Nonlinear BVPs Research Articles

A modified fixed point iterative scheme based on Green’s function with application to BVPs

This paper presents a new faster iteration approach that is based upon the well-known three-step fixed point iteration scheme called SP-iterative scheme for a broad and larger class of non-linear BVPs. The scheme is constructed using Green’s function based integral operator and the new modified scheme in the paper will be called SP–Green’s iterative scheme. We present a fixed point convergence theorem using some assumption on the sequences of scalars in our scheme. We also prove the weak [Formula: see text]-stability for our scheme and present a numerical example to support our scheme. We provide eventually applications of our scheme to a second-order BVP and a singular BVP of isothermal gas sphere.

Employing CNPS and CPS approaches to calculate numerical roots of ninth-order linear and nonlinear boundary value problems

The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of [Formula: see text] and [Formula: see text]. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods.

Existence results for singular strongly non-linear integro-differential BVPs on the half line

This work is devoted to the study of singular strongly non-linear integro-differential equations of the type (Φ(k(t)v′(t)))′=ft,∫0tv(s)ds,v(t),v′(t),a.e.onR0+:=[0,+∞[,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (\\Phi (k(t)v'(t)))'=f\\left( t,\\int _0^t v(s)\\, \ extrm{d}s,v(t),v'(t) \\right) , \ ext{ a.e. } \ ext{ on } {\\mathbb {R}}^{+}_0 := [0, + \\infty [, \\end{aligned}$$\\end{document}where f is a Carathéodory function, Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} is a strictly increasing homeomorphism, and k is a non-negative integrable function, which is allowed to vanish on a set of zero Lebesgue measure, such that 1/k∈Llocp(R0+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/k \\in L^p_\ extrm{loc}({\\mathbb {R}}^{+}_0)$$\\end{document} for a certain p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}. By considering a suitable set of assumptions, including a Nagumo–Wintner growth condition, we prove existence and non-existence results for boundary value problems associated with the non-linear integro-differential equation of our interest in the sub-critical regime on the real half line.

Existence results for a class of four point nonlinear singular BVP arising in thermal explosion in a spherical vessel

In this article, the following class of four-point singular non-linear boundary value problem (NLBVP) is considered which arises in thermal explosion in a spherical vessel −(s2y′(s))′=s2f(s,y,s2y′),s∈(0,1),y′(0)=0,y(1)=δ1y(η1)+δ2y(η2), where Ω=(0,1)×R2, f:Ω→R is continuous on Ω as well as satisfy Lipschitz condition with respect to y and y′ (one sided), δ1, δ2>0 are constants, and 0<η1≤η2<1. We provide an estimation of the region of existence of a solution of above singular NLBVP. We extend the theory of monotone iterative technique (MIT) which provides computable monotone sequences that converge to the solutions of the nonlinear four point BVPs.

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.

Existence results for singular nonlinear BVPs in the critical regime

We study the existence of solutions for a class of boundary value problems on the half line, associated to a third order ordinary differential equation of the type ( Φ ( k ( t , u ′ ( t ) ) u ″ ( t ) ) ) ′ ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.a. t ∈ R 0 + . The prototype for the operator Φ is the Φ -Laplacian; the function k is assumed to be continuous and it may vanish in a subset of zero Lebesgue measure, so that the problem can be singular; finally, f is a Carathéodory function satisfying a weak growth condition of Winter–Nagumo type. The approach we follow is based on fixed point techniques combined with the upper and lower solutions method.

Numerical Approximations of a Class of Nonlinear Second-Order Boundary Value Problems using Galerkin-Compact Finite Difference Method

In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results using the Galerkin weighted residual approach with Bernstein polynomials. The generation of residuals is brought on by the fact that our first approximation is computed using numerical methods. To minimize these residuals, we use the compact finite difference scheme of 4th-order convergence to solve the error differential equations in accordance with the error boundary conditions. We also introduce the formulation of the compact finite difference method of fourth-order convergence for the nonlinear BVPs. The improved approximations are produced by adding the error values derived from the approximations of the error differential equation to the weighted residual values. Numerical results are compared to the exact solutions and to the solutions available in the published literature to validate the proposed scheme, and high accuracy is achieved in all cases.

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

The significance of our work is to solve some second-order nonlinear boundary value problems. To do this, we take into account the equivalence of the problems considered with certain integral equations, we will obtain a fixed-point-type result for these integral equations. This result provides us the existence and uniqueness of solutions for the second-order nonlinear boundary value problems considered. As a novelty, we will use for this fixed-point-type result a family of third order iterative processes to approximate the solution, instead of the usually considered method of Successive Approximations of linear convergence.

Spectral Solutions of Even-Order BVPs Based on New Operational Matrix of Derivatives of Generalized Jacobi Polynomials

The primary focus of this article is on applying specific generalized Jacobi polynomials (GJPs) as basis functions to obtain the solution of linear and non-linear even-order two-point BVPs. These GJPs are orthogonal polynomials that are expressed as Legendre polynomial combinations. The linear even-order BVPs are treated using the Petrov–Galerkin method. In addition, a formula for the first-order derivative of these polynomials is expressed in terms of their original ones. This relation is the key to constructing an operational matrix of the GJPs that can be used to treat the non-linear two-point BVPs. In fact, a numerical approach is proposed using this operational matrix of derivatives to convert the non-linear differential equations into effectively solvable non-linear systems of equations. The convergence of the proposed generalized Jacobi expansion is investigated. To show the precision and viability of our suggested algorithms, some examples are given.

The reproducing kernel method for nonlinear fourth-order BVPs

&lt;abstract&gt;&lt;p&gt;Based on the reproducing kernel theory, we solve the nonlinear fourth order boundary value problem in the reproducing kernel space $ W_{2}^{5}[0, 1] $. Its approximate solution is obtained by truncating the n-term of the exact solution and using the $ \varepsilon $-best approximate method. Meanwhile, the approximate solution $ u^{(i)}_{n}(x) $ converges uniformly to the exact solution $ u^{(i)}(x), (i, 0, 1, 2, 3, 4) $. The validity and accuracy of this method are verified by some examples.&lt;/p&gt;&lt;/abstract&gt;

A high accuracy variable mesh numerical approximation for two point nonlinear BVPs with mixed boundary conditions

A high accuracy variable mesh numerical approximation for two point nonlinear BVPs with mixed boundary conditions

Approximating positive solutions of nonlinear BVPs of ordinary second order hybrid differential equations

In this paper we prove the existence and approximation of solution for a nonlinear two point boundary value problem of ordinary second order hybrid differential equations with Dirichlet boundary conditions via construction of an algorithm for the solution. The nonlinearity present on the right hand side of the differential equation is assumed to be Caratho\`eodory and the proof is based on a monotone iteration method of Dhage (2014) in an ordered Banach space.

A different monotone iterative technique for a class of nonlinear three-point BVPs

A different monotone iterative technique for a class of nonlinear three-point BVPs

On nonlinear singular BVPs with nonsmooth data. Part 2: Convergence of collocation methods

We discuss numerical solution of boundary value problems for systems of nonlinear ordinary differential equations with a time singularity,x′(t)=M(t)tx(t)+f(t,x(t))t,t∈(0,1],b(x(0),x(1))=0, where M:[0,1]→Rn×n and f:[0,1]×Rn→Rn are continuous matrix-valued and vector-valued functions, respectively. Moreover, b:Rn×Rn→Rn is a continuous nonlinear mapping which is specified according to a spectrum of the matrix M(0) to guarantee the BVP to be well-posed. For the case where M(0) has eigenvalues with nonzero real parts, we prove new convergence results for the collocation method and analytical results about the necessary smoothness of the solution to the problem required in the numerical analysis.We illustrate the theory by means of numerical examples.

On solutions of nonlinear BVPs with general boundary conditions by using a generalized Riesz–Caputo operator

In this work, we study the existence, uniqueness, and continuous dependence of solutions for a class of fractional differential equations by using a generalized Riesz fractional operator. One can view the results of this work as a refinement for the existence theory of fractional differential equations with Riemann–Liouville, Caputo, and classical Riesz derivative. Some special cases can be derived to obtain corresponding existence results for fractional differential equations. We provide an illustrated example for the unique solution of our main result.

.An optimal Galerkin-homotopy asymptotic method applied to the nonlinear second-order BVPs

.An optimal Galerkin-homotopy asymptotic method applied to the nonlinear second-order BVPs

On some computational aspects of Hermite &amp; Haar wavelets on a class of nonlinear singular BVPs

We propose a new class of SBVPs which deals with exothermic reactions. We also propose four computationally stable methods to solve singular nonlinear BVPs by using Hermite wavelet collocation which are coupled with Newton?s quasilinearization and Newton-Raphson method. We compare the results which are obtained by using Hermite wavelets with the results obtained by using Haar wavelets. The efficiency of these methods are verified by applying these four methods on Lane-Emden equations. Convergence analysis is also presented.

A system of nonlinear fractional BVPs with ϕ-Laplacian operators and nonlocal conditions

This work investigates the existence of multiple positive solutions for a system of two nonlinear higher-order fractional differential equations with ϕ-Laplacian operators and nonlocal conditions. A degenerate nonlinearity which obeys some general growth conditions is considered. The singularities are dealt with by approximating the fixed point operator. New existence results are presented by using the fixed point index theory. Examples of applications illustrate the theoretical results.

A new implicit-explicit local differential method for boundary value problems

In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear and singularly perturbed BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

A new fixed point iteration method for nonlinear third-order BVPs

In this article, we shall present a novel approach based on embedding Green's function into Ishikawa fixed point iterative procedure for the numerical solution of a broad class of boundary value problems of third order. A linear integral operator expressed in terms of Green's function is constructed, then the well-known Ishikawa fixed point iterative scheme is applied to obtain a new iterative scheme. The aim of our alternative strategy is to overcome the major deficiency of other iterative schemes that usually result in the deterioration of the error as the domain increases. Furthermore the proposed strategy will improve the rate of convergence of other existing methods that are based on Picard's and Mann's iterative schemes. Convergence results of the iterative algorithm have been proved. A number of numerical examples shall be solved to illustrate the method and demonstrate its reliability and accuracy. Moreover, we shall compare our results with both the analytical and the numerical solutions obtained by other methods that exist in the literature.