Related papers: Prelle-Singer Approach and Liouvillian Solutions: …

The concept of fractional order derivative can be found in extensive range of many different subject areas. For this reason, the concept of fractional order derivative should be examined. After giving different methods mostly used in…

We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $\alpha$ ($0 < \alpha < 1$) is in the Riemann-Liouville sense. The technique we…

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four…

Many special functions are solutions of first order linear systems $y_n'(x)=a_n(x)y_n(x)+d_n(x)y_{n-1}(x)$, $y_{n-1}'(x)=b_n(x)y_{n-1}(x)+e_{n}(x)y_n(x)$. We obtain bounds for the ratios $y_n(x)/y_{n-1}(x)$ and the logarithmic derivatives…

In this study, we give the variation of parameters method from a different viewpoint for the Nth order inhomogeneous linear ordinary difference equations with constant coefficient by means of delta exponential function . Advantage of this…

The 1st level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann-Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the…

Solution of fractional differential equations is an emerging area of present day research because such equations arise in various applied fields. In this paper we have developed analytical method to solve the system of fractional…

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…

The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the…

This paper provides the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant…

Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike…

Small perturbation of the Liouville equation under singular initial data is considered. An asymptotics of the singular solution is constructed by the method which is similar to Bogolubov -- Krylov one. The main object is an asymptotics of…

The goal of this work is to discuss how should we impose initial values in fractional problems to ensure that they have exactly one smooth unique solution, where smooth simply means that the solution lies in a certain suitable space of…

In this paper we solve a L\'evy driven linear stochastic first order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic…

We consider the case of exceptional Laguerre polynomials $X_1$ of type I, II and III, their ordinary differential equations and the problem of finding general solution beside the polynomial part. We will develop an algebraic approach based…

We study the fourth order Schr\"odinger type differential inequality $-\Delta^2 u + \lambda V(x)u \geq a(x)u^q$ with $a,V\in L^1_{loc}(\mathbf{R}^N)$, both nonnegative, and $\lambda>0$. We consider nonnegative solutions without making any…

To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving…

In this paper, we present a new approachment for Sturm-Liouville problem having special potentials. We acquire the representations of solutions and asymptotic formulas for solutions with regard to initial conditions. Also, a few…

A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.

We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0,…