Siddiqi and Muzammal Iftikhar. This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The University of Auckland Rights: The author Rights URI: The coefficients in a mathematical model of a physical problem typically correspond to the material parameters of the problem.
The governing equation of a physical problem in heterogeneous media is therefore likely to involve variable coefficients. For this reason the solution of variable coefficient partial differential equations PDEs is an important engineering problem.
In this thesis ways of solving linear variable coefficient PDEs using the boundary element method have been investigated. The application of the boundary element method to these equations is hampered by the difficulty of finding a fundamental solution.
In the literature several methods have been proposed to overcome this problem. A survey of these methods has been undertaken in this study from which it is concluded that the most promising approach is the dual reciprocity boundary element method DR-BEM.
However, in some cases, such as problems in highly heterogeneous media, it is found that a large number of internal solution nodes are necessary to ensure accurate results. For problems in highly heterogeneous media a subregion approach is recommended. It is found that by combining the DR-BEM with the coupled finite difference — boundary element method a wide range of parabolic problems can be solved without requiring domain integration.
This time-stepping approach can become expensive for variable coefficient PDEs particularly for large-time solutions as it requires the solution of a large number of associated elliptic problems with large numbers of internal nodes.
Also, it is found that for some problems in highly heterogeneous media the error at each time-step can accumulate leading to poor large-time solutions.
To avoid these limitations semi-analytic approaches for solving parabolic equations are investigated. A new semi-analytic method - the separation of variables dual reciprocity method SOV-DRM - is proposed which constructs the solution as an eigenfunction expansion.
This method allows parabolic problems to be solved without requiring time-stepping or domain integration.
This method is found to produce accurate results for a range of problems including some problems involving heterogeneous media. Two other semi-analytic methods are also investigated. It is concluded that each method has specific strengths and weaknesses and that the choice of method is largely problem dependent.Abstract Lie symmetry method is applied to find analytic solutions of initial-boundary-value problems of transient conduction in semi-infinite solid with constant surface .
This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions.
In this paper, we investigate the floor semi-periodic boundary value problem for a system of hyperbolic equations with mixed derivative. An algorithm is implemented for finding an approximate solution of this problem. This article is an example, where the algorithm is implemented.
Finite-difference time-domain or Yee's method (named after the Chinese American applied mathematician Kane S. Yee, born ) is a numerical analysis technique used for modeling computational electrodynamics (finding approximate solutions to the associated system of differential equations).Since it is a time-domain method, FDTD .
A method for turning data into value Data science is a method for transforming business data into assets that help organizations improve revenue, reduce costs, seize business opportunities. A new semi-analytic solution method is proposed for solving linear parabolic problems using the boundary element method.
This method constructs a solution as an .