Numerical solution of special 12th-order boundary value problems using differential transform method

The non-polynomial spline technique is used for the numerical solution of eighth-order linear special case boundary value problems. The method presented in this paper has also been proven to be second-order convergent. To compare the method developed in this paper with those developed by Inc and Evans, and Siddiqi and Twizell, two examples are considered and it is observed that our method is more efficient.

A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

On the coefficients of differentiated expansions and derivatives of chebyshev polynomials of the third and fourth kinds

Background Second-order linear boundary value problems arise in many applications of science and engineering and are commonly treated by reducing them to first-order systems, which increases the computational cost. Direct high-order methods that preserve the second-order structure are therefore of interest. Methods This study proposes fixed-step, high-order diagonally implicit Runge–Kutta–Nyström (DIRKN) methods for the direct numerical solution of special second-order linear boundary value problems. Two schemes, DIRKN(3,4) and DIRKN(4,5), are constructed and combined with a linear shooting technique to handle both Dirichlet and Neumann boundary conditions without transforming the problem into a first-order system. Results The performance of the proposed methods is evaluated using benchmark test problems and application models, including rod heat conduction and reaction–diffusion equations. Numerical results demonstrate that both schemes achieve high accuracy and good efficiency, with DIRKN(4,5) generally producing smaller errors for a comparable number of function evaluations. A linear stability analysis shows absolute stability intervals of (−1.96, 0) for DIRKN(3,4) and (−7.24, 0) for DIRKN(4,5). Conclusions The proposed DIRKN methods provide accurate and efficient solvers for second-order linear boundary value problems with different types of boundary conditions, while preserving the original problem structure and reducing computational cost.

Septic spline solutions of sixth-order boundary value problems

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.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

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

Representations of the solutions of two-dimensional boundary value problems of coupled electroelasticity

On the state of stress of an orthotropic elastic plane region in the neighborhood of a corner point

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.

This chapter is dedicated to the case of neutral type linear systems with distributed delay. The structure of quadratic functionals that have prescribed time derivatives along the solutions of such a system is defined, and the corresponding Lyapunov matrices are introduced. A system of matrix equations that defines Lyapunov matrices is given. It is proven that under some conditions this system admits a unique solution. A general class of system with distributed delay for which Lyapunov matrices are solutions of special standard boundary value problems for an auxiliary system of linear matrix ordinary differential equations is presented. Complete type functionals are defined. It is shown that these functionals can be presented in a special form that is more convenient for the computation of lower and upper bounds for the functionals.

Numerical solution of two-point boundary value problems

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.

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