Related papers: General Airy-type equations, heat-type equations a…
We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional…
For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_n$ related to…
As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does…
This article contains a new discussion for the generalized fractional Cauchy-type problem involving Hilfer-Katugampola-type fractional derivative. We study an existence and continuation of its solution. Firstly, we establish a new theorems…
The method of separation of variables can be used to solve many separable linear partial differential equations (LPDEs). Moreover, variable separation solutions usually are some trigonometric series. In the paper, base on some ideas of this…
The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…
Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…
In this study, a recursive solution technique in conjunction with generalized integrating factors is presented and applied to address first and second order linear differential equations. This approach demonstrates practical utility in…
We give explicit expressions for higher order convolutions of Cauchy numbers, either as one single integral or in terms of the Stirling numbers of the first and second kinds.
This paper provides a probabilistic approach to solve linear equations involving Caputo and Riemann-Liouville type derivatives. Using the probabilistic interpretation of these operators as the generators of interrupted Feller processes, we…
Line integration of generalized functions is studied. Second order partial differential equations with piecewise continuous and generalized variable coefficients over Cayley-Dickson algebras are investigated. Formulas for integrations of…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of…
In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial…
In this article we study the Cauchy problem for a new class of parabolic-type pseudodifferential equations with variable coefficients for which the fundamental solutions are transition density functions of Markov processes in the four…
In the present article an endeavor is made to solve the variable order fractional diffusion equations using a powerful method viz., Homotopy Analysis method. It is demonstrated how the method can be used while solving approximately two…