Dirichlet–Neumann problem for high-order partial differential equations with constant coefficients

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

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.

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, 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 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.

The initial-boundary value problem for higher-order delay partial differential equations is considered. We study the existence of its solution and also propose a method for finding approximate solutions. We are established a sufficient conditions for the existence and uniqueness of the solution to the problem under consideration. Introducing new unknown functions, we reduce the considered problem to an equivalent problem consisting of a family of problems for delay differential equation with functional parameters and integral relations. An algorithm for finding an approximate solution to the problem under study is proposed and its convergence is proved. Sufficient conditions for the existence of a unique solution to an equivalent problem with parameters are established. The conditions for the unique solvability of the initial-boundary value problem for delay partial differential equations are obtained in terms of the initial data. Unique solvability to the initial-boundary value problem for higher-order delay partial differential equations is interconnected with unique solvability to the family of problems with parameters for delay differential equations.

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.

Integral Equations for Nonlinear Problems in Partial Differential Equations

A new method is introduced for studying boundary-value problems for a class of linear partial differential equations (PDEs) with variable coefficients. This method is based on ideas recently introduced by the author for the study of boundary-value problems for PDEs with constant coefficients. As illustrative examples the following boundary-value problems are solved: a Dirichlet and a Neumann problem on the half line for the time-dependent Schrödinger equation with a space-dependent potential; and a Poincaré problem on the quarter plane for a variable coefficient generalization of the Laplace equation.

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

Convergence analysis of an iterative scheme for solving initial value problem for multidimensional partial differential equations

In this work, we investigate a unique solvability of a direct and inverse source problem for a time-fractional partial differential equation with the Caputo and Bessel operators. Using spectral expansion method, we give explicit forms of solutions to formulated problems in terms of multinomial Mittag-Leffler and first kind Bessel functions.

The usual solution methods of discretised partial differential equations are based exclusively on matrix-vector multiplications as basis operation. On the one hand, this is supported by the use of sparse matrices (cf. §1.3.2.5); on the other hand, one tries to apply fast iterative methods (e.g., the multigrid method [124], [119, §12]) whose basic steps are matrix-vector multiplications. Krylov methods are based on the same concept. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation1 and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. The $$\mathcal{H}$$ -matrix arithmetic allows the solution of these matrix equations efficiently. Here, the use of hierarchical matrix operations and matrix-valued functions is only one part of the solution method. Another essential fact is that the solution $$ X \in \mathbb{R}^{I \times I}(n==\!\!\!\!\!/ \!\!\!\!/I)$$ can be replaced by an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ . If one considers the equation $$f(X)\;=\;0$$ as a system of n 2 equation for the n 2 components of X, even an optimal solution method has complexity $$\mathcal{O}(n^2)$$ , since this is linear complexity in a number of unknowns (cf. Remark 1.1). Using traditional techniques, the solution of large-scale matrix equations is not feasible. Only an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ with $$\mathcal{O}(n\; \mathrm{log}^*n)$$ data instead of n 2 admits a solution with a cost almost linear with respect to n. Section 15.1 introduces Lyapunov and Sylvester equations and discusses their solution. In Section 15.2 we consider quadratic Riccati equation. An interesting approach uses the matrix version of the sign function from §14.1.1. General nonlinear matrix equations may be solved iteratively by Newton’s method or related methods (cf. Section 15.3). As an example, computing the square root of a positive definite matrix is described in §15.3.1. The influence of the truncation error introduced by H-matrix arithmetic is investigated in Section 15.3.

The initial‐boundary value problem for partial differential equations of higher‐order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right‐hand side of the equation are established.

The initial-boundary value problem for higher-order partial differential equations is considered. We study the existence of its classical solutions, and also propose a method for finding approximate solutions. Paper establishes sufficient conditions for the existence and uniqueness of the classical solution of the problem under consideration. Introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for second-order hyperbolic equations with functional parameters and integral relations. An algorithm for finding an approximate solution to the problem under study is proposed and its convergence is proved. Sufficient conditions for the existence of a unique solution to an equivalent problem with parameters are established. The conditions for the unique solvability of the initial-boundary value problem for higher-order partial differential equations are obtained in terms of the initial data. Solvability of the initial-boundary value problem for higher-order partial differential equations is connected with solvability of the nonlocal problem for second-order hyperbolic equations.