scholarly journals l-problem of moments for one-dimensional integro-differential equations with Erdélyi-Kober operators

2019 ◽  
Vol 486 (6) ◽  
pp. 659-662
Author(s):  
Sergey S. Postnov
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Linear Equations ◽  
Integral Operators ◽  
Analytical Investigation ◽  
Explicit Solutions ◽  
One Dimensional ◽  
Problem Of Moments

Purpose: to investigate the possibility of statement of l-problem of moments for one-dimensional linear equations of three types, which contain Erdélyi-Kober differential and integral operators of fractional order. Methods: formulation of l-problem of moments for each type of investigated equations, analytical investigation and solution of problem formulated Results. Conditions derived that determine the possibility and solvability of the problem stated. In some cases an explicit solutions of l-problem of moments obtained. Conclusions. The possibility of statement of formulated l-problem of moments shown in cases that defined by conditions obtained in paper. Some analytical solutions of investigated problem obtained.

scholarly journals Solution of Fractional Autonomous Ordinary Differential Equations

2021 ◽  
Author(s):  
Rami AlAhmad ◽  
Qusai AlAhmad ◽  
Ahmad Abdelhadi
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Singular Kernel ◽  
Autonomous Differential Equations ◽  
Implicit Analytical Solution

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

scholarly journals Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Wave Equations ◽  
Separation Of Variables ◽  
Linear Equations ◽  
Second Order ◽  
Nonlinear Wave Equations ◽  
One Dimensional ◽  
Exponential Nonlinearity ◽  
Second Order Differential Equations

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

scholarly journals On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

2021 ◽  
Vol 5 (3) ◽  
pp. 109
Author(s):  
Batirkhan Kh. Turmetov ◽  
Kairat I. Usmanov ◽  
Kulzina Zh. Nazarova
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Fractional Order ◽  
Operator Method ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Type Problem ◽  
Volterra Type ◽  
Cauchy Type Problem ◽  
Conformable Derivatives

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed, and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

scholarly journals The Analytical Solution of Parabolic Volterra Integro- Differential Equations in the Infinite Domain

2016 ◽  
Author(s):  
Yun Zhao ◽  
Feng-Qun Zhao
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Kernel Functions ◽  
Infinite Domain ◽  
Memory Kernel ◽  
One Dimensional ◽  
Different Types ◽  
Frictional Memory Kernel

This article focuses on obtaining the analytical solutions for parabolic Volterra integro- differential equations in d-dimensional with different types frictional memory kernel. Based on theories of Laplace transform, Fourier transform, the properties of Fox-H function and convolution theorem, analytical solutions of the equations in the infinite domain are derived under three frictional memory kernel functions respectively. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, Fox-H function and convolution form of Fourier transform. In addition, the graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equation with power-law memory kernel. It can be seen that the solution curves subject to Gaussian decay at any given moment.

scholarly journals New method to solve certain differential equations

2016 ◽  
Vol 15 (1) ◽  
pp. 107-111
Author(s):  
Kazimierz Rajchel ◽  
Jerzy Szczęsny
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Schroedinger Equation ◽  
New Method ◽  
Winding Number ◽  
One Dimensional

AbstractA new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.

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