Nonlinear Two-point Boundary Value Problems Research Articles

Bernstein polynomial basis for numerical solution of boundary value problems

The purpose of this paper is to propose a computational method for the approximate solution of linear and nonlinear two-point boundary value problems. In order to approximate the solution, the expansions in terms of the Bernstein polynomial basis have been used. The properties of the Bernstein polynomial basis and the corresponding operational matrices of integration and product are utilized to reduce the given boundary value problem to a system of algebraic equations for the unknown expansion coefficients of the solution. On this approach, the problem can be solved as a system of algebraic equations. By considering a special case of the problem, an error analysis is given for the approximate solution obtained by the present method. At last, five examples are examined in order to illustrate the efficiency of the proposed method.

Numerical solution of second-order fuzzy nonlinear two-point boundary value problems using combination of finite difference and Newton’s methods

In this paper, we discuss the numerical solution of second-order nonlinear two-point fuzzy boundary value problems (TPFBVP) by combining the finite difference method with Newton’s method. Numerical example using the well-known nonlinear TPFBVP is presented to show the capability of the new method in this regard and the results are satisfied the convex triangular fuzzy number. We also compare the numerical results with the exact solution, and it shows that the proposed method is good approximation for the analytic solution of the given TPFBVP.

Existence and Uniqueness of Solutions for a Class of Singular Nonlinear Two-Point Boundary Value Problems with Sign-Changing Nonlinear Terms

ABSTRACTIn this paper, a constructive proof under some conditions is presented for existence and uniqueness of the solutions to the singular problem with the boundary conditionsIn general, f(x,y(x)) is assumed to be nonsingular with respect to the independent variable x but it is allowed be singular with respect to y and moreover, f(x,y(x)) can be sign changing as well. The Picard iterative sequence is then constructed based on integral equation with the help of positive Green’s function. The convergence of this iterative sequence is controlled by an adjustable parameter so that it converges to the unique solution in the finite region in which is supposed to be nonnegative and bounded. Some examples also are given to demonstrate the trustworthiness of our constructive theory.

Partial slip and Joule heating on magnetohydrodynamic radiated flow of nanoliquid with dissipation and convective condition

Numerical investigation of three-dimensional flow of an electrically conducting nanofluid over a bidirectional stretching surface is proposed here. The slip flow over a convectively stretching sheet is considered. The flow is caused due to a non-linear stretching surface and Lorenz force. Water and copper nanoparticles are used to form nanoliquid. Suitable transformations are employed to reduce the conservation equations into nonlinear coupled, multidegree ordinary differential equations. Resultant nonlinear two-point boundary value problem is numerically integrated using Runge-Kutta-Fehlberg fourth-fifth order method. Computed results are verified with existing results under limiting cases. The influences of pertinent parameters on different flow fields are evaluated and presented via graphical and tabular form. It is found that the thermal radiation and convective heating at boundary stabilizes the thermal boundary layer growth.

New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the Duan-Rach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn) or the error remainder functions (ERn(x)) of each problem are calculated.

A Combined Analytic and Numerical Procedure for Stagnation-Point Flow

The two dimensional vector form of the Navier-Stokes equation is reduced to a fourth–order equation for the streamfunction. Boundary conditions arise from considerations of the no-slip constraint at the surface as well the interaction of viscous flow with potential-flow at the edge of the boundary layer. By employing a separation of variables technique and introducing certain dimensionless variables, the stream function equation is converted into its dimensionless analog with appropriate boundary conditions. The resulting quasi-linear third-order ordinary differential equation facilitates the numerical computation of the velocity and the pressure terms. This is achieved by solving the nonlinear two-point boundary-value problem with a time-marching method involving a Crank-Nicolson and Newton-linearization schemes until steady-state solution is obtained. The velocity, stream-function and pressure profiles are discussed with reference to various computation parameters and are found to be in good agreement with the physics of the problem. It was also found that there is no penalty in accuracy for a broad range of CFL numbers. However as the CFL number exceeds a certain threshold, the approach to convergence becomes erratic as indicated by the spurious results produced by the solution residuals.

Control of nonlinear vibrations using the adjoint method

In this paper, a new methodology is proposed to address the problems of suppressing structural vibrations and attenuating contact forces in nonlinear mechanical systems. The computational algorithms developed in this work are based on the mathematical framework of the calculus of variation and take advantage of the numerical implementation of the adjoint method. To this end, the principal aspects of the optimal control theory are reviewed and employed to derive the adjoint equations which form a nonlinear differential-algebraic two-point boundary value problem that defines the mathematical solution of the optimal control problem. The adjoint equations are obtained and solved numerically for the optimal design of control strategies considering a twofold control structure: a feedforward (open-loop) control architecture and a feedback (closed-loop) control scheme. While the feedforward control strategy can be implemented using only the active control paradigm, the feedback control method can be realized employing both the active and the passive control approaches. For this purpose, two dual numerical procedures are developed to numerically compute a set of optimal control policies, namely the adjoint-based control optimization method for feedforward control actions and the adjoint-based parameter optimization method for feedback control actions. The computational methods developed in this work are suitable for controlling nonlinear nonautonomous dynamical systems and feature a broad scope of application. In particular, it is shown in this paper that by setting an appropriate mathematical form of the cost functional, the proposed methods allow for simultaneously solving the problems of suppressing vibrations and attenuating interaction forces for a general class of nonlinear mechanical systems. The numerical example described in the paper illustrates the key features of the adjoint method and demonstrates the feasibility and the effectiveness of the proposed adjoint-based computational procedures.

All roots spectral methods: Constraints, floating point arithmetic and root exclusion

Abstract The nonlinear two-point boundary value problem (TPBVP for short) u x x + u 3 = 0 , u ( 0 ) = u ( 1 ) = 0 , offers several insights into spectral methods. First, it has been proved a priori that ? 0 1 u ( x ) d x = p / 2 . By building this constraint into the spectral approximation, the accuracy of N + 1 degrees of freedom is achieved from the work of solving a system with only N degrees of freedom. When N is small, generic polynomial system solvers, such as those in the computer algebra system Maple, can find all roots of the polynomial system, such as a spectral discretization of the TPBVP. Our second point is that floating point arithmetic in lieu of exact arithmetic can double the largest practical value of N . (Rational numbers with a huge number of digits are avoided, and eliminating M symbols like 2 and p reduces N + M -variate polynomials to polynomials in just the N unknowns.) Third, a disadvantage of an “all roots” approach is that the polynomial solver generates many roots – ( 3 N - 1 ) for our example – which are genuine solutions to the N -term discretization but spurious in the sense that they are not close to the spectral coefficients of a true solution to the TPBVP. We show here that a good tool for “root-exclusion” is calculating ? = ? n = 1 N b n 2 ; spurious roots have ? larger than that for the physical solution by at least an order of magnitude. The ? -criterion is suggestive rather than infallible, but root exclusion is very hard, and the best approach is to apply multiple tools with complementary failings.

Computational methods for solving the steady flow of a third grade fluid in a porous half space

An important class of fluids commonly used in industries is non-Newtonian fluids. In this paper, two numerical techniques based on rational Legendre functions and Chebyshev polynomials are presented for solving the flow of a third-grade fluid in a porous half space. This problem can be reduced to a nonlinear two-point boundary value problem on semi-infinite interval. Our methods are utilized to reduce the computation of this problem to some algebraic equations. The comparison of the results with the other methods and residual norm show very good accuracy and rate of convergence of our approach.

Weak solutions nonlinear fractional integrodifferential equations in nonreflexive Banach spaces

The aim of this paper is to discuss the existence of weak solutions for a nonlinear two-point boundary value problem of integrodifferential equations of fractional order $\alpha\in(1,2]$ . Our analysis relies on the Krasnoselskii fixed point theorem combined with the technique of measure of weak noncompactness.

PERFORMANCE OF MODIFIED NON-LINEAR SHOOTING METHOD FOR SIMULATION OF 2ND ORDER TWO-POINT BVPS

In this research article, numerical solution of nonlinear 2nd order two-point boundary value problems (TPBVPs) is discussed by the help of nonlinear shooting method (NLSM), and through the modified nonlinear shooting method (MNLSM). In MNLSM, fourth order Runge-Kutta method for systems is replaced by Adams Bashforth Moulton method which is a predictor-corrector scheme. Results acquired numerically through NLSM and MNLSM of TPBVPs are discussed and analyzed. Results of the tested problems obtained numerically indicate that the performance of MNLSM is rapid and provided desirable results of TPBVPs, meanwhile MNLSM required less time to implement as comparable to the NLSM for the solution of TPBVPs.

Quarter-sweep Nonlocal Discretization Scheme with QSSOR Iteration for Nonlinear Two-point Boundary Value Problems

The aim of this paper is to consider the Quarter-sweep Successive Over Relaxation (QSSOR) iteration for solving nonlinear two-point boundary value problems. The second order finite difference (FD) method is applied to derive the quarter-sweep nonlocal discretization scheme for the sake of transforming the system of nonlinear approximation equations into the corresponding system of linear equations. The formulation and the implementation of the methods are discussed. In addition, the numerical results by solving the proposed problems using QSSOR method are included and compared with the Full-sweep Successive Over Relaxation (FSSOR) and Half-sweep Successive Over Relaxation (HSSOR) methods.

A Numerical Investigation to Viscous Flow Over Nonlinearly Stretching Sheet with Chemical Reaction, Heat Transfer and Magnetic Field

In this research, we aim to propose a new numerical scheme to solve a non-linear system two-point boundary value problem on semi-infinite interval, describing the flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. This scheme is the combination of the Ritz and collocation methods based on radial basis functions. These methods reduce the solution of the problem to the solution of a system of algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.

New Analytic Technique for the Solution of Nth Order Nonlinear Two-point Boundary Value Problems

In this article, a new analytic technique based on Parker-Sochacki iteration is introduced for computing series solution of a general nonlinear two-point boundary value problems with Dirichlet and Neumann boundary conditions. For problems with or without analytic solution, we found out that this easy-to-implement method produced highly accurate results without linearization when compared with their closed form solutions.

High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form

In this article, using three grid points, we discuss variable mesh method of order three for the numerical solution of nonlinear two-point boundary value problems:(p(x)y′)′=f(x,y),y(0)=A,y(1)=B.We first establish an identity from which general three-point finite difference approximation of various order can be obtained. We obtain a family of third order discretization using variable mesh for the differential equations. We select the free parameter available in this discretization which leads to the simplest third order method. Numerical results are provided to illustrate the proposed method and their convergence.

Marangoni boundary layer flow and heat transfer of copper-water nanofluid over a porous medium disk

In this paper we present a study of the Marangoni boundary layer flow and heat transfer of copper-water nanofluid over a porous medium disk. It is assumed that the base fluid water and the nanoparticles copper are in thermal equilibrium and that no slippage occurs between them. The governing partial differential equations are transformed into a set of ordinary differential equations by generalized Kármán transformation. The corresponding nonlinear two-point boundary value problem is solved by the Homotopy analysis method and the shooting method. The effects of the solid volume fraction, the permeability parameter and the Marangoni parameter on the velocity and temperature fields are presented graphically and analyzed in detail.

On The Existence of Multiple Solutions of a Class of Third-Order Nonlinear Two-Point Boundary Value Problems

A general approach is presented for proving existence of multiple solutions of the third-order nonlinear differential equation $$Au^{\prime\prime\prime}(x) + u^{\prime\prime}(x)u^\prime(x) + u^\prime(x)f(u(x))=0,\quad x \in [0,1] ,$$ subject to given proper boundary conditions. The proof is constructive in nature, and could be used for numerical generation of the solution or closed-form analytical solution by introducing some special functions. The only restriction is about f(u), where it is supposed to be differentiable function with continuous derivative. It is proved the problem may admit no solution, may admit unique solution or may admit multiple solutions.

Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

Existence and multiplicity for a system of fractional higher-order two-point boundary value problem

The purpose of this paper is to establish some results on the existence of multiple positive solutions for a system of nonlinear fractional order two-point boundary value problem. The main tool is a fixed point theorem of the cone expansion and compression of functional type and five functional fixed point theorem. Some examples are also presented to illustrate the availability of the main results.

Inflating a flat toroidal membrane

A flat toroidal geometry, composed of a stack of two flat annular membranes bonded together at the inner and outer boundaries (equators), is proposed. The finite inflation mechanics of the composite membrane under uniform internal pressure is considered. The membranes are assumed to be identical homogeneous and isotropic solids described by the Gent hyperelastic model with a relaxed strain energy density function. The variational inflation mechanics problem yields the governing equations and boundary conditions, which are recast as a nonlinear two-point boundary value problem in a suitable form for efficient computation. The effects of the inflation pressure and material properties on the state of stretch and geometry of the inflated toroidal membrane are obtained, which exhibit a number of desirable features. The phenomenon of wrinkling instability is observed near the outer equator of the torus at low pressures (in spite of a positive Gaussian curvature), which is exactly opposite to the behavior observed in a curved torus (wrinkling at the inner equator at high pressure). Semi-analytical expressions of the Cauchy stress resultants are obtained which match well with the numerical computations. Further, the limit-point instability pressure of the toroidal membrane is shown to have a remarkable power-law dependence on a geometric parameter of the uninflated flat torus and the material chain extensibility parameter involving two universal constants.