Order BVPs Research Articles

A novel fixed point approach based on Green’s function for solution of fourth order BVPs

A novel fixed point approach based on Green’s function for solution of fourth order BVPs

Infinite system of nonlinear tempered fractional order BVPs in tempered sequence spaces

This paper deals with the existence results of the infinite system of tempered fractional BVPs Drϱ,λ0Rzj(r)+ψj(r,z(r))=0,0<r<1,zj(0)=0,0RDrm,λzj(0)=0,b1zj(1)+b20RDrm,λzj(1)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned}& {}^{\\mathtt{R}}_{0}\\mathrm{D}_{\\mathrm{r}}^{\\varrho , \\uplambda} \\mathtt{z}_{\\mathtt{j}}(\\mathrm{r})+\\psi _{\\mathtt{j}}\\bigl(\\mathrm{r}, \\mathtt{z}(\\mathrm{r})\\bigr)=0,\\quad 0< \\mathrm{r}< 1, \\\\& \\mathtt{z}_{\\mathtt{j}}(0)=0,\\qquad {}^{\\mathtt{R}}_{0} \\mathrm{D}_{ \\mathrm{r}}^{\\mathtt{m}, \\uplambda} \\mathtt{z}_{\\mathtt{j}}(0)=0, \\\\& \\mathtt{b}_{1} \\mathtt{z}_{\\mathtt{j}}(1)+\\mathtt{b}_{2} {}^{ \\mathtt{R}}_{0}\\mathrm{D}_{\\mathrm{r}}^{\\mathtt{m}, \\uplambda} \\mathtt{z}_{\\mathtt{j}}(1)=0, \\end{aligned}$$ \\end{document} where j∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathtt{j}\\in \\mathbb{N}$\\end{document}, 2<ϱ≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2<\\varrho \\le 3$\\end{document}, 1<m≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1<\\mathtt{m}\\le 2$\\end{document}, by utilizing the Hausdorff measure of noncompactness and Meir–Keeler fixed point theorem in a tempered sequence space.

Solving coupled non-linear higher order BVPs using improved shooting method

Abstract The purpose of this work is to propose a new efficient initial guess algorithm rather than the conventional Newton method to meet the adjoint terminal conditions, rapidly. Thus, the focus of this paper is to use better algorithms for obtaining refined initial guesses with shooting method to solve the coupled boundary value problems. The boundary value problem is formulated as a system of equations i.e. initial value problems with one unknown initial condition. The efficiency and accuracy of the Shooting method is enhanced by improving the initial guess and then solving the problem iteratively.

Solving 2nd order BVPs in planar irregular domains

This paper presents a pseudo-spectral method to solve PDEs in univocally describable irregular planar domains in polar coordinates, using a simple relationship that transforms the problem for the classic case in the unit circle, equipped with an orthogonal bivariate basis and complete in Sobolev sense. Given its generality, it applies to elliptical, parabolic or hyperbolic PDEs in Robin’s condition. The choice of Radau points applied here allows the exclusion of the singularity in the origin and a posteriori error estimate also here developed provides us sharp error bounds of the numerical solution.

On The Solution And Stability Analysis Of 6thorder Bvp By Special Multistep Methods

An effective method of solving a 6th order nonlinear BVP based on the numerical differentiation is presented in this article. The Uniqueness and existence properties of the solution are established. A more accurate and reliable process is derived to know the solution of 6th order BVPs. The procedure is verified on nonlinear problem. The solutions are matched with exact solutions and absolute errors obtained in this method are compared with that of Galerkin method.

Existence of positive solutions for half-linear fractional order BVPs by application of mixed monotone operators

In this paper we developed tripled fixed point theorems of ternary operator on partially ordered metric spaces. As an application we established existence of positive solutions for half-linear fractional order boundary value problem.

A note on the convergence of fuzzy transformed finite difference methods

In this paper, we develop numerical methods based on the fuzzy transform methods (FTMs). In this approach we apply fuzzy transforms on discrete version of the derivatives and use it to derive FTMs. We also establish convergence of the proposed FTMs. To test the efficiency of the proposed FTMs, we apply the FTM schemes on the second order nonlinear singular boundary value problems and fourth order BVPs. We allow the source term of the differential equation to have jump discontinuity and study the effect of jump on FTMs and finite difference methods. The work shows that FTMs are better for both class of BVPs considered in this paper, having singularity, nonlinearity and jump discontinuity.

Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

In the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

Countably many positive solutions for a coupled system of fractional order BVPS

In this paper we establish the existence of countably infinitely many positive solutions for coupled system of two point Riemann-Liouville integral fractional order boundary value problems. The main results are established by using the fixed point index theory in cone on a Banach space with increasing homeomorphism and positive homomorphism.

Numerical Solution of the Boundary Value Problems Arising in Magnetic Fields and Cylindrical Shells

This paper is devoted to the study of the Cubic B-splines to find the numerical solution of linear and non-linear 8th order BVPs that arises in the study of astrophysics, magnetic fields, astronomy, beam theory, cylindrical shells, hydrodynamics and hydro-magnetic stability, engineering, applied physics, fluid dynamics, and applied mathematics. The recommended method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 8th order BVPs using Cubic-B spline but it also describes the estimated derivatives of 1st order to 8th order of the analytic solution. The strategy is effectively applied to numerical examples and the outcomes are compared with the existing results. The method proposed in this paper provides better approximations to the exact solution.

Solvability of positive solutions for a systems of nonlinear fractional order BVPs with p-Laplacian

AbstractValues of the parameters λ and μ are determined for which there exist positive solutions for the system of fractional order withp-Laplacian two-point boundary value problems. The well-known Guo–Krasnosel’skii fixed point theorem is applied.

Numerical solution of third order boundary value problems using one-step hybrid block method

Abstract Numerical hybrid block methods have been thought to be an appropriate method for solving ordinary differential equation. Therefore, this paper introduces a one-step hybrid block method of order five for directly solving third order boundary value problems. Power series of order seven is used for the development of the new method by using collocation and interpolation approach. The proposed method together with shooting technique will solve third order BVPs. Some numerical examples are involved to show the efficiency of the method. The numerical results clearly show that the developed method is able to generate good results when it is compared with existing method in terms of error. It has been noted, the new proposed method outperforms the existing methods in terms of accuracy.

A Block Hybrid Method for Non-linear Second Order Boundary Value Problems

In this paper, we propose a new method of hybrid type, for non-linear second order BVPs, to overcome some drawbacks of classic difference finite methods. An estimation of the local and global error is also given. Numerical tests on BVPs which is well known in the literature confirm the accuracy of the theoretical results.

Necessary and sufficient conditions for the existence of non-constant solutions generated by impulses of second order BVPs with convex potential

Necessary and sufficient conditions for the existence of non-constant solutions generated by impulses of second order BVPs with convex potential

Cubic Lidstone-Spline for numerical solution of BVPs

A new collocation method based on cubic Lidstone Splines is introduced for solving second order BVPs. It derives directly from piecewise Lidstone polynomials of degree 3 by requiring the continuity of the first derivative at the nodal points. For equally spaced nodes it reduces to a classical finite difference method of second order. The estimation of local and global error is given. Finally we solve some numerical problems and we compare the results with those of other methods.

A novel fixed point scheme: Proper setting of variational iteration method for BVPs

The aim of this article is to introduce a novel approach based on embedding Green’s function into a tailored linear integral operator then apply a well-known fixed point iterative scheme, such as Picard’s and Mann’s, for the numerical solution of initial and boundary value problems (IVPs and BVPs). The strategy provides insight into the proper setting of the variational iteration method (VIM) for BVPs. A convergence analysis of the method is included. A number of examples consisting of second and third order BVPs are presented to demonstrate the efficiency and applicability of the scheme.

Numerical solution of third order BVPs by using non-polynomial spline with FDM

Numerical solution of third order BVPs by using non-polynomial spline with FDM

Numerical Solutions of Fifth and Sixth Order Nonlinear Boundary Value Problems by Daftardar Jafari Method

Fifth and sixth order boundary value problems are solved using Daftardar Jafari method (DJM). DJM is introduced by Daftardar-Gejji and Jafari (2006). The approach provides the solution in the form of a rapidly convergent series. The comparison among Daftardar Jafari method (DJM), Adomian decomposition method (ADM), homotopy perturbation method (HPM), variation iteration method (VIM), and the iterative method (ITM) are displayed in table form which shows the efficiency of DJM for the solution of fifth and sixth order BVPs.

Numerical Solutions of Sixth Order Linear and Nonlinear Boundary Value Problems

The aim of paper is to find the numerical solutions of sixth order linear and nonlinear differential equations with two point boundary conditions. The well known Galerkin method with Bernstein and modified Legendre polynomials as basis functions is exploited. In this method, the basis functions are transformed into a new set of basis functions, which satisfy the homogeneous form of Dirichlet boundary conditions. A rigorous matrix formulation is derived for solving the sixth order BVPs. Several numerical examples are considered to verify the efficiency and implementation of the proposed method. The numerical results are compared with both the exact solutions and the results of the other methods available in the literature. The comparison shows that the performance of the present method is more efficient and yields better results.

Second order BVPs with state dependent impulses via lower and upper functions

AbstractThe paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.