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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.
3Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions
Boundary shape function iterative method for nonlinear second-order boundary value problems with nonlinear boundary conditions
Read more28A collocation method for solving a class of singular nonlinear two-point boundary value problems
A collocation method for solving a class of singular nonlinear two-point boundary value problems
18The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem
The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem
Read more28Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems
Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems
A second-order periodic boundary value problem with singular and discontinuous nonlinearity
In this paper, we study a nonlinear second-order periodic boundary value problem, in which the equation has a singular and discontinuous nonlinearity. By using perturbation techniques and comparison principles, we obtain the existence of solutions for this problem.
Read more131A new spectral-homotopy analysis method for solving a nonlinear second order BVP
A new spectral-homotopy analysis method for solving a nonlinear second order BVP
26Numerical solution of sixth order boundary value problems with sixth degree B-spline functions
Numerical solution of sixth order boundary value problems with sixth degree B-spline functions
3New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales
<p/> Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained, and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods.
Read more3TYPES AND MULTIPLICITY OF SOLUTIONS TO STURM–LIOUVILLE BOUNDARY VALUE PROBLEM
We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.
Read more3Existence of positive pseudo-symmetric solution for second-order three-point boundary value problems
The existence results of at least one positive pseudo-symmetric solution for a nonlinear second-order three-point boundary value problem are established by means of the fixed point theorem due to Avery, Anderson and Krueger.
Read more3The Numerical Solution of Third Order Boundary Value Problems Using Piecewise Bernstein polynomials by the Galerkin Weighted Residual Method
There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.
Read more2Numerical 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.
Read more2Existence of positive solutions for a system of nonlinear second-order integral boundary value problems
We study the existence and multiplicity of positive solutions of a system of nonlinear second-order ordinary differential equations subject to Riemann-Stieltjes integral boundary conditions.
Read more9Iterative procedures for nonlinear second order boundary value problems
Questions of the existence of positive solutions of second order nonlinear boundary value problems with separated boundary conditions are investigated. The nonlinearities are such that the linearization about the trivial solution does not exist or is trivial. The methods are thus applicable when shooting methods or ordinary bifurcation techniques cannot be applied. The conditions on the nonlinearity are quite modest and both super and sublinear problems can be included.
Read moreComparison of Shear Stresses of Different Fluid Flows Arising in the Boundary Layer Theory
The existence of positive solution for the singular second-order nonlinear boundary value problem arising in the boundary layer theory for the strong suction is studied. Then we compared the shear stress of the strong suction with the shear stress of the Homann flow. Also we compared the shear stress of the strong suction with the shear stress of the convergent flow. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21975 Dhaka Univ. J. Sci. 62(2): 115-118, 2014 (July)
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