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Application of the Partial Differential Equations using the D' Alembert's Formula
Shipra
Abstract:
In this paper, we consider several situations stemming from the applications, and the mathematical modeling of which involves partial differential equation problems. Our primary focus in these research projects is on the good qualities and consequences of a specific partial differential equation's solution. The homogeneous one-dimensional wave equation in particular piques our interest in the mathematical modelling of the consistency and well-posedness of the solution or solutions to certain PDEs. A function u = u(x, y, z, t) will be used to measure different physical quantities. We examine the homogeneous one-dimensional wave equation via the lens of mathematical modelling of partial differential equations. Specifically, we investigate the solution's well-posedness and consistency (Guo and Zhang, 2007). The method of change of variable is to be used to derive the d' Alembert's general solution, which will ultimately lead us to the d' Alembert's formula for the wave equation solution. Though the classical theory of partial differential equations deals almost completely with the well-posed, ill posed problems can be mathematically and scientifically interesting. After that, we analyzed the results using the answer we had acquired, displayed the behavior of our results in a table, and came to the conclusion that the idea of a well-posed issue is crucial in applied mathematics.