Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

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.

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.

This study proposes the Chebyshev Wavelet Colocation method for solving a class of rth-order Boundary-Value Problems (BVPs) with nonlocal boundary conditions. This method is an extension of the Chebyshev wavelet method to the linear and nonlinear BVPs with a class of nonlocal boundary conditions. In this study, the method is tested on second and fourth-order BVPs and approximate solutions are compared with the existing methods in the literature and analytical solutions. The proposed method has promising results in terms of the accuracy.

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.

This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We will prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.

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.

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

In this paper, parametric quintic spline method is presented to solve a linear special case sixth order two point boundary value problems with two different cases of boundary conditions. The method presented in this paper has been shown to be second and fourth order accurate. Boundary equations are derived for both the cases of boundary conditions. Convergence analysis of these methods are discussed. The presented method is tested on four numerical examples of linear sixth order boundary value problems. Comparison is made to show the practical usefulness of the presented method.

In this paper we study the nonlinear fourth order differentiai equation uiv±F(u, v)u=0, where F is a pssitive monotone function of u. Asymptotic behaviour of certain solutions are treated in sections 2 and 4 while a two point boundary value problem is studied in section 3.

Numerical solution of sixth order boundary value problems with sixth degree B-spline functions

In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions

In this study, we applied the homotopy perturbation (HP) method for solving linear and nonlinear fourth-order boundary value problems. The analytical results of the boundary value problems have been obtained in terms of a convergent series with easily computable components. Comparisons between the results of the HP method and the analytical solution showed that this method gives very precise results with a few terms. In the implied HP method, some unknown parameters in the initial guess are introduced, which are identified after applying boundary conditions. This improvement results in higher accuracy.

In this paper we consider the following class of four point boundary value problems—y"(x) = f (x, y), 0 less than x lessthan 1, y'(0) = 0, y(1) = 1y(1) + 2)7(2)’where 1, 2  0 lesstahn 1, 2 less than 1, and f (x, y), is continuous in one sided Lipschitz in y. We propose a monotone iterative scheme and show that under some sufficient conditions this scheme generates sequences which converges uniformly to solution of the nonlinear multipint boundary value problem.

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