Solving Generalized Heat and Generalized Laplace Equations Using Fractional Fourier Transform

In the present work, the main objective is to find the solution of the generalized heat and generalized Laplace equations using the fractional Fourier transform, which is a general form of the solution of the heat equation and Laplace equation using the classical Fourier transform. We also formulate its solution using a sampling formula related to the fractional Fourier transform. The fractional Fourier transform is introduced, and related theorems and essential properties are collected. Several results related to the sampling formula are derived. A few examples are presented to illustrate the effectiveness and powerfulness of the proposed method compared to the classical Fourier transform method.

Automated summary

The main objective of this study is to find the solution of the generalized heat and generalized Laplace equations using the fractional Fourier transform, which is a general form of the solution of these equations using the classical Fourier transform. The solution is also formulated using a sampling formula related to the fractional Fourier transform. The study introduces the fractional Fourier transform, collects related theorems and essential properties, and derives several results related to the sampling formula. Several examples are presented to demonstrate the effectiveness and powerfulness of the proposed method compared to the classical Fourier transform method.