A new iterative computational scheme for solving second order (1 + 1) boundary value problems with non-homogeneous Dirichlet conditions

ABSTRACT

This paper is a result of an investigation to come up with a new hybrid scheme for solving second order $(1+1)$ boundary value problems of linear as well as nonlinear partial differential equations with non-homogeneous Dirichlet boundary conditions. Such an innovation is significant since there are not many analytical methods for solving partial differential equations with boundary data. The scheme involves the coupling of the ModDTM, a modified form of the Differential Transform Method, and the Adomian Decomposition Method. ModDTM is chosen because the differential transform used in the method is suitable to be applied to boundary value problems. It is the concept of decomposing the initial terms of the series-solution that is borrowed from the Decomposition Method. Solutions are obtained in the form of partial sums of their series representation. These are determined by the application of the mathematical software SageMath. The procedures of application are illustrated by solving linear and nonlinear versions of the classical equations, viz., heat, telegraph, Klein-Gordon and the Fishers equations. To validate the convergence of the solutions obtained with the exact ones two dimensional graphs plotted by SageMath software are used. This is performed in two stages. It is verified that successive partial sums of the series-solutions converge. As well as that, it is shown that the partial sum with the appropriate number of terms converges to the exact and closed form solution. This innovative method is found to be powerful, yet simple and uncomplicated, for solving Dirichlet boundary value problems of second order partial differential equations.

KEYWORDS:

• Boundary value problems
• non-homogeneous dirichlet boundary conditions
• linear and nonlinear partial differential equations
• homogeneous and non-homogeneous partial differential equations
• analytic methods
• Differential transform
• heat equation
• telegraph equation
• Klein-Gordon equation
• fishers equation
• convergence
• 2d-plots

Disclosure statement

No potential conflict of interest was reported by the author(s).

Funding

The authors declare that funding of any kind was not received for this research or for the publication of this paper.

Author details

This work was carried out in collaboration between both the authors as part of a study about various analytic methods for solving partial differential equations in general. It forms part of a project undertaken to find simple procedures for solving different types of boundary value problems in particular.

Data availability statement

No data were used to support the study.