Multiple solutions of nonlinear boundary value problems for two-dimensional differential systems

Positive solutions of nonlinear singular third-order two-point boundary value problem

Positive solutions to a multi-point boundary value problem with delay

We have obtained the conditions of existence and a scheme for constructing solutions of a weakly nonlinear boundary value problem for a degenerate differential-algebraic system in the noncritical case. The boundary condition is determined by a weakly nonlinear vector functional. The linear part of the problem is a linear boundary value problem for a degenerate differential-algebraic system. Linear differential-algebraic boundary value problems have been studied in monographs by S. Campbell, J.R. Magnus, A.M. Samoilenko and V.P. Yakovets. In the works of A.M. Samoilenko and O.A. Boichuk, using the central canonical form, the necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic boundary value problems were obtained. We have obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems without using the central canonical form, which allows us to study the solvability of differential-algebraic boundary value problems that depend on arbitrary continuous functions. This approach significantly varies the classification of nonlinear differential-algebraic boundary value problems in critical and noncritical cases. Our formulation of the weakly nonlinear differential-algebraic boundary value problem generalises the boundary value problems studied in the works of Yu.O. Mitropolsky, A.M. Samoilenko, and O.A. Boichuk. The case when a differential-algebraic system is not solvable with respect to the derivative is considered, and substitutions of the unknown are proposed. It leads the original system to a nonlinear differential-algebraic system solvable with respect to the derivative. Finally, we present an example of a nonlinear differential-algebraic antiperiodic boundary value problem for a Riccati-type equation, which demonstrates the constructiveness of the obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems. The obtained results can be transferred to the problems of finding conditions for the existence and schemes for constructing solutions of nonlinear degenerate differential-algebraic boundary value problems in critical cases, as well as to the problems of finding conditions for the stability of such solutions.

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

Solution of a nonlinear boundary value problem for a second-order ordinary differential equation by the method of finite differences

Numerical solution of two-point boundary value problems

The article proposes original solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem. And we use the matrix pseudo-inversion technique of Moore-Penrose. The posed problem in the article continues the study of conditions of solvability and schemes for finding solutions of the nonlinear Noetherian boundary-value problems given in the monographs by A. Poincare, A.M. Lyapunov, I.G. Malkin, J. Hale, Yu.A. Ryabov, A.M. Samoylenko, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina and A.A. Boychuk. We studied a general case, when a linear bounded operator corresponding to the homogeneous part of the linear Noetherian differential-algebraic boundary value problem has no inverse. Sufficient conditions for reducibility of the differential algebraic equation to the system uniting a differential and algebraic equation are found. Thus, the differential-algebraic boundary value problem is reduced to the nonlinear Noetherian boundary value problem for the system of ordinary differential equations. We studied the case of the presence of simple roots of the equation for generating amplitudes. Constructive necessary and sufficient conditions of existence were obtained to find solutions to the problem in the critical case, and the converging iterative scheme was constructed. The proposed solvability conditions, and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem are illustrated in detail by the example from the nonlinear Noetherian differential-algebraic boundary value problem for Duffing type equations. For control of the rate of the iterative scheme convergence to the exact solution of the differential-algebraic boundary value problem for the Duffing type equation, we used the residuals of the obtained approximations in the Duffing type equation in the space of continuous functions.

B-spline collocation for solution of two-point boundary value problems

Solution of nonlinear boundary value problems — I method of third order convergence for solution of nonlinear two-point boundary value problems single second order equation

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

The theorem of the existence and uniqueness of the solution of the boundary value problem for the equation in partial derivatives of the fourth order with variable coefficients containing the product of the mixed parabolic-hyperbolic operator and the differential operator of the oscillation string with discontinuous conditions of gluing in the pentagon to the plane is proved. By the method of reducing the order of equations, the solvability of the boundary value problem is reduced to the solution of the Tricomi problem for the mixed parabola-hyperbolic equation with variable coefficients and discontinuous gluing conditions. Solving this problem is reduced to the solution of Fredholm’s integral equation of the second order relative to the trace of the derivative function on y along the line of variation of the equation type. In the hyperbolic part of the domain, the representation of the solution of the problem for the hyperbolic equation with the smallest terms was obtained by using the Riemann function method. In the parabolic part of the domain, the solution of the first boundary value problem for the parabolic equation with the smallest terms is obtained by the method of successive approximations and the Green’s function. As a result, the solution of the problem is realized by the method of solving the Gursa problem and the first boundary value problem for the equation of string oscillation.

The main objective of this paper is to develop new spectral solutions for linear and nonlinear fifth-order boundary value problems. For this aim, a new operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The key idea for obtaining the suggested spectral solutions is to convert the fifth-order linear and nonlinear boundary value problems into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. A theoretical error bound is derived and the convergence analysis of the proposed expansion is proved. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the suggested algorithms.

The main goal of this study is to find the solution of initial boundary value problem for the one-dimensional time and space-fractional diffusion equation which is a very intriguing topic for many researchers. With the aim of newly defined inner product, which is the main contribution of this study, the analytic solution of the boundary value problem is obtained. The time and space-fractional derivatives are defined in the Caputo sense which is more suitable than Riemann-Liouville sense. We apply the separation of variables method to reduce the problem to two separate fractional ODEs. The generalized solution is constructed/formed in the form of a Fourier series with respect to the eigenfunctions of a certain eigenvalue problem. In order to obtain the coefficients of the Fourier series for the solution, we define a new inner product which is the key point of study.

The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domainΩand if solutions of the quasilinear problem are inΩ, then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

We prove the existence of solutions of a boundary value problem for a system of five nonlinear second-order partial differential equations with given nonlinear boundary conditions, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with free edges in the Timoshenko shear model referred to isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.