Nonlinear Third-order Boundary Value Problem Research Articles

Galerkin-Bernstein Approximations for the System of Third-Order Nonlinear Boundary Value Problems

Galerkin-Bernstein Approximations for the System of Third-Order Nonlinear Boundary Value Problems

Legendre wavelet operational matrix method for the analysis of thermal radiation effect on natural convection of a vertical plate embedded in a saturated porous medium

In this paper, the heat transfer characteristics of natural convection around a vertical flat surface embedded in a saturated porous medium are studied by taking into account the thermal radiation effect. With the aid of the similarity transformations, the governing equations of the problem are reduced to the system of ordinary differential equations over the infinite domain. Moreover, the Legendre wavelet operational matrix method (LWOMM) is hired to find the numerical solution to the considered problem. Initially, to ensure the efficiency of the proposed strategy, a third-order nonlinear boundary value problem having the exact solution is considered a test problem. The numerical findings elucidate that the solutions obtained by the LWOMM are in good agreement with the exact solution as compared with the solution obtained by the Haar wavelet operational matrix method, with the consumption of less computational CPU time. Also, the temperature and velocity profiles are discussed graphically.

On Novel Spline Function Method for Solving Third-Order Boundary Value Problems

In this paper, a numerical method is suggested for solving general a nonlinear third order boundary value problem (BVP). In this method, the given nonlinear third-order BVP will be transformed into two third-order initial value problems (IVPs), then spline function approximations are applied to both two IVP for finding the Spline solution and its derivatives up to third order of the given BVP. The study shows that the spline solution of the BVP is existent and unique, and the convergence order of the spline method is fourth with a local truncation error . The presented algorithm is designed for solving a general BVP, where it is applied to some types of nonlinear third-order differential equations. Comparisons of the results obtained by spline method with other methods show the efficiency and highly accurate of the proposed method.

A faster iterative method for solving nonlinear third-order BVPs based on Green’s function

In this article, we propose an iterative method, called the GA iterative method, to approximate the fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. Further, we obtain some convergence results of the new iterative method. Also, we provide a nontrivial example of a generalized α-nonexpansive mapping and with the example, we carry out a numeral experiment to show that our new iterative algorithm is more efficient than some existing iterative methods. Again, we present an interesting strategy based on the GA iterative method to solve nonlinear third-order boundary value problems (BVPs). For this, we derive a sequence named the GA–Green iterative method and show that the sequence converges strongly to the fixed point of an integral operator. Finally, the approximation of the solution for a nonlinear integrodifferential equation via our new iterative method is considered. We present some illustrative examples to validate our main results in the application sections of this article. Our results are a generalization and an extension of several prominent results of many well-known authors in the literature.

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

In recent times, numerical approximation of 3rd-order boundary value problems (BVPs) has attracted great attention due to its wide applications in solving problems arising from sciences and engineering. Hence, A higher-order block method is constructed for the direct solution of 3rd-order linear and non-linear BVPs. The approach of interpolation and collocation is adopted in the derivation. Power series approximate solution is interpolated at the points required to suitably handle both linear and non-linear third-order BVPs while the collocation was done at all the multiderivative points. The three sets of discrete schemes together with their first, and second derivatives formed the required higher-order block method (HBM) which is applied to standard third-order BVPs. The HBM is self-starting since it doesn’t need any separate predictor or starting values. The investigation of the convergence analysis of the HBM is completely examined and discussed. The improving tactics are fully considered and discussed which resulted in better performance of the HBM. Three numerical examples were presented to show the performance and the strength of the HBM over other numerical methods. The comparison of the HBM errors and other existing work in the literature was also shown in curves.

Qualitative analysis of magnetohydrodynamics stagnation point flow with a novel numerical approach through Shifted Chebyshev collocation method

This paper investigates the third-order nonlinear boundary value problem, resulting from the exact reduction of the Navier-Stokes equation caused by the magnetohydrodynamics boundary layer flow near a stagnation point on a rough plate. The governing partial differential equations are transformed into a nonlinear ordinary differential equation and partial slip boundary conditions by an appropriate similarity transformation. In this previous work, the boundary value problem (BVP) was investigated numerically, and a lot of speculations regarding the existence and behavior of the solutions were carried out. The primary objective of this article is to verify these speculations mathematically. In this work, we have proved that there is a unique solution for all parameters values, and further, the solution has monotonic increasing first derivative. Moreover, the resulting nonlinear boundary value problem is solved by shifted Chebyshev collocation method. We compare the present numerical results with the previous results for the particular physical parameters, concluding that the results are highly accurate. The velocity profiles and streamlines are also plotted to address the significance of the parameters. Our manuscript is a judicial mix between mathematical and numerical methods.

Positive solutions of nonlinear third-order boundary value problems involving Stieltjes integral conditions

Positive solutions of nonlinear third-order boundary value problems involving Stieltjes integral conditions

Solving nonlinear third-order boundary value problems based-on boundary shape functions

Abstract For a third-order nonlinear boundary value problem (BVP), we develop two novel methods to find the solutions, satisfying boundary conditions automatically. A boundary shape function (BSF) is created to automatically satisfy the boundary conditions, which is then employed to develop new numerical algorithms by adopting two different roles of the free function in the BSF. In the first type algorithm, we let the BSF be the solution of the BVP and the free function be a new variable. In doing so, the nonlinear BVP is certainly and exactly transformed to an initial value problem for the new variable with its terminal values as unknown parameters, whereas the initial conditions are given. In the second type algorithm, let the free functions be a set of complete basis functions and the corresponding boundary shape functions be the new bases. Since the solution already satisfies the boundary conditions automatically, we can apply a simple collocation technique inside the domain to determine the expansion coefficients and then the solution is obtained. For the general higher-order boundary conditions, the BSF method (BSFM) can easily and quickly find a very accurate solution. Resorting on the BSFM, the existence of solution is proved, under the Lipschitz condition for the ordinary differential equation system of the new variable. Numerical examples, including the singularly perturbed ones, confirm the high performance of the BSF-based numerical algorithms.

A two-step hybrid block method with fourth derivatives for solving third-order boundary value problems

This manuscript proposes an implicit two-step hybrid block method which incorporates fourth derivatives, for solving linear and non-linear third-order boundary value problems in ODEs. The derivation of the present method is based on collocation and interpolation techniques, and the convergence analysis of the new strategy is proved to be seventh-order convergent. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. Numerical experiments are studied to show the performance and viability of the proposed approach. The numerical results demonstrated that the new technique gives accurate approximations, which are better than some existing strategies in the available literature and also found to be in good agreement with known analytical solutions.

A class of nonlinear third-order boundary value problem with integral condition at resonance

A class of nonlinear third-order boundary value problem with integral condition at resonance

Simple numerical methods of second- and third-order convergence for solving a fully third-order nonlinear boundary value problem

In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both continuous and discrete levels. We prove that the discrete methods are of second order and third accuracy due to the use of appropriate formulas for numerical integration and obtain estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method.

A new class of A-stable numerical techniques for ordinary differential equations: Application to boundary-layer flow

The present attempt is made to propose a new class of numerical techniques for finding numerical solutions of ODE. The proposed numerical techniques are based on interpolation of a polynomial. Currently constructed numerical techniques use the additional information(s) of derivative(s) on particular grid point(s). The advantage of the presently proposed numerical techniques is that these techniques are implemented in one step and can provide highly accurate solution and can be constructed on fewer amounts of grid points but has the disadvantage of finding derivative(s). It is to be noted that the high order techniques can be constructed using just two grid points. Presently proposed fourth order technique is A-stable but not L-stable. The order and maximum absolute error are found for a fourth order technique. The fourth order technique is employed to solve the Darcy-Forchheimer fluid-flow problem which is transformed further to a third-order non-linear boundary value problem on the semi-infinite domain.

Positive solutions of nonlinear third-order boundary value problem with integral boundary conditions

Positive solutions of nonlinear third-order boundary value problem with integral boundary conditions

障碍带条件下两类三阶非线性边值问题解的存在性

障碍带条件下两类三阶非线性边值问题解的存在性

Application of the mixed monotone operator to a nonlinear third-order boundary value problem

In this paper, we use the mixed monotone operator method to study the following nonlinear boundary value problem $$\begin{aligned} \left\{ \begin{array}{ll} -u'''(t)=f(t,u(t),u(\varrho t))+g(t,u(t)),&{}\quad 0<t<1,\,\varrho \in (0,1), \\ u(0)=u''(0)=u(1)=0. \end{array} \right. \end{aligned}$$ An example is provided to illustrate the results.

Solutions of the Blasius and MHD Falkner-Skan boundary-layer equations by modified rational Bernoulli functions

PurposeThis paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.Design/methodology/approachThe approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.FindingsThe method is computationally very attractive and gives very accurate results.Originality/valueMany problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.

An approximate solution of the MHD flows of UCM fluids over porous stretching sheets by rational Legendre collocation method

Purpose The purpose of this paper is to develop an efficient method for solving the magneto-hydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous isothermal stretching sheet. Design/methodology/approach The paper applied a collocation approach based on rational Legendre functions for solving the third-order non-linear boundary value problem, describing the MHD boundary layer flow of an UCM fluid over a porous isothermal stretching sheet. This method solves the problem on the semi-infinite domain without transforming domain of the problem to a finite domain. Findings This approach reduces the solution of a problem to the solution of a system of algebraic equations. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem. The authors also compare the results of this work with some recent results and show that the new method is efficient and applicable. Originality/value The method solves this problem without use of discrete variables and linearization or small perturbation. Also it was confirmed by the theorem and figure of absolute coefficients that this approach has exponentially convergence rate.

A Subdivision Based Iterative Collocation Algorithm for Nonlinear Third-Order Boundary Value Problems

We construct an algorithm for the numerical solution of nonlinear third-order boundary value problems. This algorithm is based on eight-point binary subdivision scheme. Proposed algorithm is stable and convergent and gives more accurate results than fourth-degree B-spline algorithm.

Multiple positive solutions of nonlinear third-order boundary value problems with integral boundary conditions on time scales

In this paper, we consider a class of nonlinear third-order boundary value problems with integral boundary conditions on time scales. By applying a generalization of the Leggett-Williams fixed point theorem, we establish the existence of at least three positive solutions. The discussed problem involves both an increasing and positive homomorphism, which generalizes the p-Laplacian operator. As an application, we give an example to illustrate our results.

Some Algorithms for Solving Third-Order Boundary Value Problems Using Novel Operational Matrices of Generalized Jacobi Polynomials

The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.