A Stable Difference Scheme for a Third-Order Partial Differential Equation

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

Surface representation as a boundary-valued problem of partial differential equations (PDE) is an important topic of computer graphics and computer aided design. In existing references, various free-from surfaces were created with a fourth order PDE which is only able to meet the tangential conditions at the surface boundaries. The need for a sixth order PDE in surface modeling arises in two situations: one is to generate surfaces with curvature continuity; and the other is to use curvature values as a user handle for surface shape manipulation. In this chapter, we introduce such a sixth order PDE for free-form surface generation and develop a finite difference method to solve this PDE. We also investigate the effects of boundary curvature and the vector-valued shape parameters on the surface shape.

A numerical method is proposed for solving nonlocal boundary value problem for the multidimensional elliptic partial differential equation with the Bitsadze‐Samarskii‐Dirichlet condition. The first and second‐orders of accuracy stable difference schemes for the approximate solution of this nonlocal boundary value problem are presented. The stability estimates, coercivity, and almost coercivity inequalities for solution of these schemes are established. The theoretical statements for the solutions of these nonlocal elliptic problems are supported by results of numerical examples.

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

In this paper, the proposed modification to Adomian decomposition method (ADM) is introduced. In addition, a new modification to solve singular and nonsingular initial-value problems (IVPs) in first order partial differential equations (PDEs) is investigated.
 Then, a discussion of how ADM can be used to solve singular and nonsingular IVPs is produced. Finally, some example are given to demonstrate the applicability of the results of paper.
 Key words; Adomian decomposition method; Nonlinear first-order PDE; Singular and nonsingular first-order partial differential equations.
 1 Introduction
 Singular nonlinear partial differential equations (PDEs) arise in various physical phe- nomena in applied sciences and engineering from such areas as fluid mechanics and heat-

In this paper we consider a boundary value problem for elliptic second order partial differential equations with highly discontinuous coefficients in a 2D polygonal region Ω. The problem is discretized by a (full) DG method on triangular elements using the space of piecewise linear functions. The goal of this paper is to study a special version of FETI-DP preconditioner, called deluxe, for the resulting discrete system of this discretization. The deluxe version for continuous FE discretization is considered in [1], for standard FETI-DP methods for composite DG method, see [4], for full DG, see [4], and for conforming FEM, see the book [5].

In the paper, we investigate the basic transmission problems arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body ( Ω + ) is embedded in an unbounded fluid domain ( Ω − ). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω + , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω − , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.

On the variational iteration method and other iterative techniques for nonlinear differential equations

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form . We prove that when the Cauchy problem in has no local-in-time weak solution for a large class of initial functions, while when there is a local weak solution.

A one-stage meshless method is devised for solving Cauchy boundary value problems of elliptic partial differential equations (PDEs) with variable coefficients. The main idea is to approximate an unknown solution using a linear combination of fundamental solutions and radial basis functions. Compared with the two-stage method of particular solution, the proposed method can deal with more general elliptic PDEs with variable coefficients. Several numerical results in both two- and three-dimensional space show that our proposed method is accurate and effective.

The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential equations. Such models are referred to as bilevel optimal control problems. Here, we first review some different features of bilevel optimal control including important applications, existence results, solution approaches, and optimality conditions. Afterwards, we focus on a specific problem class where parameters appearing in the objective functional of an optimal control problem of partial differential equations have to be reconstructed. After verifying the existence of solutions, necessary optimality conditions are derived by exploiting the optimal value function of the underlying parametric optimal control problem in the context of a relaxation approach.

We consider a finite-difference approximation to the Cauchy problem for a firstorder hyperbolic partial differential equation using different mesh spacings in different portions of the domain. By reformulating our problem as a difference approximation to an initial-boundary value problem, we are able to use the theory of H. O. Kreiss and S. Osher to analyze the L 2 {L_2} stability of our scheme.