Even Higher Order Fractional Initial Boundary Value Problem with Nonlocal Constraints of Purely Integral Type

Abstract

In this paper, the a priori estimate method, the so-called energy inequalities method based on some functional analysis tools is developed for a Caputo time fractional 2 m th order diffusion wave equation with purely nonlocal conditions of integral type. Existence and uniqueness of the solution are proved. The proofs of the results are based on some a priori estimates and on some density arguments.

Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

“…By using Cauchy inequality, Inequality (12), Conditions (16) and Lemma 1, we infer from Equation 29that…”

“…For the proof of the existence and the uniqueness of the solution of the posed problem, we use the energy inequality method based mainly on some a priori estimates and on the density of the range of the operator generated by the considered problem. In the literature, there are few articles using the method of energy inequalities for the proof of existence and uniqueness of fractional initial-boundary value problems in the fractional case (see [16][17][18][19]).…”

Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems

“…Several numerical methods, namely local adaptive differential quadrature have been implemented in various studies. A numerical method namely Adomian decomposition methods [ 13 , 14 ], energy inequalities method [15] , Galerkin based Septic B-Splines technique [16] , splines and non-polynomial spline techniques [17] , [18] , [19] , Local adaptive differential quadrature method [20] , reproducing kernel Hilbert space method [21] to solve the boundary value problems of higher order with multi-boundary conditions. The authors of [3] used both analytical and numerical methods to investigate the linear Electro-hydrodynamic stability problem of an eighth order differential equation.…”

On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate

“…It seems that the functional analysis method we apply in this paper is very efficient to solve some nonlocal fractional initial boundary value problems for single and systems of some different classes of partial differential equations. We can find only a few papers that use the previous method in the literature, and we can cite, for example, [23][24][25][26][27][28].…”