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The problem of solving the linear diffusion equation by a method related to the Restrictive Pade Approximation (RPA) is considered. The advantage is that it has the exact value at certain r. This method will exhibit several advantages for example highly accurate, fast and with good results, etc. The absolutely error is still very small. The obtained results are compared with the exact solution and the other methods. The numerical results are in agreement with the exact solution.

In this paper, we apply a new implicit method of high accuracy and the number of linear systems which to be solved are smaller than that for many famous known implicit methods of small step length. Therefore, our required machine time is less than that for the other implicit methods.

Restrictive Pade Approximation (RPA) for parabolic Partial Differential Equation (PDE) and Partial Difference Equations is a new technique done by İsmail and Elbarbary [

In this work, we consider the following one dimensional diffusion equation;

Subject to the initial condition

and boundary conditions

The functions

The Restrictive Pade Approximation (RPA) of function

where the positive integer α doesn’t exceed the degree of the numerator N,

Let

From Equations (6)-(8) we get, [

The varishing of the first

Hence we can determine the coefficient,

Note:

The local truncation error form the RPA can be summarized by the following theorem.

THEOREM: If the function

where

The exponential matrix exp(rA) can be formally defined by the convergent power series,

where A is

In the case of Restrictive Pade Approximation of single function the term

for example,

We consider the diffussion for

Subject to the initial condition;

and boundary conditions;

The exact solution of the Equation (16) is given by:

Condiser the diffusion Equation (16) with the initial and boundary condition. The open rectangular domain is covered by a rectangular grid with spacing h and k in the x and t directions respectively. The grid point

The exact solution of grid representation of (6) is given by [

The approximation of the partical derivatives

and according to central finite difference for mulation

The result of making this approximation is to replace (20) by the following equation

where

We use the

Equation to approximate the exponential matrix in Equation (23), then the approximate solution of grid representation of Equation (16) can take the form.

The constrat matrix

Other implicit methods can be derived if we use the possible Restrictive Pade Approximation RPA [M/N], with non-negative integers M and N.

The accuracy of Restrictive Pade Approximation method are compared in tables for various values of the time t. Tables give exact value, approximate value for compact finite difference method, approximate value for Restrictive Taylor Approximation, Restrictive Pade Approximation and absolute error for ε = 0.0032408523. Comparison of the RPA results with RTA method for k = 0.0001, N = 6, r = 0.0036 given below in tables.

The absolute error (AE) is give by the following formula:

We tabulated all AE values at

t | x | Exact | RPA | RTA | AE [Present] |
---|---|---|---|---|---|

0.01 | 1/6 | 0.162611 | 0.162611 | 0.162611 | 1.02E−10 |

2/6 | 0.320715 | 0.320715 | 0.320715 | 1.01E−10 | |

3/6 | 0.469932 | 0.469932 | 0.469932 | 1.01E−11 | |

4/6 | 0.606125 | 0.606125 | 0.606125 | 2.06E−11 | |

5/6 | 0.725520 | 0.725520 | 0.725520 | 1.12E−10 | |

1 | 0.824808 | 0.824808 | 0.824808 | 1.15E−10 |

t | x | Exact | RPA | RTA | AE [Present] |
---|---|---|---|---|---|

0.1 | 1/6 | 0.135824 | 0.135824 | 0.135824 | 1.26E−9 |

2/6 | 0.267884 | 0.267884 | 0.267884 | 1.16E−9 | |

3/6 | 0.392520 | 0.392520 | 0.392520 | 2.01E−11 | |

4/6 | 0.506278 | 0.506278 | 0.506278 | 1.25E−10 | |

5/6 | 0.606005 | 0.606005 | 0.606005 | 1.21E−9 | |

1 | 0.688938 | 0.688938 | 0.688938 | 1.66E−9 |

t | x | Exact | RPA | RTA | AE [Present] |
---|---|---|---|---|---|

1.00 | 1/6 | 0.022451 | 0.022451 | 0.022451 | 2.54E−10 |

2/6 | 0.044280 | 0.044280 | 0.044280 | 2.12E−10 | |

3/6 | 0.064883 | 0.064883 | 0.064883 | 4.25E−12 | |

4/6 | 0.083687 | 0.083687 | 0.083687 | 1.92E−11 | |

5/6 | 0.100172 | 0.100172 | 0.100172 | 2.04E−10 | |

1 | 0.113880 | 0.113880 | 0.113880 | 2.52E−10 |

In this article, a numerical algorithm was applied in the one dimensional diffusion equation. Computed results were compared with other paper results in

Boz, A. and Gülsever, F. (2016) Numerical Solution of the Diffusion Equation with Restrictive Pade Approximation. Journal of Applied Mathematics and Physics, 4, 2031-2037. http://dx.doi.org/10.4236/jamp.2016.411202