Nonlinear Two-point Boundary Value Problems Research Articles

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional integral equations.

A finite difference method on quasi-uniform grids for the fractional boundary-layer Blasius flow

In this paper, we propose a fractional formulation, in terms of the Caputo derivative, of the Blasius flow described by a non-linear two-point fractional boundary value problem on a semi-infinite interval. We develop a finite difference method on quasi-uniform grids, based on a suitable modification of the classical L1 approximation formula and show the consistency, the stability and the convergence. The numerical results confirm the theoretical ones. Comparisons with some recently proposed results are carried out to validate the accuracy of the obtained numerical results, and to show the efficiency and the reliability of the proposed numerical method.

Indirect sequential convex programming for mars entry terminal altitude maximization

The uniform trigonometrization method (UTM) faces two difficulties in solving the Mars entry problem with maximum terminal altitude. One is that two possible solutions need to be evaluated by Pontryagin's minimum principle (PMP). The other one is that using numerical continuation logic to solve the Hamiltonian two-point boundary value problem (TPBVP) involves a significant amount of computation. To overcome these two challenges, indirect sequential convex programming (ISCP), which combines the UTM and sequential convex programming (SCP), is developed in this article. The proposed method first modifies the two control options into a single solution, which eliminates the requirement for PMP evaluation. The nonlinear TPBVP resulted from the improved UTM is then relaxed into a convex programming problem that is established by introducing the virtual control and buffer. Subsequently, the SCP process generates a numerical solution to the Mars entry problem within a short time. Compared to the general pseudospectral method and UTM, simulations demonstrate the effectiveness and efficiency of the ISCP method for longitudinal Mars entry missions with or without path constraints.

Existence of solutions to the iterative system of nonlinear two-point tempered fractional order boundary value problems

Existence of solutions to the iterative system of nonlinear two-point tempered fractional order boundary value problems

A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model that consists of a nonlinear PDE for the temperature along the solid body, coupled to a linear ODE for the temperature in the tank, the boundary and the initial conditions. All equations are converted into a dimensionless form reducing the input parameters to three dimensionless numbers and a dimensionless function. A steady-state analysis is performed. To solve the transient problem, a nontrivial numerical approach is proposed whereby the differential equations are first discretized in time. This reduces the problem to a sequence of nonlinear two-point boundary value problems (TPBVP) and a sequence of linear algebraic equations coupled to it. We show that knowing the temperature in the system at time level n − 1 allows us to decouple the TPBVP and the corresponding algebraic equation at time level n. Thus, starting from the initial conditions, the equations are decoupled and solved sequentially. The TPBVPs are solved by FDM with the Newtonian method.

An efficient technique based on Green’s function for solving two-point boundary value problems and its convergence analysis

This study proposes an accurate approximation to the solution of second-order nonlinear two-point boundary value problems, including the well-known Bratu problem, using an iterative technique based on Green’s function. The approach relies on constructing an equivalent integral representation of the problem incorporating Green’s function. The proposed methodology provides a reliable approximate solution and takes just a few iterations to achieve good accuracy. The mathematical formulation is further supported by discussing in detail the convergence analysis of this approach. Different numerical examples are used to check the robustness and effectiveness of the scheme. The numerical testing for nonlinear problems with nonlinear boundary conditions demonstrates that the proposed method outperforms other existing methods, including the finite element method, the finite volume method, the finite difference method, the B-spline method, and the Adomain’s decomposition method.

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

A reliable algorithm for a class of singular nonlinear two-point boundary value problems arising in physiology

In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessing Michaelis–Menten uptake kinetics. This approach is based on a modified concept of Adomian polynomials (AP), and the two-step Adomian decomposition method (TSADM) merged with Padé approximants. Furthermore a Maple package ADMP is applied to solve various problems, which is very easy to use and efficient and needed to input the system of equations with initial or boundary conditions and diverse essential parameters to deliver the analytic approximate solutions within a few seconds. The suggested scheme does not require linearization, perturbations, guessing the initial terms, a set of basis function or other limiting presumptions, which yields the solutions in closed form. Many examples are examined to make clear the scope and validity of the package ADMP.

Solution of Second- and Higher-Order Nonlinear Two-Point Boundary-Value Problems Using Double Decomposition Method

The present paper makes use of the efficient double decomposition method to propose a method for solving two-point boundary-value problems, featuring second- and higher-order nonlinear ordinary differential equations. The efficacy of the proposed method is demonstrated on numerous test problems. In the end, a high level of exactitude between the obtained approximate solution and the available exact solution is achieved.

Numerical treatment of fractal boundary value problems for heat conduction in polar bear with spatial variation of thermal conductivity

A high-resolution compact discretization scheme for the numerical approximation of two-point nonlinear fractal boundary value problems is presented to study the stationary anomalous diffusion process. Hausdorff derivative is applied to derive the models in fractal media. The proposed scheme solves the nonlinear fractal model and achieves an accuracy of order four by employing only three mesh points in a stencil and consumes short computing time. Numerical simulations with heat conduction in polar bear, convection–diffusion, boundary layer, Bessel’s and Burgers equation in a fractal medium are carried out to illustrate the utility of the scheme and their numerical rate of convergence.

Existence of solutions of discrete fractional problem coupled to mixed fractional boundary conditions

In this paper, we introduce a two-point nonlinear boundary value problem for a finite fractional difference equation. An associated Green’s function is constructed as a series of functions and some of its properties are obtained. Some existence results are deduced from fixed point theory and lower and upper solutions.

Certain Analysis of Solution for the Nonlinear Two-Point Boundary Value Problem with Caputo Fractional Derivative

In this paper, the existence and uniqueness of solutions for a nonlinear fractional differential equation with a two-point boundary condition in a Banach space are investigated by using the contraction mapping principle and the Brouwer fixed-point theorem with Bielecki norm. The iterative scheme of the numerical solution for the nonlinear two-point boundary value problem will be discussed and illustrated by solving some problems. The well-known Ulam-Hyers and Ulam-Hyers-Rassias stability theorems are employed to establish the stability of solutions to the boundary value problem. In the end, we provided a couple of examples to support our results.

A numerical iterative method for solving two-point BVPs in infinite-horizon nonzero-sum differential games: Economic applications

In this work, the Nash equilibrium solution of nonlinear infinite-horizon nonzero-sum differential games with open-loop information is investigated numerically. For this class of games, some difficulties are involved in finding an open-loop Nash equilibrium. For instance, we must solve a nonlinear differential equations system with split boundary conditions, namely the two-point boundary value problem (TPBVP), in which some of the boundary conditions are specified at the initial time and some at the infinite final time. In the current study, we provide a combined numerical algorithm based on a new set of basis functions on the half-line, called the exponential Chelyshkov functions (ECFs), and quasilinearization (QL) method to solve TPBVPs. In the first step, we reduce the nonlinear TPBVP to a sequence of linear differential equations by using the QL method. Although we have now a linearized system, another difficulty is finding the approximate solution of this linear system such that the transversality conditions at the infinite final time are satisfied. So, in the second step, we apply a collocation method based on the ECFs to solve the obtained linear system in the semi-infinite domain. The convergence of the proposed method is discussed in detail. To confirm the validity and efficiency of the proposed scheme, we compute the approximate solution of TPBVP as well as the open-loop Nash equilibrium for four applications of differential games in economics and management science.

Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.

Numerical solution of two-point nonlinear boundary value problems via Legendre–Picard iteration method

The aim of the present work is to introduce an effective numerical method for solving two-point nonlinear boundary value problems. The proposed iterative scheme, called the Legendre–Picard iteration method is based on the Picard iteration technique, shifted Legendre polynomials and Legendre–Gauss quadrature formula. In the Legendre–Picard iteration method, the boundary value problem is reduced to an iterative formula for updating the coefficients of the approximate solution in each step and with a straightforward manner, the integrals of the shifted Legendre polynomials are calculated. In addition, to reduce the CPU time, a vector–matrix scheme of the Legendre–Picard iteration method is constructed. The convergence analysis of the method is studied. Five nonlinear boundary value problems are given to illustrate the validity of the Legendre–Picard iteration method. Numerical results indicate the good performance and the precision of the proposed procedure.

Nonlinear two-point iterative functional boundary value problems on time scales

Nonlinear two-point iterative functional boundary value problems on time scales

A new hybrid collocation method for solving nonlinear two-point boundary value problems

A new hybrid collocation method for solving nonlinear two-point boundary value problems

A new hybrid collocation method for solving nonlinear two-point boundary value problems

A new hybrid collocation method for solving nonlinear two-point boundary value problems