Abstract
Several mathematical models that explain natural phenomena are mostly formulated in terms of nonlinear differential equations. Many problems in applied sciences such as nuclear physics, engineering, thermal management, gas dynamics, chemical reaction, studies of atomic structures and atomic calculations lead to singular boundary value problems and often only positive solutions are vital. However, most of the methods developed in mathematics are used in solving linear differential equations. For this reason, this research considered a model problem representing temperature distribution in heat dissipating fins with triangular profiles using MATLAB codes. MADM was used with a computer code in MATLAB to seek solution for the problem involving constant and a power law dependence of thermal conductivity on temperature governed by linear and nonlinear BVPs, respectively, for which considerable results were obtained. A problem formulated dealing with a triangular silicon fin and more examples were solved and analyzed using tables and figures for better elaborations where appreciable agreement between the approximate and exact solutions was observed. All the computations were performed using MATHEMATICA and MATLAB.
Highlights
A little attention was devoted for its application in solving the singular two-point boundary value problems (STPBVPs). This project treated some classes of singular second-order two-point boundary value problems both analytically and numerically using the Adomian decomposition method (ADM) and the modifications Improved Adomian Decomposition Method (IADM) and modified Adomian decomposition method (MADM) focusing on the Dirchlet and mixed boundary conditions; and applied the symbolic softwares MATLAB and MATHEMATICA to facilitate computing
The results show that the rate of convergence of the modified decomposition method, MADM is higher than the standard ADM for initial value problems
For fast convergence to the exact solution, MADM by Wazwaz, A.-M. helps to rearrange the result obtained above so that 4 assumes to be zero and all the existing terms obtained to be added to the following way [21]: For 5 = 0, = −7 $$ $ * + )4 The Adomian polynomial )4 can be determined either using the Adomian formula (3) or the results provided by MATHEMATICA at appendix C to be:
Summary
Several mathematical models that explain natural phenomena are mostly formulated in terms of nonlinear differential equations, both ordinary and partial. Extended surfaces are governed by BVPs; and they are widely used in many engineering appliances which include, but are not limited to, air conditioning, refrigeration, automobile and chemical processing equipments For this reason this research tried to consider both invariant and power-law dependence of thermal conductivities governed respectively by linear and nonlinear BVPs. The primary objective of using extended surfaces is to enhance the heat transfer rate between a solid and an adjoining fluid. The method is a powerful technique, which provides efficient algorithms for analytic approximate solutions and numeric simulations for real-world applications in the applied sciences and engineering It permits to solve both nonlinear IVPs and BVPs without restrictive assumptions such as required by linearization, perturbation, discretization, guessing the initial term or a set of basis functions, and so forth. This project treated some classes of singular second-order two-point boundary value problems both analytically and numerically using the ADM and the modifications Improved Adomian Decomposition Method (IADM) and MADM focusing on the Dirchlet and mixed boundary conditions; and applied the symbolic softwares MATLAB and MATHEMATICA to facilitate computing
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