Boundary Value Problems For Equations Research Articles

Common Best Proximity Point Theorems for Generalized Dominating with Graphs and Applications in Differential Equations

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations.

Asymptotic spatial behavior for the heat equation on noncompact regions

AbstractWe consider the isotropic initial boundary value problem for the heat equation on open regions with noncompact boundary and construct differential inequalities for a generalized heat flow measure defined over a spherical cross section. Under suitable assumptions, integration of the differential inequality leads to spatial growth and decay rate estimates for mean‐square cross‐sectional measures of the time‐weighted temperature spatial gradient. The estimates are then used to obtain similar results for the time‐weighted temperature. In particular, when the base heat flow measure is positive, the time‐weighted temperature becomes pointwise unbounded at large spatial distance.

A boundary value problem for a non-linear difference equation

A boundary value problem for a non-linear difference equation

A novel extended integrated radial basis functions meshfree method for crack analysis in plate problem

In this paper, a novel extended meshfree method is presented based on the integrated radial basis functions (iRBF) for crack analysis in bending plate problems. The new method is named XiRBF. The First-order Shear Deformation Theory (FSDT) is adopted to describe the deformation and stresses in cracked plate structures. The advantage of the iRBF approach is that it starts the approximation process with the highest-order derivative of the primary function first and then the lower-order derivatives are obtained by integration. This approximation style can give more accurate results for derivative values that appear a lot in governing equations of boundary value problems. In addition, the iRBF shape function also satisfies the Kronecker delta property, so essential boundary conditions can be applied as simply as the finite element method. In fracture analysis, to model the discontinuous behavior of crack surfaces, the Heaviside function is used, and to capture the singularity at the crack tip, the enriched functions are selected suitably based on the asymptotic analysis from the analytical solution. Various numerical tests are performed in which the stress resultant intensity factors (SRIFs) are calculated and the crack paths are visualized for crack propagation in the plate problems. The accuracy and efficiency of the proposed method are confirmed by comparing the present results with those obtained by other existing methods.

RESEARCH OF THE STATIONARY PHENOMENON OF HEAT TRANSFER IN A SPHERICAL CONDENSED MEDIA

A method for investigating approximate analytical solutions of boundary value problems of singularly perturbed equations of stationary heat transfer processes in a spherical condensed medium, close to solving the boundary value problem for an undisturbed equation, is presented. The main task in which the structure of the article is defined is the development of mathematical and computer models of the stationary phenomenon of heat transfer in flat condensed media and a model of diffusion-reactive energy transfer depending on the thermal conductivity of materials.

Characteristic boundary layers in the vanishing viscosity limit for the Hunter-Saxton equation

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method.

Numerical approaches for solution of hyperbolic difference equations on circle

Abstract The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 \mathbb{T}^{1} . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.

Monotone iterative method for boundary value problem with linear condition of first order delay differential equation

In this paper we discuss the boundary value problem for a first order delay differential equation of the type, \(y'(t) + \lambda y(t) = f(t, y(t - r))\). We prove the existence of solution between weakly coupled lower and upper solution by assuming \(f\) to be a non-decreasing function in the second coordinate. Further, we use this existence result to establish monotone iterative method, where we obtain increasing as well as decreasing sequence of functions whose limits are a solution of the boundary value problem. The sequence of functions obtained are solutions of some defined boundary value problem with linear condition of linear delay differential equation.

Application of branching processes to solve boundary value problems for the semi-linear Helmholtz equation

This work devoted to construct a probabilistic model for solving boundary value problems for the semi-linear Helmholtz equation. Probabilistic representations of problem solving in the form of the mathematical expectation of some random variable are obtained. In accordance with probabilistic representation, branching processes are constructed and modeling formulas are specified for each of the branching processes. It has been proven that the constructed branching processes degenerate with probability one and the average number of particles of each n-th generation is less than or equal to one. For the considered problems, unbiased estimates were constructed on the trajectories of the corresponding random processes. Unbiased estimates which constructed on the trajectories of a branching process with a limited average number of branches, easily modeled in computer and has limited variance. Based on the proposed estimates, computational experiments were carried out. The results of the computational experiment show that using the constructed algorithm it is possible to solve semi-linear problems that occur in practice.

Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

<p>In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving $ L^2 $-contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.</p>

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

A MULTIPOINT BOUNDARY VALUE PROBLEM IN TIME FOR A $2B$-PARABOLIC EQUATION WITH DEGENERACY

One of the most important issues in the general theory of differential equations with partial derivatives is establishing the solvability of boundary value problems. Among the boundary value problems for equations with partial derivatives, problems with nonlocal boundary conditions occupy an important place. Such interest in such problems is caused both by their rich practical application (the process of diffusion, moisture distortion in soils, plasma physics, etc.), and by the needs of the general theory of boundary value problems. A general multipoint boundary value problem for nonuniformly $2b$-parabolic equations with degeneracy is studied. The coefficients of parabolic equations and boundary conditions allow power degeneracy of arbitrary order in terms of time variable and spatial variables at some set of points. To solve the given multipoint boundary value problem, solutions of problems with smooth coefficients in Hölder spaces with the appropriate norm are studied. With the help of interpolation inequalities and a priori estimates, estimates of the solution of auxiliary problems and their derivatives in special Gelder spaces are established. Using the theorems of Ross and Archel, a convergent sequence is distinguished from the compact sequence of solutions of the auxiliary problems, the limiting value of which is the solution of the multipoint boundary value problem in time for the $2b$-parabolic equation with degeneracy. Estimates of the solution of the given problem are established in Hölder spaces with power-law weights. The order of the power weight is determined by the order of features of the coefficients of the equations and the boundary conditions. With certain restrictions on the right-hand side of the equation and boundary conditions, an integral image of the given problem is obtained.

Boundary value problems of degenerate integro-differential equations on the boundary of the region with application in filtration

Boundary value problems of degenerate integro-differential equations on the boundary of the region with application in filtration

Existence Result for Fractional Differential Equation on Unbounded Domain

In this article, we establish certain sufficient conditions to show the existence of solutions of boundary value problem for fractional differential equations on the half-line in a Fréchet space. The main result is based on Tykhonoff fixed point theorem combining with a suitable measure of non-compactness. An example is given to illustrate our approach.

Positive solutions for integral boundary value problems of nonlinear fractional differential equations

In this paper, we consider integral boundary value problems of nonlinear fractional differential equations. Existence results of positive solutions for the problem are obtained based on the Guo-Krasnoselskii theorem and the Five functional fixed point theorem. Simple examples follow the main results in successive sections.

Local Existence and Blow up for a Nonlinear Viscoelastic Kirchhoff-type Equation with Logarithmic Nonlinearity

The aim of this paper is to consider the initial boundary value problem of nonlinear viscoelastic Kirchhoff-type equation with logarithmic source term. Firstly, we prove the local existence of weak solution by applying Banach fixed theorem. Later, we derive the blow-up results by the combination of the perturbation energy method, concavity method and differential-integral inequality technique.

A Study of The Scattering Properties of Eigenparameter- Dependent Matrix Difference Operator With Transmission Condition

In this paper, we set a transmission boundary value problem for a matrix valued difference equation on the semi axis. The main purpose of this study is to examine the properties of scattering solutions and scattering functions of this problem. Firstly, by giving the Jost solution and scattering solutions of this problem, we obtain the Jost function and the scattering function of the problem. We also investigate eigenvalues, spectral singularities, resolvent operator and continuous spectrum of this problem.

Legendre operational differential matrix for solving ordinary differential equations

In this paper, we used the Legendre operational differential matrix method based on the Tau method to find the approximate analytical solutions to the initial value problems and boundary value problems of ordinary differential equations. This method allows the solution of the ordinary differential equation to be computed in the form of an infinite series in which the components can be easily calculated. We introduced a comparison between the approximate solution that we computed and the exact solution of the selected problem, as we found the absolute error. According to the numerical results, the series of solutions we found are accurate and very close to the exact analytical solutions.

Local well-posedness of 1D degenerate drift diffusion equation

<abstract><p>This paper proves the well-posedness of locally smooth solutions to the free boundary value problem for the 1D degenerate drift diffusion equation. At the free boundary, the drift diffusion equation becomes a degenerate hyperbolic-Poisson coupled equation. We apply the Hardy's inequality and weighted Sobolev spaces to construct the appropriate a priori estimates, overcome the degeneracy of the system and successfully establish the existence of solutions in the Lagrangian coordinates.</p></abstract>

Two-point boundary value problems for 4th order ordinary differential equations

The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations u^(4)(t) = p(t)u(t) + q(t) for t ∈ [a,b], u^(4)(t) = p(t)u(t) + f(t, u(t)) for t ∈ [a,b], under the following two-point boundary conditions u^(i)(a) = 0, u^(i)(b) = 0 (i = 0, 1), and u^(i)(a) = 0 (i = 0, 1, 2), u(b) = 0, where p ∈ L([a,b]; R) is a nonconstant sign function and f ∈ K([a,b] × R; R).