scholarly journals Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

2021 ◽  
Vol 13 (2) ◽  
pp. 57-71
Author(s):  
Wakjira Tolassa Gobena ◽  
Gemechis File Duressa
Keyword(s):  
Boundary Condition ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Parabolic Differential Equations ◽  
Parabolic Boundary ◽  
Delay Term ◽  
Local Boundary ◽  
Backward Euler ◽  
Local Boundary Condition ◽  
Non Local

The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

scholarly journals Accelerated fitted operator finite difference method for singularly perturbed delay differential equations with non-local boundary condition

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Habtamu Garoma Debela ◽  
Gemechis File Duressa
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Singularly Perturbed ◽  
Local Boundary ◽  
Local Boundary Condition ◽  
Numerical Experimentation ◽  
Non Local ◽  
Delay Differential

Abstract In this paper, accelerated fitted finite difference method for solving singularly perturbed delay differential equation with non-local boundary condition is considered. To treat the non-local boundary condition, Simpson’s rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter ε and mesh size h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result, and it also improves the results of the methods existing in the literature.

A METHOD FOR THE TREATMENT OF A SLOPING SEA BOTTOM IN THE PARABOLIC APPROXIMATION

1996 ◽  
Vol 04 (01) ◽  
pp. 89-100 ◽  
Author(s):  
J. S. PAPADAKIS ◽  
B. PELLONI
Keyword(s):  
Boundary Condition ◽  
Helmholtz Equation ◽  
Pressure Field ◽  
Impedance Boundary Condition ◽  
Parabolic Approximation ◽  
Impedance Boundary ◽  
Local Boundary ◽  
Sea Bottom ◽  
Local Boundary Condition ◽  
Non Local

The impedance boundary condition for the parabolic approximation is derived in the case of a sea bottom profile sloping at a constant angle, as a non-local boundary condition imposed exactly at the interface. This condition is integrated into the IFD code for the numerical computation of the pressure field and implemented to test its accuracy in some benchmark cases, for which the backscattered field is negligible. It is shown that by avoiding the sloping interface, the results obtained are closer to the benchmark results given by normal mode codes solving the full Helmholtz equation, such as the 2-way COUPLE code, than those of the standard IFD or other 1-way codes, at least for problems that do not have significant backscattering effects.

scholarly journals 2nd Order Parallel Splitting Methods for Heat Equation

2017 ◽  
Vol 1 (1) ◽  
pp. 01-10
Author(s):  
M. Aziz ◽  
M. A. Rehman
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Two Dimensions ◽  
Splitting Technique ◽  
Local Boundary ◽  
Local Boundary Condition ◽  
Non Local ◽  
Parallel Splitting ◽  
Matrix Exponential Function ◽  
Order Parallel

In this paper, heat equation in two dimensions with non local boundary condition is solved numerically by 2nd order parallel splitting technique. This technique used to approximate spatial derivative and a matrix exponential function is replaced by a rational approximation. Simpson’s 1/3 rule is also used to approximate the non local boundary condition. The results of numerical experiments are checked and compared with the exact solution, as well as with the results already existed in the literature and found to be highly accurate.

scholarly journals Finite difference approximations for a class of non-local parabolic equations

1997 ◽  
Vol 20 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Yanping Lin ◽  
Shuzhan Xu ◽  
Hong-Ming Yin
Keyword(s):  
Finite Difference ◽  
Parabolic Equations ◽  
The Maximum Principle ◽  
Local Boundary ◽  
Finite Difference Approximations ◽  
Fully Implicit ◽  
Difference Approximations ◽  
Backward Euler ◽  
Local Boundary Condition ◽  
Non Local

In this paper we study finite difference procedures for a class of parabolic equations with non-local boundary condition. The semi-implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully-implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero ast→∞exponentially. The numerical results of some examples are presented, which support our theoretical justifications.

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