Boundary Value Conditions Research Articles

The trace of uin W^{1,1}_{mathrm{loc}}(Omega )bigcap L^{infty}(Omega ) and its applications

This paper is concerned with the well-posedness problem of a doubly degenerate parabolic equation with variable exponents. By the parabolically regularized method, the existence of local solution is proved. Moreover, the trace of uin W^{1,1}_{0}(Omega ) is generalized to uin W^{1,1}_{mathrm{loc}}(Omega )bigcap L^{infty}(Omega ) in a rational way. Then, a partial boundary value condition matching up with the stability theorem is found.

Existence of weak solutions for a $p(x)$-Laplacian equation via topological degree

We consider the $p(x)$-Laplacian equation with a Dirichlet boundary value condition $$ \begin{cases} -\Delta_{p(x)}(u)+|u|^{p(x)-2}u= g(x,u,\nabla u), &x\in\Omega,\\ u=0, &x\in\partial\Omega. \end{cases} $$ Using the topological degree constructed by Berkovits, we prove, under appropriate assumptions, the existence of weak solutions for this equation.

The partial boundary value conditions of nonlinear degenerate parabolic equation

The stability of the solutions to a parabolic equation ∂u∂t=ΔA(u)+∑i=1Nbi(x,t)Diu−c(x,t)u−f(x,t)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\frac{\\partial u}{\\partial t} = \\Delta A(u) +\\sum_{i=1}^{N}b_{i}(x,t)D_{i}u-c(x,t)u-f(x,t) $$\\end{document} with homogeneous boundary condition is considered. Since the set {s: A'(s)=a(s)=0} may have an interior point, the equation is with strong degeneracy and the Dirichlet boundary value condition is overdetermined generally. How to find a partial boundary value condition to match up with the equation is studied in this paper. By choosing a suitable test function, the stability of entropy solutions is obtained by Kruzkov bi-variables method.

Well-posed and stable problems for Prandtl's boundary layer system

We are concerned with the well-posed problem of a strongly degenerate parabolic equation. A kind of entropy solutions in the BV sense is introduced and a reasonable partial boundary value condition is provided. By means of Kruzkov's bi-variables method, the stability of entropy solutions is proved under this partial boundary value condition. Through the Crocco transformation, the Prandtl boundary layer system is transformed to an anisotropic degenerate parabolic equation with a nonlinear boundary value condition. We propose a new analytical method by restricting some geometric properties on the spatial domain and establish the well-posedness of the BV solution to this anisotropic degenerate parabolic equation with the initial value condition.

Double Phase Implicit Obstacle Problems with Convection and Multivalued Mixed Boundary Value Conditions

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.

Existence and uniqueness solution for integral boundary value problem of fractional differential equation

In this work, we will study the existence and uniqueness results of a class of nonlinear fractional differential equations with integral boundary value conditions. By using Leray-Schauder nonlinear alternative and the Banach contraction mapping principle. As an application, an example is given to prove our conclusions.

New semi-analytical solutions of the time-fractional Fokker–Planck equation by the neural network method

This paper suggests a neural network method for solving the time-fractional Fokker–Planck equation. An energy function is constructed by means of initial–boundary value conditions. The Caputo derivative is approximated by L1 numerical scheme and an unconstrained discretization minimization problem is presented. Gradient descent algorithm is adopted for neural network training. Semi-analytical solutions are obtained where two numerical examples demonstrate the method’s efficiency.

Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented.

Fixed Point Theory and the Liouville–Caputo Integro-Differential FBVP with Multiple Nonlinear Terms

This work is reserved for the study of a special category of boundary value problems ( BVP s) consisting of Liouville–Caputo integro-differential equations with multiple nonlinear terms. This fractional model and its boundary value conditions (BVCs) involve different simple BVP s, in which the second BVC as a linear combination of two Caputo derivatives of the unknown function equals a nonzero constant. The Banach principle gives a unique solution for this Liouville–Caputo BVP . Further, the Krasnoselskii and Leray–Schauder criteria give the existence property regarding solutions of the mentioned problem. For each theorem, we provide an example based on the required hypotheses and derive numerical data in the framework of tables and figures to show the consistency of results from different points of view.

Discrete Maximum Principle and Energy Stability of the Compact Difference Scheme for Two-Dimensional Allen-Cahn Equation

The Allen-Cahn model is discussed mainly in the phase field simulation. The compact difference method will be used to numerically approximate the two-dimensional nonlinear Allen-Cahn equation with initial and boundary value conditions, and then, a fully discrete compact difference scheme with second-order accuracy in time and fourth-order in space is established. And its numerical solution satisfies the discrete maximum principle under the constraints of reasonable space and time steps. On this basis, the energy stability of the scheme is investigated. Finally, numerical examples are given to illustrate the theoretical results.

Cold-preventing simulation of low-temperature protective clothing based on an unsteady one-dimensional heat conduction model

With the development of science and technology, people are constantly exploring the extreme cold regions. In order to enable people to work normally in extremely cold weather, scientists have been studying low-temperature protective composite materials to protect workers in ultra-low ambient temperatures. This paper studies the cold-proof effect of low-temperature protective clothing composed of three materials. First of all, this paper uses the principle of heat transfer to establish an unsteady one-dimensional heat conduction partial differential equation model for the experimenter, cryogenic protective clothing and the system composed of the external air environment. Secondly, this paper determines the boundary value conditions and boundary value conditions of the definite solution, as well as the continuous conditions of the contact surface between the materials in the protective clothing. On this basis, this paper uses the finite difference method to convert the differential equation into a display format to solve the problem. The final result is that the experimenter’s persistence time with 15°Cas the holding limit outdoors is 662.5s, and 10°Cas the holding limit. It is 675.3s. Finally, the sensitivity analysis of the model is carried out by setting the temper-ature fluctuation.

Existence of multiple positive solutions to the Caputo-type nonlinear fractional differential equation with integral boundary value conditions

Existence of multiple positive solutions to the Caputo-type nonlinear fractional differential equation with integral boundary value conditions

On the solutions of fractional integro-differential equations involving Ulam-Hyers-Rassias stability results via $\psi$-fractional derivative with boundary value conditions

In this paper, we study boundary value problems for the impulsive integro-differential equations via $\psi$-fractional derivative. The contraction mapping concept and Schaefer's fixed point theorem are used to produce the main results. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam-Hyers-Rassias stability of the solution to the proposed system.

EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF CAPUTO-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS WITH <i>P</i>-LAPLACIAN OPERATOR

In this paper, by using the Schauder fixed point theorem and Banach contraction mapping principle, the existence and uniqueness of solutions for a coupled system of Caputo-Hadamard fractional differential equations with <i>p</i>-Laplacian operator are established. As applications, two examples are given to illustrate the main results. The interesting point of this article is that the boundary value conditions contain integrals, and the approximate solutions are given by using the iterative method.

A Semi-Analytical Solution Approach for Solving Constant-Coefficient First-Order Partial Differential Equations

Simulation and control of many dynamic systems involve solving partial differential equations (PDE). This letter proposes a semi-analytical solution (SAS) approach for fast and high-quality solution of first-order PDEs. The region of interest of the studied PDE is divided into a grid, and an SAS is derived for each grid cell in the form of the multivariate polynomials, of which the coefficients are identified using initial value and boundary value conditions. The solutions are solved in a “time-stepping” manner, i.e., within one time step, the coefficients of the SAS are identified and the initial value of the next time step is evaluated. This approach achieves a significantly larger grid cell than the widely used finite difference method, and thus enhances the computational efficiency significantly. The simulation result on the natural gas pipeline model demonstrates the advantages of SAS in accuracy and computational efficiency.

EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR CAPUTO FRACTIONAL DIFFERENTIAL EQUATION WITH INFINITE-POINT BOUNDARY VALUE CONDITIONS

<abstract> In this paper, the existence of positive solutions for Caputo fractional differential equations boundary value problem with infinite points contained in the boundary value condition, and based on Green's function and its properties, the existence of multiple positive solutions are obtained by Avery-Peterson fixed point theorem. </abstract>

On a class of double phase problem with nonlinear boundary conditions

&lt;abstract&gt;&lt;p&gt;The existence of nontrivial solutions of the double phase problem with nonlinear boundary value condition is an important quasilinear problem: we use variational techniques and sum decomposition of a space $ W_0^{1, \xi}(\Omega) $ to prove the existence of infinitely many solutions of the problem considered. Moreover, our conditions are suitable and different from those considered previously.&lt;/p&gt;&lt;/abstract&gt;

On a non-Newtonian fluid type equation with variable diffusion coefficient

&lt;abstract&gt;&lt;p&gt;Since the non-Newtonian fluid type equations arise from a broad and in-depth background, many research achievements have been gained from 1980s. Different from the usual non-Newtonian fluid equation, there is a nonnegative variable diffusion in the equations considered in this paper. Such a variable diffusion reflects the characteristic of the medium which may not be homogenous. By giving a generalization of the Gronwall inequality, the stability and the uniqueness of weak solutions to the non-Newtonian fluid equation with variable diffusion are studied. Since the variable diffusion may be degenerate on the boundary $ \partial \Omega $, it is found that a partial boundary value condition imposed on a submanifold of $ \partial\Omega\times (0, T) $ is enough to ensure the well-posedness of weak solutions. The novelty is that the concept of the trace of $ u(x, t) $ is generalized by a special way.&lt;/p&gt;&lt;/abstract&gt;

Unified Approach to the Existence of Solutions for a Coupled System of Fractional Differential‐Integral Equations with Infinite Points and Riemann–Stieltjes Integral Boundary Conditions

In this article, by using the Schauder fixed point theorem, we first study the existence of solutions for a new coupled system of Caputo fractional differential equations with multipoint boundary value conditions under the assumption that the nonlinear term satisfies two types of the Carathéodory conditions. Using this result, we provide the existence of solutions of the system with infinite points and Riemann–Stieltjes integral boundary conditions, respectively, instead of doing it directly. Finally, we give three examples to illustrate the feasibility of main results.

Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in ℝn

In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable x∈Rn and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained.