A New Numerical Method Combining the Homotopy Perturbation Method with Multiscale Functions for Second-Order Nonlinear Boundary Value Problems

Error analysis of modified Runge–Kutta–Nyström methods for nonlinear second-order delay boundary value problems

PurposeThe purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.Design/methodology/approachThe authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.FindingsWhen the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.Research limitations/implicationsBasically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.Practical implicationsStarting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.Originality/valueThrough the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.

The localized method of fundamental solutions for 2D and 3D second-order nonlinear boundary value problems

About the existence and uniqueness of solutions for some second-order nonlinear BVPs

In this paper, we introduce an improved variational iteration method (VIM) for nonlinear second-order boundary value problems. The main advantage of this modification is that it can avoid additional computation in determining the unknown parameters in initial approximation when solving boundary value problems using the conventional VIM. Also, iterative sequences obtained using the improved VIM do satisfy the boundary conditions while iterative sequences obtained using conventional VIM may not, in general, satisfy the boundary conditions. Numerical results reveal that the improved method is accurate and efficient.

Existence and approximation of solutions for nonlinear second-order four-point boundary value problems

Based upon He's homotopy perturbation and variational iteration methods, we present a method for approximate solutions of nonlinear second-order multi-point boundary value problems (BVPs) in bridge design. Two numerical experiments are carried out to demonstrate the efficiency of the present method. The results reveal that the proposed method is very effective for second-order multi-point BVPs in bridge design.

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem

Successive iteration and positive solution for nonlinear second-order three-point boundary value problems

Under suitable conditions on, the nonlinear second-order m-point boundary value problem has at least one positive solution. In this paper, the authors examine the positive solutions of nonlinear second-order m-point boundary value problem.

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

It is difficult to exactly and automatically satisfy nonseparable multipoint boundary conditions by numerical methods. With this in mind, we develop a novel algorithm to find solution for a second-order nonlinear boundary value problem (BVP), which automatically satisfies the multipoint boundary conditions prescribed. A novel concept of boundary shape function (BSF) is introduced, whose existence is proven, and it can satisfy the multipoint boundary conditions a priori. In the BSF, there exists a free function, from which we can develop an iterative algorithm by letting the BSF be the solution of the BVP and the free function be another variable. Hence, the multipoint nonlinear BVP is properly transformed to an initial value problem for the new variable, whose initial conditions are given arbitrarily. The BSF method (BSFM) can find very accurate solution through a few iterations.

Owing to their enhanced expressive capability in handling uncertainty, q-rung orthopair fuzzy (q-ROF) metric spaces provide a flexible framework for analyzing nonlinear problems under vagueness. In this article, we introduce a new class of q-ROF fuzzy contractive mappings and establish a corresponding fixed point theorem in complete q-ROF metric spaces. The uniqueness and existence of the fixed point are proved under suitable contractive conditions. To demonstrate the practical relevance of the theoretical results, an application to a second-order nonlinear boundary value problem with homogeneous boundary conditions is presented. By transforming the boundary value problem into an equivalent integral equation and employing the proposed fixed point theorem, the existence of a unique continuous solution is obtained. Illustrative examples are also included to validate the applicability and effectiveness of the proposed approach.

In the numerical integration of the second-order nonlinear boundary value problem (BVP), the right boundary condition plays the role as a target equation, which is solved either by the half-interval method (HIM) or a new derivative-free Newton method (DFNM) to be presented in the paper. With the help of a boundary shape function, we can transform the BVP to an initial value problem (IVP) for a new variable. The terminal value of the new variable is expressed as a function of the missing initial value of the original variable, which is determined through a few integrations of the IVP to match the target equation. In the new boundary shape function method (NBSFM), we solve the target equation to obtain a highly accurate missing initial value, and then compute a precise solution. The DFNM can find more accurate left boundary values, whose performance is superior than HIM. Apparently, DFNM converges faster than HIM. Then, we modify the Lie-group shooting method and combine it to the BSFM for solving the nonlinear BVP with Robin boundary conditions. Numerical examples are examined, which assure that the proposed methods together with DFNM can successfully solve the nonlinear BVPs with high accuracy.