Solution of some types of differential equations: operational calculus and inverse differential operators.

K Zhukovsky

doi:10.1155/2014/454865

Abstract

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.

Highlights

IntroductionMost of physical systems can be described by appropriate sets of differential equations, which are well suited as models for systems
We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations
Most of physical systems can be described by appropriate sets of differential equations, which are well suited as models for systems

Summary

Introduction

Most of physical systems can be described by appropriate sets of differential equations, which are well suited as models for systems. It is common knowledge that expansion into series of Hermite, Laguerre, and other relevant polynomials [1] is useful when solving many physical problems (see, e.g., [2, 3]). Generalised forms of these polynomials exist with many variables and indices [4, 5]. We develop an analytical method to obtain solutions for various types of partial differential equations on the base of operational identities, employing expansions in series of Hermite, Laguerre polynomials, and their modified forms [1, 6]. We will demonstrate in what follows that when used properly and combined, in particular, with integral transforms, such an approach leads to elegant analytical solutions with transparent physical meaning without cumbersome calculations

Inverse Derivative

Operational Approach and Other Types of Differential Equations

Conclusions