Dr. Ashok Kumar

https://doi.org/10.64823/ijter.2605033

In this paper we compare the inapplicable cases of the Method of Undetermined Coefficients and Method of Variation of Parameters in solving second order differential equations with constant coefficients. We first analyse the cases of in applicability for each method to identify the in applicability, investigate their respective limitations and the underlying reasons responsible for it .The applicability of Method of undetermined coefficients clearly depends upon the forcing functions on the other hand the method of variation of parameter limits itself over computational complexities over non elementary integrals. Then finally we carried out comparative analysis to highlight the differences and sameness between the limitations of these methods.

https://doi.org/10.64823/ijter.2605032

The Differential equations play a very important role in the mathematical modelling of physical phenomena by explaining how physical quantities differ with respect to space and time. These applications like, heat conduction is one of the most important process modeled using partial differential equations, particularly the heat equation. This study throws light on the analysis of differential equations arising in heat conduction and their numerical treatment using the matrix method. The continuous heat conduction model is discretized into a system of algebraic equations through finite difference approximation, which is then represented in matrix form for accomplished calculation. The matrix method presents an orderly approach for solving large systems of equations and analyzing temperature distribution within conductive media under various boundary and initial conditions. This approach shows that the physical importance of differential equations in engineering and applied sciences while throw light on the effectiveness of matrix techniques in obtaining approximate solutions to heat transfer problems.In this problem we find general solution of heat conduction equation by using matrix method