Boundary Value Problems For Equations Research Articles

A study on the existence results of boundary value problems of fractional relaxation integro-differential equations with impulsive and delay conditions in Banach spaces

<abstract><p>The aim of this paper was to provide systematic approaches to study the existence of results for the system fractional relaxation integro-differential equations. Applied problems require definitions of fractional derivatives, allowing the utilization of physically interpretable boundary conditions. Impulsive conditions serve as basic conditions to study the dynamic processes that are subject to sudden changes in their state. In the process, we converted the given fractional differential equations into an equivalent integral equation. We constructed appropriate mappings and employed the Schaefer's fixed-point theorem and the Banach fixed-point theorem to show the existence of a unique solution. We presented an example to show the applicability of our results.</p></abstract>

Some new methods for studying boundary value problems for general partial differential equations

Some new methods for studying boundary value problems for general partial differential equations

ON DIFFERENT EXTENSIONS WITH EXIT FROM SPACE OF MINIMAL OPERATOR ASSOCIATED WITH SECOND ORDER OPERATOR DIFFERENTIAL EXPRESSION

Abstract. Our aim in that paper is defining domains of definitions of accumulative, dissipative and self-adjoint extensions of minimal operator associated with boundary value problems for operator Sturm-Liouville equation with boundary conditions, containing unbounded operator and spectral parameter λ.

Uniqueness of Solutions to Boundary Value Problems of Higher Order Fractional Differential Equations

Uniqueness of Solutions to Boundary Value Problems of Higher Order Fractional Differential Equations

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

APPLICATION OF THE POTENTIAL THEORY TO SOLVING A MIXEDPROBLEM FOR A MULTIDIMENSIONAL ELLIPTIC EQUATION WITHONE SINGULAR COEFFICIENT

Potential theory has played a paramount role in both analysis and computation for boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using some properties of Gaussian hypergeometric function, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find a solution of the mixed problem for the multidimensional singular elliptic equation in the half-space

Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

OPERATOR-THEORETICAL FORMULATLONS OF BOUNDARY VALUE PROBLEM FOR STURM–LIOUVILLE OPERATOR EQUATlON

Abstract. Our aim in that paper is to set the operator theoretical formulation of boundary value problem for Sturm-Liovuille equation with unbounded operator coefficient and boundary condition involving Herglotz-Nevanlinna function of spectral parameter in boundary condition. We define minimal operator and give description of maximal operator.

Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator

<p>In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.</p>

LANDAU LEGENDRE WAVELET GALERKIN METHOD APPLIED TO STUDY TWO-PHASE MOVING BOUNDARY PROBLEM OF HEAT TRANSFER IN FINITE REGION

In this paper, we developed a mathematical model of solidification where specific heat and thermal conductivity are temperature dependent. This model is a two-phase moving boundary problem (MBP) of heat transfer in finite region and represents as MBP of system of parabolic nonlinear second-order partial differential equations (PDEs). We developed a Landau Legendre wavelet Galerkin method for finding the solution of the problem. The MBP of a system of PDEs is transformed into a variable boundary value problem of nonlinear ordinary defferential equations (ODEs) by the use of dimensionless variables and the Landau transform. The problem is converted into a system of algebraic equations with the application of Legendre wavelet Galerkin method. In particular case, we compared present solution with Laplace transform solution and found approximately the same. The whole investigation has been done in dimensionless form. When the specific heat and thermal conductivity exponentially vary in temperatures, it is the effect of dimensionless parameters: thermal diffusivity (α<sub>12</sub>), ratio of thermal conductivity (<i>k</i><sub>12</sub>), dimensionless temperature (θ<sub><i> f</i></sub>), Fourier number (F<sub>0</sub>), Stefan number (Ste), and ratio of densities (ρ<sub>1</sub> / ρ<sub>2</sub>) are discussed in detail.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

Existence of nodal solutions to some nonlinear boundary value problems for ordinary differential equations of fourth order

Existence of nodal solutions to some nonlinear boundary value problems for ordinary differential equations of fourth order

Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity

Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity

AVERAGING METHOD FOR MULTI-POINT BOUNDARY VALUE PROBLEMS OF SET-VALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

AVERAGING METHOD FOR MULTI-POINT BOUNDARY VALUE PROBLEMS OF SET-VALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

Solution approximation of fractional boundary value problems and convergence analysis using AA-iterative scheme

<abstract><p>Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature.</p></abstract>

ON THE CORRECT STATEMENTS OF BOUNDARY VALUE PROBLEMS FOR PDE'S IN BANACH-SOBOLEV SPACES

Abstract. In this report we consider the Banach Functional Space    X , defined on measure space   m ;  , where n R   is bounded domain, m is the Lebesgue measure in n R . It is defined the Banach Sobolev Space    m X W , generated by norm of    X . Regarding these spaces the elliptic equation is considered and the concepts of classical, local, weak and strong solutions are given. One method for defining the trace, trace operator and trace space generated by space    1 X W are introduced. These concepts allow us to consider the boundary value problems for differential equations regarding the spaces    m X W . Finally, it is considered the Dirichlet problem for polyharmonic equations in Sobolev spaces, generated by norm of symmetric spaces and the correct solvability of the corresponding problem is proved.

Solutions of initial and boundary value problems using invariant curves

<p>The purpose of this study is to investigate the solutions of initial and boundary value problems of ordinary differential equations by employing Lie symmetry generators. In this investigation, it shown that invariant curves, which obtained by symmetry generators, also be utilized to find solutions to initial and boundary value problems. A method, involving invariant curves, presented to find solutions to initial and boundary value problems. Solutions to many linear and nonlinear initial and boundary value problems discussed by applying the proposed method.</p>

A fundamental sequences method for an inverse boundary value problem for the heat equation in double-connected domains

A fundamental sequences method for an inverse boundary value problem for the heat equation in double-connected domains

Study on coupled problems of thermoelasticity in Strains

In the work, within the framework of the strain compatibility conditions of Saint-Venant, two equivalent dynamic boundary value problems of thermoelasticity with respect to strains are formulated. In the case of the first boundary value problem, the dynamic equations of thermoelasticity are obtained from the compatibility conditions, in the second case, instead of the first three equations of thermoelasticity, the equations of motion expressed with respect to deformations are considered. Discrete analogues of boundary value problems are constructed using the finite-difference method in the form of explicit and implicit schemes. The solution of explicit schemes is reduced to recurrent relations with respect to deformations and temperature. Implicit schemes are solved by sequential application of the elimination method. The validity of the formulated thermoelastic boundary value problems is substantiated by comparing the numerical results of the problem of a thermoelastic parallelepiped obtained by different methods, as well as solving a similar problem in displacements.