# An Efficient Local Formulation for Time–Dependent PDEs

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Partial Differential Equation Models

## 3. Local Meshless Numerical Scheme

#### 3.1. 1D Fifth Order Kortewege-de Vries Equation

## 4. Results and Discussion

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

**Problem**

**5.**

**Problem**

**6.**

**Problem**

**7.**

**Problem**

**8.**

**Problem**

**9.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

^{th}Anniversary Commemorative Fund”.

## Conflicts of Interest

## References

- Yamaoto, Y.; Takizawa, E. On a Solution on Non-Linear Time-Evolution Equation of Fifth-Order. J. Phys. Soc. Jpn.
**1981**, 50, 1421–1422. [Google Scholar] [CrossRef] - Djidjeli, K.; Price, W.G.; Twizell, E.H.; Wang, Y. Numerical Methods for the Soltution of the Third and Fifth-Order Disprsive Korteweg-de Vries Equations. J. Comput. Appl. Math.
**1995**, 58, 307–336. [Google Scholar] [CrossRef] - Bakodah, H.O. Modified Adomain Decomposition Method for the Generalized Fifth Order KdV Equations. Am. J. Comput. Math.
**2013**, 3, 53–58. [Google Scholar] [CrossRef] - Kaya, D. An Explicit and Numerical Solutions of Some Fifth-Order KdV Equation by Decomposition Method. Appl. Math. Comput.
**2003**, 144, 353–363. [Google Scholar] [CrossRef] - Goswami, A.; Singh, J.; Kumar, D. Numerical simulation of fifth order KdV equations occurring in magneto- acoustic waves. Ain Shams Eng. J.
**2017**, 9, 2265–2273. [Google Scholar] [CrossRef] - Helal, M.A.; Mehanna, M.S. A Comparative Study between Two Different Methods for Solving the General Korteweg-de Vries Equation (GKDV). Chaos Solitons Fractals
**2007**, 33, 725–739. [Google Scholar] [CrossRef] - Pomeau, Y.; Ramani, A.; Grammaticos, B. Structural stability of the Korteweg-de Vries solitons under a singular perturbation. Phys. D
**1988**, 31, 127–134. [Google Scholar] [CrossRef] - Soliman, A.A. A numerical simulation and explicit solutions of KdVBurgers’ and Lax’s seventh-order KdV equations. Chaos Solitons Fractals
**2006**, 29, 294–302. [Google Scholar] [CrossRef] - Darvishia, M.T.; Kheybari, S.; Khani, F. A Numerical Solution of the Lax’s 7th-order KdV Equation by Pseudospectral Method and Darvishi’s Preconditioning. Int. J. Contemp. Math. Sci.
**2007**, 2, 1097–1106. [Google Scholar] [CrossRef] - El-Sayed, S.M.; Kaya, D. An application of the ADM to seven-order Sawada-Kotara equations. Appl. Math. Comput.
**2004**, 157, 93–104. [Google Scholar] [CrossRef] - Satsuma, J. Topics in Soliton Theory and Exactly Solvable Nonlinear Equations; World Scientific: Singapore, 1987. [Google Scholar]
- Al-Rozbayani, A.M.; Al-Amr, M.O. Discrete Adomian Decomposition Method for Solving Burgers’-Huxley Equation. Int. J. Contemp. Math. Sci.
**2013**, 8, 623–631. [Google Scholar] [CrossRef] - Celik, I. Haar wavelet method for solving generalized Burgers’-Huxley equation. Arab J. Math. Sci.
**2012**, 18, 25–37. [Google Scholar] - Ismail, H.N.A.; Raslan, K.R.; Rabboh, A.A.A. Adomian decomposition method for Burgers’-Huxley and Burgers’-Fisher equations. Appl. Math. Comput.
**2004**, 159, 291–301. [Google Scholar] - Khattak, A.J. A computational meshless method for the generalized Burgers’-Huxley equation. Appl. Math. Model.
**2009**, 33, 3718–3729. [Google Scholar] [CrossRef] - Bukhari, M.; Arshad, M.; Batool, S.; Saqlain, S.M. Numerical solution of generalized Burger’s-Huxley equation using local radial basis functions. Int. J. Adv. Appl. Sci.
**2017**, 4, 1–11. [Google Scholar] [CrossRef] [Green Version] - Hodgkin, A.L.; Huxley, A.F. A Quantitative Description of Ion Currents and Its Applications to Conduction and Excitation in Nerve Membranes. J. Physiol.
**1952**, 117, 500–544. [Google Scholar] [CrossRef] [PubMed] - Babolian, E.; Saeidian, J. Analytic approximate solutions to Burgers’, Fisher, Huxley equations and two combined forms of these equations. Commun. Nonlinear Sci. Numerical Simul.
**2009**, 14, 1984–1992. [Google Scholar] [CrossRef] - Ashrafi, S.; Alineia, M.; Kheiri, H.; Hojjati, G. Spectral Collocation Method for the Numerical Solution of the Gardner and Huxley Equations. Int. J. Nonlinear Sci.
**2014**, 18, 71–77. [Google Scholar] - Fisher, R.A. The Wave of Advance of Advantageous Genes. Ann. Eugenics
**1937**, 7, 355–369. [Google Scholar] [CrossRef] - Moghimia, M.; Hejazi, F.S.A. Variational iteration method for solving generalized Burgers’-Fisher and Burgers’ equations. Chaos Solitons Fractals
**2007**, 33, 1756–1761. [Google Scholar] [CrossRef] - Javidi, M. Modified pseudospectral method for generalized Burgers’-Fisher equation. Int. Math. Forum
**2006**, 32, 1555–1564. [Google Scholar] [CrossRef] - Mittal, R.C.; Tripathi, A. Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines. Int. J. Comput. Math.
**2015**, 92, 1053–1077. [Google Scholar] [CrossRef] - Bhrawy, A.H. A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients. Appl. Math. Comput.
**2013**, 222, 255–264. [Google Scholar] [CrossRef] - Abbasbandy, S. Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method. Appl. Math. Model.
**2008**, 32, 2706–2714. [Google Scholar] [CrossRef] - Hariharan, G.; Kannan, K. Haar wavelet method for solving FitzHugh-Nagumo equation. World Acad. Sci.
**2010**, 43, 560–564. [Google Scholar] - Jiwari, R.; Gupta, R.K.; Kumar, V. Polynomial differential quadrature method for numerical solutions of the generalized Fitzhugh-Nagumo equation with time-dependent coefficients. Ain Shams Eng. J.
**2014**, 5, 1343–1350. [Google Scholar] [CrossRef] - Hirota, R.; Satsuma, J. Soliton solutions of a coupled Kortewege-de Vries equation. Phys. Lett. A
**1981**, 85, 407–418. [Google Scholar] [CrossRef] - Haq, S.; Uddin, M. A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations. Eng. Anal. Boundary Elements
**2009**, 33, 399–409. [Google Scholar] - Haq, S.; Hussain, A.; Uddin, M. RBFs Meshless Method of Lines for the Numerical Solution of Time- Dependent Nonlinear Coupled Partial Differential Equations. Appl. Math.
**2011**, 2, 414–423. [Google Scholar] [CrossRef] - Assas, L.M.B. Variational Iteration Method for Solving Coupled-KdV Equations. Chaos Solitons Fractals
**2008**, 38, 1225–1228. [Google Scholar] [CrossRef] - Khater, A.H.; Temsah, R.S.; Callebaut, D.K. Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method. Math. Comput. Model.
**2008**, 48, 1237–1253. [Google Scholar] [CrossRef] - Wu, Y.T.; Geng, X.G.; Hu, X.B.; Zhu, S.M. Generalized Hirota-Satsuma coupled Korteweg-de Vries equation and Miura transformations. Phys. Lett. A
**1999**, 64, 255–259. [Google Scholar] [CrossRef] - Uddin, M.; Haq, S. Application of a Numerical Method Using Radial Basis Functions to Nonlinear Partial Differential Equations. Selcuk J. Appl. Math.
**2011**, 12, 77–93. [Google Scholar] - Ganji, D.D.; Nourollahi, M.; Rostamian, M. A Comparison of Variational Iteration Method with Adomian’s Decomposition Method in Some Highly Nonlinear Equations. Int. J. Sci. Technol.
**2007**, 2, 179–188. [Google Scholar] - Lefever, R.; Nicolis, G. Chemical instabilities and sustained oscillations. J. Theor. Biol.
**1971**, 30, 267–284. [Google Scholar] [CrossRef] - Tyson, J. Some further studies of nonlinear oscillations in chemical systems. J. Chem. Phys.
**1973**, 58, 3919. [Google Scholar] [CrossRef] - Ali, A.; Haq, S. A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system. Appl. Math. Model.
**2010**, 34, 3896–3909. [Google Scholar] - Wazwaz, A.M. The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. Math. Comput.
**2000**, 110, 251–264. [Google Scholar] [CrossRef] - Adomian, G. The diffusion Brusselator equation. Comput. Math. Appl.
**1995**, 29, 1–3. [Google Scholar] [CrossRef] - Whye-Teong, A. The two-dimensional reaction-diffusion Brusselator system: a dual-reciprocity boundary element solution. Eng. Anal. Boundary Elements
**2003**, 27, 897–903. [Google Scholar] - Thounthong, P.; Khan, M.N.; Hussain, I.; Ahmad, I.; Kumam, P. Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations. Mathematics
**2018**, 6, 327. [Google Scholar] [CrossRef] - Ahmad, I. Local meshless method for PDEs arising from models of wound healing. Appl. Math. Model.
**2017**, 48, 688–710. [Google Scholar] - Ahmad, I.; Riaz, M.; Ayaz, M.; Arif, M.; Islam, S.; Kumam, P. Numerical Simulation of Partial Differential Equations via Local Meshless Method. Symmetry
**2019**, 11, 257. [Google Scholar] [CrossRef] - Aziz, I. Meshless methods for multivariate highly oscillatory Fredholm integral equations. Eng. Anal. Boundary Elements
**2015**, 53, 100–112. [Google Scholar] - Din, Z.-u. Meshless methods for one-dimensional oscillatory Fredholm integral equations. Appl. Math. Comput.
**2018**, 324, 156–173. [Google Scholar] - Din, Z.-u. Meshless methods for two-dimensional oscillatory Fredholm integral equations. J. Comput. Appl. Math.
**2018**, 335, 33–50. [Google Scholar] - Ahmad, I.; Khaliq, A.Q.M. Local RBF method for multi-dimensional partial differential equations. Comput. Math. Appl.
**2017**, 74, 292–324. [Google Scholar] [CrossRef] - FitzHugh, R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J.
**1961**, 1, 445–466. [Google Scholar] [CrossRef] - Nagumo, J.S.; Arimoto, S.; Yoshizawa, S. An active pulse transmission line simulating nerve axon. Proc. IRE
**1962**, 50, 2061–2070. [Google Scholar] [CrossRef] - Ahmad, I. A comparative analysis of local meshless formulation for multi-asset option models. Eng. Anal. Boundary Elements
**2016**, 65, 159–176. [Google Scholar] - Wang, X.Y.; Zhu, Z.S.; Lu, Y.K. Solitary wave solutions of the generalized Burgers’-Huxley equation. J. Phys. A Math. Theor.
**1990**, 23, 271–274. [Google Scholar]

**Figure 1.**Comparing the curves of the numerical and analytical solutions for $N=41$, $q=4$ for Test Problem 6.

**Figure 2.**(

**a**) Numerical solutions for $q=0.75$; (

**b**) Numerical solutions for $q=4$ using the EEM for $N=41$, $t=1$ for Test Problem 6.

**Figure 3.**Computational domain (

**a**), numerical solution (

**b**) and absolute error (

**c**) by using the EEM for Test Problem 9.

**Figure 4.**Computational domain (

**a**), numerical solution (

**b**) and absolute error (

**c**) by using the EEM for Test Problem 9.

**Figure 5.**Computational domain (

**a**), numerical solution (

**b**) and absolute error (

**c**) by using the EEM for Test Problem 9.

**Figure 6.**Computational domain (

**a**), numerical solution (

**b**) and absolute error (

**c**) by using the EEM for Test Problem 9.

**Table 1.**Comparisons between the results obtained by the LMM with those adapted from [3] for Test Problem 1.

x = 0.2 | x = 1 | x = 5 | ||||
---|---|---|---|---|---|---|

$\mathit{t}$ | EEM | [3] | EEM | [3] | EEM | [3] |

2 | 5.2899 × ${10}^{-15}$ | 5.7619 × ${10}^{-14}$ | 2.6450 × ${10}^{-14}$ | 2.8786 × ${10}^{-13}$ | 1.2669 × ${10}^{-13}$ | 1.4210 × ${10}^{-12}$ |

4 | 1.0580 × ${10}^{-14}$ | 1.1528 × ${10}^{-13}$ | 5.2900 × ${10}^{-14}$ | 5.7577 × ${10}^{-13}$ | 2.5337 × ${10}^{-13}$ | 2.8421 × ${10}^{-12}$ |

6 | 1.5869 × ${10}^{-14}$ | 1.7298 × ${10}^{-13}$ | 7.9350 × ${10}^{-14}$ | 8.6372 × ${10}^{-13}$ | 3.8006 × ${10}^{-13}$ | 4.2632 × ${10}^{-12}$ |

8 | 2.1159 × ${10}^{-14}$ | 2.3073 × ${10}^{-13}$ | 1.0580 × ${10}^{-13}$ | 1.1517 × ${10}^{-12}$ | 5.0676 × ${10}^{-13}$ | 5.6844 × ${10}^{-12}$ |

10 | 2.6448 × ${10}^{-14}$ | 2.8851 × ${10}^{-13}$ | 1.3225 × ${10}^{-13}$ | 1.4397 × ${10}^{-12}$ | 6.3345 × ${10}^{-13}$ | 7.1056 × ${10}^{-12}$ |

t | k = 0.1 | k = 0.01 | k = 0.001 | CPU Time |
---|---|---|---|---|

1 | 5.8157 × ${10}^{-10}$ | 1.2395 × ${10}^{-15}$ | 3.3881 × ${10}^{-19}$ | 0.03 |

10 | 5.8205 × ${10}^{-9}$ | 1.2395 × ${10}^{-14}$ | 3.3881 × ${10}^{-18}$ | 0.21 |

20 | 1.1652 × ${10}^{-8}$ | 2.4790 × ${10}^{-14}$ | 6.7763 × ${10}^{-18}$ | 0.46 |

30 | 1.7494 × ${10}^{-8}$ | 3.7185 × ${10}^{-14}$ | 1.0164 × ${10}^{-17}$ | 0.71 |

50 | 2.9211 × ${10}^{-8}$ | 6.1975 × ${10}^{-14}$ | 1.6941 × ${10}^{-17}$ | 1.07 |

t | x | RK4 | [15] | [14] | [12] |
---|---|---|---|---|---|

0.05 | 0.1 | 6.30 × ${10}^{-12}$ | 1.0 × ${10}^{-9}$ | 1.93 × ${10}^{-7}$ | 1.87 × ${10}^{-8}$ |

0.5 | 4.42 × ${10}^{-12}$ | 1.0 × ${10}^{-9}$ | 1.93 × ${10}^{-7}$ | 1.87 × ${10}^{-8}$ | |

0.9 | 2.55 × ${10}^{-12}$ | 1.0 × ${10}^{-9}$ | 1.93 × ${10}^{-7}$ | 1.87 × ${10}^{-8}$ | |

0.1 | 0.1 | 1.23 × ${10}^{-11}$ | 1.0 × ${10}^{-9}$ | 3.87 × ${10}^{-7}$ | 3.75 × ${10}^{-8}$ |

0.5 | 8.62 × ${10}^{-12}$ | 1.0 × ${10}^{-9}$ | 3.87 × ${10}^{-7}$ | 3.75 × ${10}^{-8}$ | |

0.9 | 4.87 × ${10}^{-12}$ | 1.0 × ${10}^{-9}$ | 3.87 × ${10}^{-7}$ | 3.75 × ${10}^{-8}$ | |

1.0 | 0.1 | 8.16 × ${10}^{-11}$ | 0.0 × ${10}^{-9}$ | 3.88 × ${10}^{-6}$ | 3.75 × ${10}^{-7}$ |

0.5 | 4.41 × ${10}^{-11}$ | 0.0 × ${10}^{-9}$ | 3.88 × ${10}^{-6}$ | 3.75 × ${10}^{-7}$ | |

0.9 | 6.61 × ${10}^{-12}$ | 0.0 × ${10}^{-9}$ | 3.88 × ${10}^{-6}$ | 3.75 × ${10}^{-7}$ |

**Table 4.**Comparisons between the results obtained by the LMM with those adapted from [13] in term of ${L}_{abs}$ error norm for Test Problem 3.

$\mathit{\delta}\mathbf{=}\mathbf{1}$ | $\mathit{\delta}\mathbf{=}\mathbf{1}$ | $\mathit{\delta}\mathbf{=}\mathbf{2}$ | $\mathit{\delta}\mathbf{=}\mathbf{2}$ | |
---|---|---|---|---|

x | RK4 | [13] | RK4 | [13] |

0.15625 | 6.8575 × ${10}^{-11}$ | 2.4648 × ${10}^{-8}$ | 5.9052 × ${10}^{-7}$ | 1.1465 × ${10}^{-6}$ |

0.28125 | 5.9200 × ${10}^{-11}$ | 3.7832 × ${10}^{-8}$ | 5.8977 × ${10}^{-7}$ | 1.7600 × ${10}^{-6}$ |

0.34375 | 5.4512 × ${10}^{-11}$ | 4.2226 × ${10}^{-8}$ | 5.8940 × ${10}^{-7}$ | 1.9644 × ${10}^{-6}$ |

0.46875 | 4.5137 × ${10}^{-11}$ | 4.6621 × ${10}^{-8}$ | 5.8866 × ${10}^{-7}$ | 2.1749 × ${10}^{-6}$ |

0.53125 | 4.0448 × ${10}^{-11}$ | 4.6622 × ${10}^{-8}$ | 5.8828 × ${10}^{-7}$ | 2.1749 × ${10}^{-6}$ |

0.65625 | 3.1070 × ${10}^{-11}$ | 4.2228 × ${10}^{-8}$ | 5.8754 × ${10}^{-7}$ | 1.9643 × ${10}^{-6}$ |

0.71875 | 2.6382 × ${10}^{-11}$ | 3.7834 × ${10}^{-8}$ | 5.8717 × ${10}^{-7}$ | 1.7603 × ${10}^{-6}$ |

0.84375 | 1.7004 × ${10}^{-11}$ | 2.4650 × ${10}^{-8}$ | 5.8642 × ${10}^{-7}$ | 1.1462 × ${10}^{-6}$ |

0.96875 | 7.6260 × ${10}^{-12}$ | 5.5962 × ${10}^{-9}$ | 5.8568 × ${10}^{-7}$ | 2.6037 × ${10}^{-7}$ |

t | x | RK4 | [15] | [14] |
---|---|---|---|---|

0.1 | 2.18 × ${10}^{-11}$ | 0.0 × ${10}^{-9}$ | 1.88 × ${10}^{-7}$ | |

0.05 | 0.5 | 1.83 × ${10}^{-11}$ | 1.0 × ${10}^{-9}$ | 1.87 × ${10}^{-7}$ |

0.9 | 1.47 × ${10}^{-11}$ | 1.0 × ${10}^{-9}$ | 1.87 × ${10}^{-7}$ | |

0.1 | 4.29 × ${10}^{-11}$ | 1.0 × ${10}^{-9}$ | 3.75 × ${10}^{-7}$ | |

0.1 | 0.5 | 3.59 × ${10}^{-11}$ | 0.0 × ${10}^{-9}$ | 3.75 × ${10}^{-7}$ |

0.9 | 2.88 × ${10}^{-11}$ | 0.0 × ${10}^{-9}$ | 3.75 × ${10}^{-7}$ | |

0.1 | 3.18 × ${10}^{-10}$ | 1.0 × ${10}^{-9}$ | 3.75 × ${10}^{-6}$ | |

1.0 | 0.5 | 2.47 × ${10}^{-10}$ | 0.0 × ${10}^{-9}$ | 3.75 × ${10}^{-6}$ |

0.9 | 1.76 × ${10}^{-10}$ | 1.0 × ${10}^{-9}$ | 3.75 × ${10}^{-6}$ |

**Table 6.**Comparisons between the results obtained by the LMM with those adapted from [19] in term of ${L}_{abs}$ error norm for Test Problem 4.

a | b | t | x | EEM | [19] |
---|---|---|---|---|---|

−2 | 2 | 0.002 | 0.050 | 1.9974 × ${10}^{-7}$ | 2.22 × ${10}^{-3}$ |

−2 | 2 | 0.100 | 0.700 | 3.0067 × ${10}^{-6}$ | 1.78 × ${10}^{-3}$ |

−5 | 5 | 0.001 | 0.500 | 2.5260 × ${10}^{-7}$ | 1.69 × ${10}^{-2}$ |

−5 | 5 | 0.001 | 2.500 | 1.7843 × ${10}^{-6}$ | 9.90 × ${10}^{-3}$ |

−10 | 10 | 0.002 | 0.010 | 3.6673 × ${10}^{-6}$ | 4.05 × ${10}^{-4}$ |

−10 | 10 | 0.100 | 1.000 | 2.7722 × ${10}^{-4}$ | 3.17 × ${10}^{-3}$ |

t | x | RK4 | [15] | [14] |
---|---|---|---|---|

0.1 | 3.2492 × ${10}^{-8}$ | 2.7 × ${10}^{-7}$ | 9.75 × ${10}^{-6}$ | |

0.005 | 0.5 | 3.2495 × ${10}^{-8}$ | 1.4 × ${10}^{-7}$ | 5.96 × ${10}^{-5}$ |

0.9 | 3.2498 × ${10}^{-8}$ | 2.7 × ${10}^{-7}$ | 9.75 × ${10}^{-6}$ | |

0.1 | 2.4835 × ${10}^{-9}$ | 2.7 × ${10}^{-7}$ | 1.90 × ${10}^{-5}$ | |

0.01 | 0.5 | 2.4895 × ${10}^{-9}$ | 1.3 × ${10}^{-7}$ | 1.90 × ${10}^{-5}$ |

0.9 | 2.4955 × ${10}^{-9}$ | 2.7 × ${10}^{-7}$ | 1.90 × ${10}^{-5}$ |

**Table 8.**Comparisons between the results obtained by the LMM with those adapted from [27] for Test Problem 6.

t | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{\mathit{rms}}$ | ${\mathit{L}}_{\mathit{\infty}}$ | ${\mathit{L}}_{\mathit{rms}}$ |
---|---|---|---|---|

EEM | EEM | [27] | [27] | |

0.2 | 1.8896 × ${10}^{-5}$ | 2.1960 × ${10}^{-7}$ | 4.7416 × ${10}^{-5}$ | 1.5880 × ${10}^{-5}$ |

0.5 | 4.1554 × ${10}^{-5}$ | 1.5696 × ${10}^{-6}$ | 1.2312 × ${10}^{-4}$ | 3.8433 × ${10}^{-5}$ |

1.0 | 6.9891 × ${10}^{-5}$ | 7.1449 × ${10}^{-6}$ | 2.6261 × ${10}^{-4}$ | 8.1870 × ${10}^{-5}$ |

1.5 | 9.1687 × ${10}^{-5}$ | 1.7262 × ${10}^{-5}$ | 4.2096 × ${10}^{-4}$ | 1.3387 × ${10}^{-4}$ |

2.0 | 1.0969 × ${10}^{-4}$ | 3.1857 × ${10}^{-5}$ | 5.9999 × ${10}^{-4}$ | 1.9433 × ${10}^{-4}$ |

3.0 | 1.3942 × ${10}^{-4}$ | 7.2878 × ${10}^{-5}$ | 1.0324 × ${10}^{-3}$ | 3.4320 × ${10}^{-4}$ |

5.0 | 1.8964 × ${10}^{-4}$ | 1.8803 × ${10}^{-4}$ | 2.3050 × ${10}^{-3}$ | 7.8638 × ${10}^{-4}$ |

**Table 9.**Comparisons between the results obtained by the LMM with those adapted from [29] in term of ${L}_{2}$ error norm for Test Problem 7.

t | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 1 | 2 |
---|---|---|---|---|---|---|---|

[29] | |||||||

U | 3.662 × ${10}^{-5}$ | 5.177 × ${10}^{-5}$ | 4.065 × ${10}^{-5}$ | 3.951 × ${10}^{-5}$ | 3.964 × ${10}^{-5}$ | 3.975 × ${10}^{-5}$ | 4.461 × ${10}^{-5}$ |

V | 2.584 × ${10}^{-6}$ | 3.648 × ${10}^{-6}$ | 2.855 × ${10}^{-6}$ | 2.769 × ${10}^{-6}$ | 2.772 × ${10}^{-6}$ | 2.749 × ${10}^{-6}$ | 3.033 × ${10}^{-6}$ |

EEM | |||||||

U | 8.3147 × ${10}^{-8}$ | 3.3251 × ${10}^{-7}$ | 7.4809 × ${10}^{-7}$ | 1.3299 × ${10}^{-6}$ | 2.0779 × ${10}^{-6}$ | 8.3112 × ${10}^{-6}$ | 3.3245 × ${10}^{-5}$ |

V | 5.8794 × ${10}^{-9}$ | 2.3512 × ${10}^{-8}$ | 5.2898 × ${10}^{-8}$ | 9.4037 × ${10}^{-8}$ | 1.4693 × ${10}^{-7}$ | 5.8769 × ${10}^{-7}$ | 2.3508 × ${10}^{-6}$ |

k | t | U | V | W |
---|---|---|---|---|

0.1 | 0.25 | 4.9097 × ${10}^{-5}$ | 1.7702 × ${10}^{-8}$ | 1.7702 × ${10}^{-8}$ |

0.50 | 9.8195 × ${10}^{-5}$ | 3.6410 × ${10}^{-8}$ | 3.6410 × ${10}^{-8}$ | |

0.75 | 1.4729 × ${10}^{-4}$ | 5.6117 × ${10}^{-8}$ | 5.6117 × ${10}^{-8}$ | |

1.00 | 1.9639 × ${10}^{-4}$ | 7.6814 × ${10}^{-8}$ | 7.6814 × ${10}^{-8}$ | |

5.00 | 9.8258 × ${10}^{-4}$ | 6.0675 × ${10}^{-7}$ | 6.0675 × ${10}^{-7}$ | |

0.05 | 0.25 | 5.2245 × ${10}^{-6}$ | 2.2740 × ${10}^{-9}$ | 2.2740 × ${10}^{-9}$ |

0.50 | 1.0464 × ${10}^{-5}$ | 4.5679 × ${10}^{-9}$ | 4.5679 × ${10}^{-9}$ | |

0.75 | 1.5720 × ${10}^{-5}$ | 6.8820 × ${10}^{-9}$ | 6.8820 × ${10}^{-9}$ | |

1.00 | 2.0991 × ${10}^{-5}$ | 9.2162 × ${10}^{-9}$ | 9.2162 × ${10}^{-9}$ | |

5.00 | 1.0741 × ${10}^{-4}$ | 4.9335 × ${10}^{-8}$ | 4.9335 × ${10}^{-8}$ |

U | V | ||||||||
---|---|---|---|---|---|---|---|---|---|

t | CPU Time | EEM | [38] | [41] | Exact | EEM | [38] | [41] | Exact |

0.30 | 0.17 | 0.3167 | 0.3168 | 0.3166 | 0.3166 | 3.1584 | 3.158 | 3.157 | 3.158 |

0.60 | 0.19 | 0.2726 | 0.2724 | 0.2725 | 0.2725 | 3.6696 | 3.669 | 3.667 | 3.669 |

0.90 | 0.23 | 0.2346 | 0.2347 | 0.2345 | 0.2346 | 4.2635 | 4.263 | 4.260 | 4.263 |

1.20 | 0.26 | 0.2019 | 0.2020 | 0.2018 | 0.2019 | 4.9534 | 4.953 | 4.950 | 4.953 |

1.50 | 0.30 | 0.1738 | 0.1739 | 0.1737 | 0.1738 | 5.7551 | 5.755 | 5.751 | 5.755 |

1.80 | 0.34 | 0.1496 | 0.1496 | 0.1495 | 0.1496 | 6.6864 | 6.686 | 6.681 | 6.686 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ahmad, I.; Ahsan, M.; Din, Z.-u.; Masood, A.; Kumam, P.
An Efficient Local Formulation for Time–Dependent PDEs. *Mathematics* **2019**, *7*, 216.
https://doi.org/10.3390/math7030216

**AMA Style**

Ahmad I, Ahsan M, Din Z-u, Masood A, Kumam P.
An Efficient Local Formulation for Time–Dependent PDEs. *Mathematics*. 2019; 7(3):216.
https://doi.org/10.3390/math7030216

**Chicago/Turabian Style**

Ahmad, Imtiaz, Muhammad Ahsan, Zaheer-ud Din, Ahmad Masood, and Poom Kumam.
2019. "An Efficient Local Formulation for Time–Dependent PDEs" *Mathematics* 7, no. 3: 216.
https://doi.org/10.3390/math7030216