scholarly journals On the Numerical Solution of Initial-Boundary Value Problems for the Convection-Diffusion Equation with a Fractional Caputo Derivative and a Nonlocal Linear Source

2021 ◽  
pp. 35-50
Author(s):  
Aslan Apekov ◽  
Murat Beshtokov ◽  
Zaryana Beshtokova ◽  
Zamir Shomakhov
Keyword(s):  
Boundary Value Problems ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Fractional Caputo Derivative ◽  
Priori Estimates ◽  
The Difference ◽  
Initial Boundary Value

In a rectangular domain the first and third initial-boundary value problems are studied for the one-dimensional with respect to the spatial variable diffusion convection equation with a fractional Caputo derivative and a nonlocal linear source of integral form. Using the method of energy inequalities, under the assumption of the existence of a regular solution, a priori estimates are obtained in differential form, which implies the uniqueness and continuous dependence of the solution on the input data of the problem. On a uniform grid, two difference schemes are constructed that approximate the first and third initial-boundary value problems, respectively. For the solution of the difference problems, a priori estimates are obtained in the difference interpretation. The obtained estimates in difference form imply uniqueness and stability, as well as convergence at a rate equal to the order of the approximation error. An algorithm for the approximate solution of the third boundary value problem is constructed, numerical calculations of test examples are carried out, illustrating the theoretical results obtained in this work.

scholarly journals Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

2021 ◽  
Vol 5 (4) ◽  
pp. 134
Author(s):  
Serik E. Aitzhanov ◽  
Abdumauvlen S. Berdyshev ◽  
Kymbat S. Bekenayeva
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Priori Estimates ◽  
Initial Boundary Value

This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.

scholarly journals SOLVABILITY OF HOMOGENIZED PROBLEMS WITH CONVOLUTIONS FOR WEAKLY POROUS MEDIA

2020 ◽  
pp. 59-70
Author(s):  
G. V. Sandrakov ◽  
A. L. Hulianytskyi
Keyword(s):  
Porous Media ◽  
Boundary Value Problems ◽  
A Priori Estimates ◽  
Initial Conditions ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
Parabolic Problems ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

Semi-discrete Galerkin approximation method applied to initial boundary value problems for Maxwell's equations in anisotropic, inhomogeneous media

1981 ◽  
Vol 89 (1-2) ◽  
pp. 125-133 ◽  
Author(s):  
Pekka Neittaanmäki ◽  
Jukka Saranen
Keyword(s):  
Boundary Value Problems ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Boundary Value ◽  
A Priori ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Standard Galerkin Method ◽  
Initial Boundary Value

SynopsisIn this paper the semi-discrete Galerkin approximation of initial boundary value problems for Maxwell's equations is analysed. For the electric field a hyperbolic system of equations is first derived. The standard Galerkin method is applied to this system and a priori error estimates are established for the approximation.

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