Order Nonlinear Boundary Value Problems Research Articles

Construction of high order numerical methods for solving fourth order nonlinear boundary value problems

Construction of high order numerical methods for solving fourth order nonlinear boundary value problems

On the existence of solutions for a class of nonlinear fully fourth-order differential systems

This paper discusses the existence of solutions to fourth order nonlinear boundary value problem involving systems of differential equations and two point boundary conditions. The nonlinearity f ∈ C ([0,1] × Rn+ × Rn × Rn × Rn, Rn)considered in this paper includes derivatives up to order three. Using recent fixed point results for the sum of two operators, we impose a growth condition on f to establish a new existence criteria that ensure the existence of at least one and the existence of at least two nonnegative solutions.

Existence and multiplicity of solutions of Stieltjes differential equations via topological methods

In this work, we use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution of a first order non-linear boundary value problem with linear boundary conditions that extend the periodic case. We also provide the Green’s function associated to the problem as well as an example of application.

New collocation approaches based on reproducing kernel functions for second order nonlinear boundary value problems

In the previous work, the reproducing kernel Hilbert space (RKHS) theory has been employed mainly to solve linear boundary value problems (BVPs). Reproducing kernel functions (RKFs) based techniques for nonlinear BVPs mainly applied iterative techniques. However, there is no error analysis for these methods. In this letter, we shall propose new collocation algorithms based on RKFs for dealing with second order nonlinear BVPs. And the detailed error analysis is provided.

Numerical method of sixth order convergence for solving a fourth order nonlinear boundary value problem

In this paper, we construct an iterative method of sixth order convergence for solving a fourth order nonlinear boundary value problem. The idea of the method is to construct difference schemes of sixth order accuracy for solving Dirichlet problems for second order equation at each iteration of the iterative method on continuous level constructed before by ourselves. Some numerical examples demonstrate the validity of obtained theoretical results.

Numerical method to solve generalized nonlinear system of second order boundary value problems: Galerkin approach

In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of 
 different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.

Numerical and Approximate Analytic Solutions of Second- order Nonlinear Boundary Value Problems

The shooting method consists of guessing unknown initial values, transforming a second-order nonlinear boundary value problem (BVP) to an initial value problem and integrating it to obtain the values at the right end to match the specified boundary condition, which acts as a target equation. In the shooting method, the key issue is accurately solving the target equation to obtain highly precise initial values. Due to the implicit and nonlinear property, we develop a generalized derivative-free Newton method (GDFNM) to solve the target equation, which offers very accurate initial values. Numerical examples are examined to show that the shooting method together with the GDFNM can generate a very accurate solution. Additionally, the GDFNM can successfully solve the three-point nonlinear BVPs with high accuracy. A new splitting-linearizing method is developed to express the approximate analytic solutions of nonlinear BVPs in terms of elementary functions, which adopts the Lyapunov technique by inserting a dummy parameter into the governing equation and the power series solution. Then, linearized differential equations are sequentially solved to derive the analytic solution.

Existence of a priori bounded solutions for discrete two-point boundary value problems

In this paper, a careful analysis on the existence of a priori bounded solutions for difference equations is provided. In particular, we study a second order nonlinear difference boundary value problem and its relationship with the continuous one and we obtain a version of these existence results with a priori bounds of the solution u and its derivative u′,u″ and we give an analytic expression of the solution u as the limit of a suitable sequence of functions.

Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems

Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems

Cubic spline solutions of the ninth order linear and non-linear boundary value problems

A lot of numerical formulations of physical phenomena contain 9th-order BVPs. The presented probe intends to consider the spline solutions of the 9th-order boundary value problems using Cubic B Spline(CBS). Ninth order boundary value problems arise in the study of laminar viscous flow in a semi-porous channel, astrophysics, hydrodynamic & hydro-magnetic stability. The derived technique is exceptionally useful and is appropriate for such kinds of linear and non-linear boundary value problems. Cubic B Spline(CBS) is successfully applied to mathematical models. The outcomes are contrasted to those presented in the literature, revealing that the introduced technique leads to an advisable estimation of the exact solution.

Galerkin Finite Element Approach to Solve System of Nonlinear Second-Order Boundary Value Problems

In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.

Galerkin Method and Its Residual Correction with Modified Legendre Polynomials

Accuracy and error analysis is one of the significant factors in computational science. This study employs the Galerkin method to solve second order linear or nonlinear Boundary Value Problems (BVPs) of Ordinary Differential Equations (ODEs) with modified Legendre polynomials to seek numerical solutions. The residual function of a differential operator is used as non-homogeneous term information of an error differential equation. The Galerkin approximation is then improved or corrected by solving the error differential equation by the Galerkin method using the same polynomials. Thus we apply the double layer Galerkin method to a variety of instances. We compare approximate solutions with exact ones and results available in the literature, and in every case, we find better accuracy.

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.

Lucas polynomials based spectral methods for solving the fractional order electrohydrodynamics flow model

The aim of this paper is to study the fractional order nonlinear singular boundary value problem arising in electrohydrodynamics flow, which describes the velocity of the ionized fluid in a circular cylindrical conduit. For this purpose, Lucas polynomials are employed via spectral collocation and Galerkin methods for reducing the nonlinear singular differential equation of fractional order to an algebraic nonlinear system, which is solved by Newton iterative method. The problem includes two important parameters, including the nonlinearity and Hartman numbers. The effect of these parameters and the influence of fractional order derivatives on the velocity of the fluid is discussed. Based on the reported results in tables and figures, for large values of Hartman number and the fractional derivatives, as well as weak nonlinearity parameter, the velocity profile was increased. The purposed schemes are simple and attractive, and their accuracy and efficiency, were confirmed by studying some cases of main problem and a comparison of the numerical results with the results obtained by some other methods.

Distributional Henstock-Kurzweil integral for fourth order nonlinear boundary value problems

In this paper existence of solution of fourth order nonlinear boundary value problems involving distributional Henstock-Kurzweil integral is obtained using fixed point theorem and distributional derivative.

Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

A superconvergent B-spline technique for second order nonlinear boundary value problems

In the present work, a high-order numerical scheme based on B-spline functions is developed for solving a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). To derive the method, we first generate a high order perturbation of the original problem by using spline alternate relations. Then, we determine the approximate solution by forcing it to satisfy the resulting perturbed problem at the grid points of the spline. Convergence analysis of the method is established through matrix approach. Four nonlinear examples are considered to demonstrate the accuracy and robustness of the method. The proposed method provides O(h6) superconvergent approximation to the solution of the problem under consideration, where h is the step size. This method produces significantly more accurate results than the two newly developed numerical schemes using the same B-spline functions as used in the present method, namely UCS method and NCS method. Moreover, the computational time of present method is compared with that of NCS method.

Numerical solutions of higher order boundary value problems via wavelet approach

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

Solution of third order linear and nonlinear boundary value problems of integro-differential equations using Haar Wavelet method

In this paper, numerical solution of third order integro-differential equation with boundary conditions is given utilizing Haar collocation technique. Both nonlinear and linear integro-differential equations are solved using this method. The third order derivative is approximated using Haar functions in both nonlinear and linear integro-differential equations. Integration is used to obtain the expression of lower order derivatives as well as the solution for the unknown function. The Gauss elimination approach is utilized for linear systems and Broyden approach is adopted for nonlinear systems. Validation and convergence of the proposed approach are illustrated using some examples. At various collocation and gauss points, the maximum absolute and root mean square errors are compared to the exact solution. The convergence rate is also measured using different numbers of nodal points, and it is nearly equal to 2.

An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, mathcal{O}(h^{2}) and mathcal{O}(h^{3}) for shooting techniques with Newton’s method and Halley’s method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.