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.

Most of physical systems can be described by appropriate sets of differential equations, which are well suited as models for systems. Hence, understanding of differential equations and finding its solutions are of primary importance for pure mathematics as for physics. With rapidly developing computer methods for the solutions of equations, the question of understanding of the obtained solutions and their application to real physical situations remains opened for analytical study. There are few types of differential equations, allowing explicit and straightforward analytical solutions. It is common knowledge that expansion into series of Hermite, Laguerre, and other relevant polynomials [

For a common differential operator

Laguerre polynomials of two variables [

In the following sections, we will investigate the possibilities to solve some partial differential equations, involving the differential operators studied above. Now we just note how the technique of inverse operator, applied for derivatives of various orders and their combinations and combined with integral transforms, allows for easy and straightforward solutions of various types of differential equations.

Heat propagation and diffusion type problems play a key role in the theory of partial differential equations. Combination of exponential operator technique and inverse derivative together with the operational identities of the previous section is useful for the solution of a broad spectrum of partial differential equations, related to heat and diffusion processes. Some of them have been already studied by operational method (see, e.g., [

The simplest example of the initial function

In conclusion of the present chapter we consider the example of the solution of a heat propagation type equation by operational method, involving the inverse derivative operator and exponential operator technique. We recall that the common heat equation with initial condition problem

Operational approach to solution of partial differential equations, demonstrated on the examples of diffusion-like and heat-like equations with

Thus, operational technique, combined with integral transforms, operational identities, and extended forms of orthogonal polynomials, represents powerful tool for finding solutions of various classes of differential equations and initial value problems. Note that within the framework of inverse differential operators, developed and described above, the usage of the evolution operator method opens new possibilities, which we will elucidate in what follows.

Let us consider the following generalization of the heat equation:

The effect, produced by the translation operator

The above result is exactly the generalization of Gleisher rule (

Another interesting example of solving (

The same operational technique as employed for the treatise of (

Now let us consider another Fokker-Plank type equation, that is, the following example:

Operational method is fast and universal mathematical tool for obtaining solutions of differential equations. Combination of operational method, integral transforms, and theory of special functions together with orthogonal polynomials closely related to them provides a powerful analytical instrument for solving a wide spectrum of differential equations and relevant physical problems. The technique of inverse operator, applied for derivatives of various orders and combined with integral transforms, allows for easy and straightforward solutions of various types of differential equations. With operational approach, we developed the methodology of inverse differential operators and derived a number of operational identities with them. We have demonstrated that using the technique of inverse derivatives and inverse differential operators, combined with exponential operator, integral transforms, and special functions, we can make significant progress in solution of various mathematical problems and relevant physical applications, described by differential equations.

In complete analogy with the heat equation solution by Gauss-Weierstrass transform [

The author declares that there is no conflict of interests regarding the publication of this paper.