scholarly journals On Some Initial and Initial Boundary Value Problems for Linear and Nonlinear Boussinesq Models

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1273
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Boussinesq Equation ◽  
A Priori ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Main Concern ◽  
Initial Boundary Value Problems ◽  
Priori Estimate ◽  
Initial Boundary Value ◽  
Laplace Decomposition Method ◽  
Boussinesq Models

The main concern of this paper is to apply the modified double Laplace decomposition method to a singular generalized modified linear Boussinesq equation and to a singular nonlinear Boussinesq equation. An a priori estimate for the solution is also derived. Some examples are given to validate and illustrate the method.

scholarly journals On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

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.

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.

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

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 305 ◽  
Author(s):  
Said Mesloub ◽  
Faten Aldosari
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Nonlocal Conditions ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Type ◽  
Uniqueness Of The Solution ◽  
Estimate Method ◽  
Initial Boundary Value

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.

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 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 Finite-difference method for the Gamma equation on non-uniform grids

2019 ◽  
pp. 3-8
Keyword(s):  
Finite Difference ◽  
A Priori ◽  
Quasilinear Parabolic Equation ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Local Approximation ◽  
Uniform Grids ◽  
The Difference ◽  
Initial Boundary Value

We propose a new monotone finite-difference scheme for the second-order local approximation on a nonuniform grid that approximates the Dirichlet initial boundary value problem (IBVP) for the quasi-linear convection-diffusion equation with unbounded nonlinearity, namely, for the Gamma equation obtained by transformation of the nonlinear Black-Scholes equation into a quasilinear parabolic equation. Using the difference maximum principle, a two-sided estimate and an a priori estimate in the C-norm are obtained for the solution of the difference schemes that approximate this equation.

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