4}$ | $5.008402\times {10}^{-4}$ | $5.008402\times {10}^{-4}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $9.509935\times {10}^{-9}$ | $9.509930\times {10}^{-9}$ | $9.048885\times {10}^{-9}$ | $9.048222\times {10}^{-9}$ |

$0.5$ | $2.641711\times {10}^{-8}$ | $2.641710\times {10}^{-8}$ | $2.513647\times {10}^{-8}$ | $2.513446\times {10}^{-8}$ | |

$0.9$ | $9.510585\times {10}^{-9}$ | $9.510580\times {10}^{-9}$ | $9.049488\times {10}^{-9}$ | $9.048757\times {10}^{-9}$ | |

10 | $0.1$ | $9.510514\times {10}^{-9}$ | $9.510509\times {10}^{-9}$ | $9.049214\times {10}^{-9}$ | $9.048760\times {10}^{-9}$ |

$0.5$ | $2.641900\times {10}^{-8}$ | $2.641898\times {10}^{-8}$ | $2.513754\times {10}^{-8}$ | $2.513624\times {10}^{-8}$ | |

$0.9$ | $9.511164\times {10}^{-9}$ | $9.511159\times {10}^{-9}$ | $9.049817\times {10}^{-9}$ | $9.049302\times {10}^{-9}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $1.901660\times {10}^{-5}$ | $1.901659\times {10}^{-5}$ | $1.809466\times {10}^{-5}$ | $1.809333\times {10}^{-5}$ |

$0.5$ | $5.282102\times {10}^{-5}$ | $5.282100\times {10}^{-5}$ | $5.026038\times {10}^{-5}$ | $5.025637\times {10}^{-5}$ | |

$0.9$ | $1.901493\times {10}^{-5}$ | $1.901492\times {10}^{-5}$ | $1.809304\times {10}^{-5}$ | $1.809158\times {10}^{-5}$ | |

10 | $0.1$ | $1.899174\times {10}^{-5}$ | $1.899173\times {10}^{-5}$ | $1.807056\times {10}^{-5}$ | $1.806967\times {10}^{-5}$ |

$0.5$ | $5.275251\times {10}^{-5}$ | $5.275249\times {10}^{-5}$ | $5.019375\times {10}^{-5}$ | $5.019115\times {10}^{-5}$ | |

$0.9$ | $1.899007\times {10}^{-5}$ | $1.899006\times {10}^{-5}$ | $1.806894\times {10}^{-5}$ | $1.806792\times {10}^{-5}$ |

**Table 5.**${L}_{1}$ and ${L}_{\infty}$ error norms with CPU time taken for the four numerical methods.

t | Schemes | ${\mathit{L}}_{1}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | ${\mathit{L}}_{\infty}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | CPU Time |
---|---|---|---|---|

1 | NSFD1 | $1.743535\times {10}^{-8}$ | $2.641711\times {10}^{-8}$ | $0.0642$ |

NSFD2 | $1.743534\times {10}^{-8}$ | $2.641710\times {10}^{-8}$ | $0.0649$ | |

EEFDM | $1.659010\times {10}^{-8}$ | $2.513647\times {10}^{-8}$ | $0.0660$ | |

FIEFDM | $1.658878\times {10}^{-8}$ | $2.513446\times {10}^{-8}$ | $0.0683$ | |

10 | NSFD1 | $1.743654\times {10}^{-8}$ | $2.641900\times {10}^{-8}$ | $0.2261$ |

NSFD2 | $1.743653\times {10}^{-8}$ | $2.641898\times {10}^{-8}$ | $0.2247$ | |

EEFDM | $1.659078\times {10}^{-8}$ | $2.513754\times {10}^{-8}$ | $0.2210$ | |

FIEFDM | $\phantom{\rule{4pt}{0ex}}1.658990\times {10}^{-8}$ | $2.513624\times {10}^{-8}$ | $0.2341$ |

t | x | Exact | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|---|

1 | $0.1$ | $5.004115\times {10}^{-4}$ | $5.003626\times {10}^{-4}$ | $5.003626\times {10}^{-4}$ | $5.003715\times {10}^{-4}$ | $5.003715\times {10}^{-4}$ |

$0.5$ | $5.004997\times {10}^{-4}$ | $5.003639\times {10}^{-4}$ | $5.003639\times {10}^{-4}$ | $5.003886\times {10}^{-4}$ | $5.003886\times {10}^{-4}$ | |

$0.9$ | $5.005880\times {10}^{-4}$ | $5.005391\times {10}^{-4}$ | $5.005391\times {10}^{-4}$ | $5.005480\times {10}^{-4}$ | $5.005480\times {10}^{-4}$ | |

10 | $0.1$ | $5.039160\times {10}^{-4}$ | $5.038671\times {10}^{-4}$ | $5.038671\times {10}^{-4}$ | $5.038760\times {10}^{-4}$ | $5.038760\times {10}^{-4}$ |

$0.5$ | $5.040428\times {10}^{-4}$ | $5.038684\times {10}^{-4}$ | $5.038684\times {10}^{-4}$ | $5.038931\times {10}^{-4}$ | $5.038931\times {10}^{-4}$ | |

$0.9$ | $5.040926\times {10}^{-4}$ | $5.040436\times {10}^{-4}$ | $5.040436\times {10}^{-4}$ | $5.040525\times {10}^{-4}$ | $5.040525\times {10}^{-4}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $4.891509\times {10}^{-8}$ | $4.891498\times {10}^{-8}$ | $4.001139\times {10}^{-8}$ | $4.000776\times {10}^{-8}$ |

$0.5$ | $1.358748\times {10}^{-7}$ | $1.358745\times {10}^{-7}$ | $1.111457\times {10}^{-7}$ | $1.111355\times {10}^{-7}$ | |

$0.9$ | $4.891869\times {10}^{-8}$ | $4.891858\times {10}^{-8}$ | $4.001405\times {10}^{-8}$ | $4.000958\times {10}^{-8}$ | |

10 | $0.1$ | $4.892413\times {10}^{-8}$ | $4.892402\times {10}^{-8}$ | $4.001053\times {10}^{-8}$ | $4.000788\times {10}^{-8}$ |

$0.5$ | $1.359053\times {10}^{-7}$ | $1.359050\times {10}^{-7}$ | $1.111440\times {10}^{-7}$ | $1.111369\times {10}^{-7}$ | |

$0.9$ | $4.892767\times {10}^{-8}$ | $4.892756\times {10}^{-8}$ | $4.001314\times {10}^{-8}$ | $4.000966\times {10}^{-8}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $9.774973\times {10}^{-5}$ | $9.774951\times {10}^{-5}$ | $7.995698\times {10}^{-5}$ | $7.994972\times {10}^{-5}$ |

$0.5$ | $2.714782\times {10}^{-4}$ | $2.714776\times {10}^{-4}$ | $2.220695\times {10}^{-4}$ | $2.220490\times {10}^{-4}$ | |

$0.9$ | $9.772245\times {10}^{-5}$ | $9.772223\times {10}^{-5}$ | $7.993409\times {10}^{-5}$ | $7.992517\times {10}^{-5}$ | |

10 | $0.1$ | $9.708786\times {10}^{-5}$ | $9.708764\times {10}^{-5}$ | $7.939920\times {10}^{-5}$ | $7.939394\times {10}^{-5}$ |

$0.5$ | $2.696512\times {10}^{-4}$ | $2.696506\times {10}^{-4}$ | $2.205220\times {10}^{-4}$ | $2.205079\times {10}^{-4}$ | |

$0.9$ | $9.706089\times {10}^{-5}$ | $9.706068\times {10}^{-5}$ | $7.937658\times {10}^{-5}$ | $7.936967\times {10}^{-5}$ |

**Table 9.**${L}_{1}$ and ${L}_{\infty}$ error norms with CPU time taken for the four numerical methods.

t | Schemes | ${\mathit{L}}_{1}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | ${\mathit{L}}_{\infty}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | CPU Time (Sec) |
---|---|---|---|---|

1 | NSFD1 | $8.967845\times {10}^{-8}$ | $1.358748\times {10}^{-7}$ | $0.0641$ |

NSFD2 | $8.967825\times {10}^{-8}$ | $1.358745\times {10}^{-7}$ | $0.0643$ | |

EEFDM | $7.335633\times {10}^{-8}$ | $1.111457\times {10}^{-7}$ | $0.0645$ | |

FIEFDM | $7.334929\times {10}^{-8}$ | $1.111355\times {10}^{-7}$ | $0.0654$ | |

10 | NSFD1 | $8.969751\times {10}^{-8}$ | $1.359053\times {10}^{-7}$ | $0.2322$ |

NSFD2 | $8.969731\times {10}^{-8}$ | $1.359053\times {10}^{-7}$ | $0.2273$ | |

EEFDM | $7.335505\times {10}^{-8}$ | $1.111440\times {10}^{-7}$ | $0.2237$ | |

FIEFDM | $7.335000\times {10}^{-8}$ | $1.111369\times {10}^{-7}$ | $0.2321$ |

t | x | Exact | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|---|

1 | $0.1$ | $5.022873\times {10}^{-4}$ | $5.016837\times {10}^{-4}$ | $5.016837\times {10}^{-4}$ | $5.020743\times {10}^{-4}$ | $5.020743\times {10}^{-4}$ |

$0.5$ | $5.024987\times {10}^{-4}$ | $5.008301\times {10}^{-4}$ | $5.008301\times {10}^{-4}$ | $5.019071\times {10}^{-4}$ | $5.019071\times {10}^{-4}$ | |

$0.9$ | $5.027102\times {10}^{-4}$ | $5.021065\times {10}^{-4}$ | $5.021065\times {10}^{-4}$ | $5.024972\times {10}^{-4}$ | $5.024972\times {10}^{-4}$ | |

10 | $0.1$ | $5.223822\times {10}^{-4}$ | $5.217618\times {10}^{-4}$ | $5.217619\times {10}^{-4}$ | $5.221696\times {10}^{-4}$ | $5.221696\times {10}^{-4}$ |

$0.5$ | $5.225932\times {10}^{-4}$ | $5.208699\times {10}^{-4}$ | $5.208699\times {10}^{-4}$ | $5.220027\times {10}^{-4}$ | $5.220026\times {10}^{-4}$ | |

$0.9$ | $5.228042\times {10}^{-4}$ | $5.221838\times {10}^{-4}$ | $5.221838\times {10}^{-4}$ | $5.225917\times {10}^{-4}$ | $5.225916\times {10}^{-4}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $6.036257\times {10}^{-7}$ | $6.036201\times {10}^{-7}$ | $2.129791\times {10}^{-7}$ | $2.129933\times {10}^{-7}$ |

$0.5$ | $1.668652\times {10}^{-6}$ | $1.668637\times {10}^{-6}$ | $5.916245\times {10}^{-7}$ | $5.917030\times {10}^{-7}$ | |

$0.9$ | $6.036927\times {10}^{-7}$ | $6.036870\times {10}^{-7}$ | $2.129928\times {10}^{-7}$ | $2.129952\times {10}^{-7}$ | |

10 | $0.1$ | $6.203521\times {10}^{-7}$ | $6.203444\times {10}^{-7}$ | $2.125747\times {10}^{-7}$ | $2.125934\times {10}^{-7}$ |

$0.5$ | $1.723336\times {10}^{-6}$ | $1.723315\times {10}^{-6}$ | $5.905050\times {10}^{-7}$ | $5.905967\times {10}^{-7}$ | |

$0.9$ | $6.204109\times {10}^{-7}$ | $6.204031\times {10}^{-7}$ | $2.125841\times {10}^{-7}$ | $2.125916\times {10}^{-7}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $1.201754\times {10}^{-3}$ | $1.201743\times {10}^{-3}$ | $4.240185\times {10}^{-4}$ | $4.240467\times {10}^{-4}$ |

$0.5$ | $3.320709\times {10}^{-3}$ | $3.320679\times {10}^{-3}$ | $1.177365\times {10}^{-3}$ | $1.177521\times {10}^{-3}$ | |

$0.9$ | $1.200876\times {10}^{-3}$ | $1.200865\times {10}^{-3}$ | $4.236891\times {10}^{-4}$ | $4.236939\times {10}^{-4}$ | |

10 | $0.1$ | $1.187545\times {10}^{-3}$ | $1.187530\times {10}^{-3}$ | $4.069333\times {10}^{-4}$ | $4.069691\times {10}^{-4}$ |

$0.5$ | $3.297662\times {10}^{-3}$ | $3.297621\times {10}^{-3}$ | $1.129952\times {10}^{-3}$ | $1.130127\times {10}^{-3}$ | |

$0.9$ | $1.186698\times {10}^{-3}$ | $1.186683\times {10}^{-3}$ | $4.066228\times {10}^{-4}$ | $4.066371\times {10}^{-4}$ |

**Table 13.**${L}_{1}$ and ${L}_{\infty}$ error norms with CPU time taken for the four numerical methods.

t | Schemes | ${\mathit{L}}_{1}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | ${\mathit{L}}_{\infty}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | CPU Time (Sec) |
---|---|---|---|---|

1 | NSFD1 | $1.102963\times {10}^{-6}$ | $1.668652\times {10}^{-6}$ | $0.0475$ |

NSFD2 | $1.102954\times {10}^{-6}$ | $1.668637\times {10}^{-6}$ | $0.0748$ | |

EEFDM | $3.904728\times {10}^{-7}$ | $4.320404\times {10}^{-7}$ | $0.1122$ | |

FIEFDM | $3.905126\times {10}^{-7}$ | $4.320876\times {10}^{-7}$ | $0.1620$ | |

10 | NSFD1 | $1.137391\times {10}^{-6}$ | $1.723336\times {10}^{-6}$ | $0.2023$ |

NSFD2 | $1.137376\times {10}^{-6}$ | $1.723315\times {10}^{-6}$ | $0.3251$ | |

EEFDM | $3.897320\times {10}^{-7}$ | $4.312213\times {10}^{-7}$ | $0.2729$ | |

FIEFDM | $3.897805\times {10}^{-7}$ | $5.905967\times {10}^{-7}$ | $0.3018$ |

t | x | Exact | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|---|

1 | $0.1$ | $5.000267\times {10}^{-4}$ | $5.000196\times {10}^{-4}$ | $5.000196\times {10}^{-4}$ | $5.000200\times {10}^{-4}$ | $5.000200\times {10}^{-4}$ |

$0.5$ | $5.000473\times {10}^{-4}$ | $5.000280\times {10}^{-4}$ | $5.000280\times {10}^{-4}$ | $5.000290\times {10}^{-4}$ | $5.000290\times {10}^{-4}$ | |

$0.9$ | $5.000680\times {10}^{-4}$ | $5.000611\times {10}^{-4}$ | $5.000611\times {10}^{-4}$ | $5.000614\times {10}^{-4}$ | $5.000614\times {10}^{-4}$ | |

10 | $0.1$ | $5.002138\times {10}^{-4}$ | $5.002121\times {10}^{-4}$ | $5.002121\times {10}^{-4}$ | $5.002124\times {10}^{-4}$ | $5.002124\times {10}^{-4}$ |

$0.5$ | $5.002397\times {10}^{-4}$ | $5.002205\times {10}^{-4}$ | $5.002205\times {10}^{-4}$ | $5.002214\times {10}^{-4}$ | $5.002214\times {10}^{-4}$ | |

$0.9$ | $5.002604\times {10}^{-4}$ | $5.002535\times {10}^{-4}$ | $5.002535\times {10}^{-4}$ | $5.002539\times {10}^{-4}$ | $5.002538\times {10}^{-4}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $6.922896\times {10}^{-9}$ | $6.922895\times {10}^{-9}$ | $6.585668\times {10}^{-9}$ | $6.585338\times {10}^{-9}$ |

$0.5$ | $1.923270\times {10}^{-8}$ | $1.923269\times {10}^{-8}$ | $1.829585\times {10}^{-8}$ | $1.829491\times {10}^{-8}$ | |

$0.9$ | $6.924789\times {10}^{-9}$ | $6.924788\times {10}^{-9}$ | $6.587425\times {10}^{-9}$ | $6.587092\times {10}^{-9}$ | |

10 | $0.1$ | $6.923333\times {10}^{-9}$ | $6.923332\times {10}^{-9}$ | $6.585922\times {10}^{-9}$ | $6.585749\times {10}^{-9}$ |

$0.5$ | $1.923411\times {10}^{-8}$ | $1.923411\times {10}^{-8}$ | $1.829667\times {10}^{-8}$ | $1.829624\times {10}^{-8}$ | |

$0.9$ | $6.925227\times {10}^{-9}$ | $6.925226\times {10}^{-9}$ | $6.587680\times {10}^{-9}$ | $6.587504\times {10}^{-9}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $1.384506\times {10}^{-5}$ | $1.384505\times {10}^{-5}$ | $1.317064\times {10}^{-5}$ | $1.316998\times {10}^{-5}$ |

$0.5$ | $3.846175\times {10}^{-5}$ | $3.846175\times {10}^{-5}$ | $3.658823\times {10}^{-5}$ | $3.658636\times {10}^{-5}$ | |

$0.9$ | $1.384770\times {10}^{-5}$ | $1.384769\times {10}^{-5}$ | $1.317306\times {10}^{-5}$ | $1.317239\times {10}^{-5}$ | |

10 | $0.1$ | $1.384060\times {10}^{-5}$ | $1.384060\times {10}^{-5}$ | $1.316608\times {10}^{-5}$ | $1.316573\times {10}^{-5}$ |

$0.5$ | $3.844979\times {10}^{-5}$ | $3.844978\times {10}^{-5}$ | $3.657580\times {10}^{-5}$ | $3.657494\times {10}^{-5}$ | |

$0.9$ | $1.384324\times {10}^{-5}$ | $1.384324\times {10}^{-5}$ | $1.316850\times {10}^{-5}$ | $1.316815\times {10}^{-5}$ |

**Table 17.**${L}_{1}$ and ${L}_{\infty}$ error norms with CPU time taken for the four numerical methods.

t | Schemes | ${\mathit{L}}_{1}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | ${\mathit{L}}_{\infty}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | CPU Time (Sec) |
---|---|---|---|---|

1 | NSFD1 | $1.269362\times {10}^{-8}$ | $1.923270\times {10}^{-8}$ | $0.0654$ |

NSFD2 | $1.269362\times {10}^{-8}$ | $1.923269\times {10}^{-8}$ | $0.0677$ | |

EEFDM | $1.207528\times {10}^{-8}$ | $1.829585\times {10}^{-8}$ | $0.0677$ | |

FIEFDM | $1.207467\times {10}^{-8}$ | $1.829491\times {10}^{-8}$ | $0.0688$ | |

10 | NSFD1 | $1.269451\times {10}^{-8}$ | $1.923411\times {10}^{-8}$ | $0.2381$ |

NSFD2 | $1.269451\times {10}^{-8}$ | $1.923411\times {10}^{-8}$ | $0.2354$ | |

EEFDM | $1.207580\times {10}^{-8}$ | $1.829667\times {10}^{-8}$ | $0.2242$ | |

FIEFDM | $1.207551\times {10}^{-8}$ | $1.829624\times {10}^{-8}$ | $0.2372$ |

t | x | Exact | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|---|

1 | $0.1$ | $2.636642\times {10}^{-1}$ | $2.619319\times {10}^{-1}$ | $2.619323\times {10}^{-1}$ | $2.620016\times {10}^{-1}$ | $2.620423\times {10}^{-1}$ |

$0.5$ | $2.733691\times {10}^{-1}$ | $2.684776\times {10}^{-1}$ | $2.684788\times {10}^{-1}$ | $2.686755\times {10}^{-1}$ | $2.688005\times {10}^{-1}$ | |

$0.9$ | $2.830035\times {10}^{-1}$ | $2.812149\times {10}^{-1}$ | $2.812153\times {10}^{-1}$ | $2.812875\times {10}^{-1}$ | $2.813362\times {10}^{-1}$ | |

10 | $0.1$ | $3.573732\times {10}^{-1}$ | $3.559870\times {10}^{-1}$ | $3.559874\times {10}^{-1}$ | $3.560367\times {10}^{-1}$ | $3.560550\times {10}^{-1}$ |

$0.5$ | $3.651974\times {10}^{-1}$ | $3.612915\times {10}^{-1}$ | $3.612927\times {10}^{-1}$ | $3.614329\times {10}^{-1}$ | $3.614903\times {10}^{-1}$ | |

$0.9$ | $3.727452\times {10}^{-1}$ | $3.713231\times {10}^{-1}$ | $3.713236\times {10}^{-1}$ | $3.713749\times {10}^{-1}$ | $3.713974\times {10}^{-1}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $1.732294\times {10}^{-3}$ | $1.731870\times {10}^{-3}$ | $1.662559\times {10}^{-3}$ | $1.621813\times {10}^{-3}$ |

$0.5$ | $4.891452\times {10}^{-3}$ | $4.890247\times {10}^{-3}$ | $4.693599\times {10}^{-3}$ | $4.568532\times {10}^{-3}$ | |

$0.9$ | $1.788565\times {10}^{-3}$ | $1.788119\times {10}^{-3}$ | $1.715994\times {10}^{-3}$ | $1.667276\times {10}^{-3}$ | |

10 | $0.1$ | $1.386199\times {10}^{-3}$ | $1.385769\times {10}^{-3}$ | $1.336481\times {10}^{-3}$ | $1.318209\times {10}^{-3}$ |

$0.5$ | $3.905879\times {10}^{-3}$ | $3.904660\times {10}^{-3}$ | $3.764468\times {10}^{-3}$ | $3.707117\times {10}^{-3}$ | |

$0.9$ | $1.422126\times {10}^{-3}$ | $1.421680\times {10}^{-3}$ | $1.370378\times {10}^{-3}$ | $1.347886\times {10}^{-3}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $6.570078\times {10}^{-3}$ | $6.568471\times {10}^{-3}$ | $6.305595\times {10}^{-3}$ | $6.151057\times {10}^{-3}$ |

$0.5$ | $1.789322\times {10}^{-2}$ | $1.788881\times {10}^{-2}$ | $1.716946\times {10}^{-2}$ | $1.671196\times {10}^{-2}$ | |

$0.9$ | $6.319940\times {10}^{-3}$ | $6.318365\times {10}^{-3}$ | $6.063510\times {10}^{-3}$ | $5.891364\times {10}^{-3}$ | |

10 | $0.1$ | $3.878856\times {10}^{-3}$ | $3.877653\times {10}^{-3}$ | $3.739734\times {10}^{-3}$ | $3.688607\times {10}^{-3}$ |

$0.5$ | $1.069525\times {10}^{-2}$ | $1.069191\times {10}^{-2}$ | $1.030804\times {10}^{-2}$ | $1.015099\times {10}^{-2}$ | |

$0.9$ | $3.815276\times {10}^{-3}$ | $3.814081\times {10}^{-3}$ | $3.676447\times {10}^{-3}$ | $3.616104\times {10}^{-3}$ |

t | Schemes | ${\mathit{L}}_{1}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | ${\mathit{L}}_{\infty}\phantom{\rule{3.33333pt}{0ex}}\mathbf{Error}$ | CPU Time (Sec) |
---|---|---|---|---|

1 | NSFD1 | $3.228089\times {10}^{-3}$ | $4.891452\times {10}^{-3}$ | $0.0621$ |

NSFD2 | $3.227293\times {10}^{-3}$ | $4.890247\times {10}^{-3}$ | $0.0637$ | |

EEFDM | $3.097547\times {10}^{-3}$ | $4.693599\times {10}^{-3}$ | $0.0659$ | |

FIEFDM | $3.014506\times {10}^{-3}$ | $4.568532\times {10}^{-3}$ | $0.0671$ | |

10 | NSFD1 | $2.576806\times {10}^{-3}$ | $3.905879\times {10}^{-3}$ | $0.2335$ |

NSFD2 | $2.576002\times {10}^{-3}$ | $3.904660\times {10}^{-3}$ | $0.2281$ | |

EEFDM | $2.483571\times {10}^{-3}$ | $3.764468\times {10}^{-3}$ | $0.2328$ | |

FIEFDM | $2.445540\times {10}^{-3}$ | $3.707117\times {10}^{-3}$ | $0.2871$ |

t | x | Exact | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|---|

1 | $0.1$ | $3.197787\times {10}^{-1}$ | $3.115395\times {10}^{-1}$ | $3.115521\times {10}^{-1}$ | $3.129688\times {10}^{-1}$ | $3.131647\times {10}^{-1}$ |

$0.5$ | $3.395893\times {10}^{-1}$ | $3.164245\times {10}^{-1}$ | $3.164602\times {10}^{-1}$ | $3.204963\times {10}^{-1}$ | $3.210964\times {10}^{-1}$ | |

$0.9$ | $3.581881\times {10}^{-1}$ | $3.498705\times {10}^{-1}$ | $3.498835\times {10}^{-1}$ | $3.513357\times {10}^{-1}$ | $3.515904\times {10}^{-1}$ | |

10 | $0.1$ | $4.976668\times {10}^{-1}$ | $4.974441\times {10}^{-1}$ | $4.974445\times {10}^{-1}$ | $4.974708\times {10}^{-1}$ | $4.974707\times {10}^{-1}$ |

$0.5$ | $4.979926\times {10}^{-1}$ | $4.975407\times {10}^{-1}$ | $4.975418\times {10}^{-1}$ | $4.976175\times {10}^{-1}$ | $4.976172\times {10}^{-1}$ | |

$0.9$ | $4.983164\times {10}^{-1}$ | $4.981586\times {10}^{-1}$ | $4.981589\times {10}^{-1}$ | $4.981856\times {10}^{-1}$ | $4.981855\times {10}^{-1}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $8.239162\times {10}^{-3}$ | $8.226605\times {10}^{-3}$ | $6.809932\times {10}^{-3}$ | $6.613999\times {10}^{-3}$ |

$0.5$ | $2.316489\times {10}^{-2}$ | $2.312918\times {10}^{-2}$ | $1.909310\times {10}^{-2}$ | $1.849292\times {10}^{-2}$ | |

$0.9$ | $8.317574\times {10}^{-3}$ | $8.304529\times {10}^{-3}$ | $6.852408\times {10}^{-3}$ | $6.597658\times {10}^{-3}$ | |

10 | $0.1$ | $1.626796\times {10}^{-4}$ | $1.622971\times {10}^{-4}$ | $1.360407\times {10}^{-4}$ | $1.361553\times {10}^{-4}$ |

$0.5$ | $4.518804\times {10}^{-4}$ | $4.507901\times {10}^{-4}$ | $3.751243\times {10}^{-4}$ | $3.754311\times {10}^{-4}$ | |

$0.9$ | $1.578207\times {10}^{-4}$ | $1.574438\times {10}^{-4}$ | $1.308204\times {10}^{-4}$ | $1.309152\times {10}^{-4}$ |

t | x | NSFD1 | NSFD2 | EEFDM | FIEFDM |
---|---|---|---|---|---|

1 | $0.1$ | $2.576520\times {10}^{-2}$ | $2.572593\times {10}^{-2}$ | $2.129576\times {10}^{-2}$ | $2.068305\times {10}^{-2}$ |

$0.5$ | $6.821441\times {10}^{-2}$ | $6.810926\times {10}^{-2}$ | $5.622408\times {10}^{-2}$ | $5.445674\times {10}^{-2}$ | |

$0.9$ | $2.322125\times {10}^{-2}$ | $2.318483\times {10}^{-2}$ | $1.913075\times {10}^{-2}$ | $1.841954\times {10}^{-2}$ | |

10 | $0.1$ | $3.269239\times {10}^{-4}$ | $3.261553\times {10}^{-4}$ | $2.733899\times {10}^{-4}$ |