A Study on the Numerical and Analytical Solutions of Complex-Variable Partial Differential Equations
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Abstract
In this work, we consider the analogue of a real-variable partial differential equation. In comparison to what has already been thoroughly investigated, recall the non-linear Schrodinger equation (NLSE). The NLSE, which is used in determining the wave equation for quantum particles, is a real-variable PDE with complex coefficients. Instead, we consider equations where both the function $\omega$ and its independent variable $z$ belong to the complex plane. We approach the complex problem by an intuitive approach of treating a one-complex variable differential equation as a two-real variable partial differential equation by analyzing the real and imaginary parts of both $\omega$ and $z$. We investigate thoroughly the first-order complex PDE case and prove the existence and uniqueness theorem for these types of equations. We also investigate the analytical solutions by considering the complex-variable Laplace transform, which can be thought of in parallel as a two-variable Laplace transform with in $\mathbb{R}^2$. Upon completion of the first-order case, we consider the higher order complex-variable PDE. We discuss both the direct way of solving higher-order equations via systems of real-variable PDE’s and also via first-order systems of complex-variable PDE’s, in which we implement the methods of the previous topics. As a direct consequence of the higher-order differential equation solution method, we also discuss an alternative method of evaluating complex contour integrals via a real-variable partial differential equation evaluation. To conclude, we consider the time-dependent complex variable PDE analogues of the advection and wave equations, we briefly discuss multi-complex variable PDE’s and methods that we plan to investigate in the near future.