# Top PDF Numerical solution of two-point boundary-value problems

### Numerical solution of two-point boundary-value problems

Employing Euler's method to approximate the differential equation (lala), this formulation of the discrete problem becomes.. This example will be used throughout to illustrate various[r]

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### A Method for Numerical Solution of Two Point Boundary Value Problems with Mixed Boundary Conditions

How to cite this paper: Pandey, P.K. (2014) A Method for Numerical Solution of Two Point Boundary Value Problems with Mixed Boundary Conditions. Open Access Library Journal, 1: e565. http://dx.doi.org/10.4236/oalib.1100565 A Method for Numerical Solution of Two Point Boundary Value Problems with Mixed Boundary Conditions

### Non Standard Difference Method for Numerical Solution of Linear Fredholm Integro Differential Type Two Point Boundary Value Problems

y = y = where f x ( ) is calculated so that the analytical solution of the problem is y x ( ) = ln 1 ( + x ) . The MAY com- puted by Method (8) for different values of N and number of iterations Iter. are presented in Table 4. We have described a numerical method for numerical solution of Fredholm integro-differential type boundary value problem and four model problems considered to illustrate the preciseness and effectiveness of the pro- posed method. Numerical results for example 1 which is presented in Table 1, for different values of N show decreases with step size maximum absolute errors in our method decrease. Similar observation can be found in result of example 2, 3 and 4. Over all Method (6) is convergent and convergence of the method does not de- pends on choice of step size h.
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### Analysis and numerical simulation of positive and dead-core solutions of singular two-point boundary value problems

Θ ( s ) ds on the mesh consisting of the points τ ˜ i to be determined accordingly, where at the same time measures are taken to ensure that the variation of the stepsizes is restricted and tolerance requirements are satisfied with small computational effort. Details of the mesh selection algorithm and a proof of the fact that our strategy implies that the global error of the numerical solution is asymptotically equidistributed are given in [ 18 ].

### Sinc-collocation solution for nonlinear two-point boundary value problems arising in chemical reactor theory

Abstract Numerical solution of nonlinear second order two-point boundary value problems based on Sinc-collocation method, developed in this work. We first apply the method to the class of nonlinear two-point boundary value problems in general and specifically solved special problem that is arising in chemical reactor theory.

### An ‎E‎ffective Numerical Technique for Solving Second Order Linear Two-Point Boundary Value Problems with Deviating Argument

where T > 0, α, β ∈ R , and ϕ : [0, T ] → R is such that 0 ≤ ϕ(t) ≤ T, ∀ t ∈ [0, T ]. We suppose that ϕ is Lipschitzian, p, q ∈ C 2 (0, a) and f ∈ L 2 w [0, T ] are sufficiently regular given functions such that Eq. (1.1) satisfies the existence and uniqueness of the solution. Without loss of generality, we can assume that the boundary conditions in Eq. (1.1) are homogeneous. In this paper, based on re- producing kernel theory, reproducing kernels with polynomial form will be constructed and a com- putational method is described in order to obtain the accurate numerical solution with polynomial form of the Eq. (1.1) in the reproducing kernel spaces spanned by the Chebyshev basis polyno- mials.
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### A numerical approach to nonlinear two-point boundary value problems for ODEs

In other cases the choice y i 0 = 0 for all i is preferred. However, as it is proved in this paper, the choice of the initial guess is strictly connected to the assigned BVP and a wrong choice could compromise the results obtained with the implementation of the finite difference method. Generally, a unified theory of stability and convergence for the nonlinear BVPs with mixed linear or nonlinear boundary conditions, is given in terms of first-order systems of ODEs (see [ 7 , 8 ]). In addition, the consistency and the order of accuracy are defined as before, whereas the stability is defined only in the proximity of an isolated solution. The known result that the convergence is assured by the consistence and stability is adopted in [ 8 ], where the authors prove the convergence theorems under the assumption that the linearized finite difference scheme, around the (unknown) isolated solution, is consistent and stable (see [ 8 ], p. 207). We rather prefer to follow Keller’s approach (see [ 7 ]) in which the consistency and convergence of the finite difference scheme is proved by the following theorem (it has been reformulated in order to fit the formulation of BVP (1) ).
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### Numerical determination of the geodesic curves: the solution of a two-point boundary value problem

In addition to approximate closed form relation represented for solving this integral (Krakiwsky & Thomson, 1978), it could be accurately solved using computing routines e.g. ODE113 in MATLAB. In Table 1, the solution of Eq. (41) is compared by that of the Eq. (22). The comparison is carried out for the pole-to-pole meridian arc length. The efficiency of the method represented for geodesic curve determination is tested on the surface of ellipsoid WGS84 (World Geodetic Datum 1984). A sub- millimeter level of accuracy in the computed geodesic curve length is considered for the maximum error tolerance. It is equal to 0.00001 " in longitudinal and latitudinal differences, i.e., the maximum size of misfit vector at the end point. The error of each iteration in case pole-to-pole geodesic determination is represented in Table 2.
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### Solution of two-point fuzzy boundary value problems by fuzzy neural networks

Abstract In this work, we have introduced a modified method for solving second-order fuzzy differential equations. This method based on the fully fuzzy neural network to find the numerical solution of the two-point fuzzy boundary value problems for the ordinary differential equations. The fuzzy trial solution of the two-point fuzzy boundary value problems is written based on the concepts of the fully fuzzy feed-forward neural networks which containing fuzzy adjustable parameters. In comparison with other numerical methods, the proposed method provides numerical solutions with high accuracy.
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### Convergence Analysis of Spline Solution of Certain Two-Point Boundary Value Problems

Convergence Analysis of Spline Solution of Certain Two-Point Boundary Value Problems J. Rashidinia 1; , R. Jalilian 2 and R. Mohammadi 1 Abstract. The smooth approximate solution of second order boundary value problems are developed by using non-polynomial quintic spline function. We obtained the classes of numerical methods, which are second, fourth and six-order. For a specic choice of the parameters involved in a non-polynomial spline, truncation errors are given. A new approach convergence analysis of the presented methods are discussed.
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### An approach for solving singular two point boundary value problems: analytical and numerical treatment

The numerical treatment of singular boundary value problems (bvps) has always been a difficult and challenging task due to the singular behaviour that occurs at a point. Various efficient numerical methods have been proposed to solve this type of problem [ 10 , 13 , 16 ]. These methods include the homo- topy analysis method [ 17 ], differential transformation method [ 16 ], modified adm [ 18 , 19 , 20 , 21 ], and an improved adm [ 13 ]. The first two of these methods obtained analytical and numerical solutions for some linear singular two-point bvps, the others successfully solved linear and nonlinear singular two-point bvps. Often the adm may require additional calculations in order to determine an unknown constant in any of its n-term approximations to the solution [ 19 ] and the accuracy of the method decreases. Accordingly, it is desirable to have a modification of the decomposition method which explicitly determines the zeroth component of the solution. One such method was suggested by Ebaid [ 13 ], which is based on the adm and a modification
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### A numerical algorithm for nonlinear multi-point boundary value problems

Multi-point boundary value problems (BVPs) arise in a variety of applied mathematics and physics. For instance, the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities can be set up as a multi-point BVP, as in [ 1 ]; also, many problems in the theory of elastic stability can be handled by the method of multi-point problems [ 2 ]. The existence and multiplicity of solutions of multi-point boundary value problems have been studied by many authors; see [ 3–6 ] and the references therein. For two-point BVPs, there are many solution methods such as orthonormalization, invariant imbedding algorithms, finite difference, collocation methods, etc. [ 7–9 ]. However, there seems to be little discussion about numerical solutions of multi-point boundary value problems. The shooting method is used to solve multi-point boundary value problems in [ 10 , 11 ]. However, the shooting method is a trial-and-error method, and it is often sensitive to the initial guess. This makes computation by the conventional shooting method expensive and ineffective. Geng [ 12 ] proposed a method for a class of second-order three-point BVPs by converting the original problem into an equivalent integro–differential equation. Lin and Lin [ 13 ] introduced an algorithm for solving a class of multi-point BVPs by constructing a reproducing kernel satisfying the multi-point boundary conditions. However, the method introduced in [ 13 , 14 ] for obtaining a reproducing kernel satisfying multi-point boundary conditions is very complicated, and the form of the reproducing kernel obtained is also very complicated. Hence, the computational cost of this method is very high. Tatari and Mehghan [ 15 ] introduced the Adomian decomposition method (ADM) for multi-point BVPs. Yao [ 16 ] proposed a successive iteration method for three-point BVPs. Li and Wu [ 17 ] developed a method for solving linear multi-point BVPs. Motivated by the interesting paper [ 17 ], we shall present an effective method for solving nonlinear multi-point BVPs.
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### Splines For Linear Two-Point Boundary Value Problems

u(a) = α and u(b) = β . This criterion is important because we are doing approximation of solution, which requires stability [7]. On the other hand, three types of splines are used in this study to approximate the solution for linear two-point boundary value problems by splines interpolation. They are cubic trigono- metric B-spline, cubic Beta-spline and extended cubic B-spline. The details on the splines as well as the reasons for choosing the splines are discussed in Chapter 3. Up to our knowledge, no work has been published pertaining to using these three splines in solving two-point bound- ary value problems. Therefore, this study is a fresh start in this direction. For that reason, the experimental results are only compared with the results from cubic B-spline interpolation method, but not with other methods that do not use splines interpolation.
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### Decomposition conditions for two point boundary value problems

Abstract. We study the solvability of the equation x = f (t,x,x ) subject to Dirichlet, Neumann, periodic, and antiperiodic boundary conditions. Under the assumption that f can be suitably decomposed, we prove approximation solvability results for the above equation by applying the abstract continuation type theorem of Petryshyn on A-proper mappings.

### ON THE SOLUTION SET OF A TWO POINT BOUNDARY VALUE PROBLEM

where I = [0, 1], F (., ., .) : I × R 2 → P(R), H(., .) : I × R → P(R), λ > 0 and c i ∈ R, i = 1, 2. When F does not depend on the last variable (1.1) reduces to x 00 − λx 0 ∈ F (t, x), a.e. (I). (1.3) Qualitative properties of the set of solutions of problem (1.3)-(1.2) may be found in [1, 2, 7, 9, 10, 11] etc. In all these papers the set-valued map F is assumed to be at least closed-valued. Such an assumption is quite natural in order to obtain good properties of the solution set, but it is interesting to investigate the problem when the right-hand side of the multivalued equation may have nonclosed values.
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### Numerical Solution for Initial and Boundary Value Problems of Fractional Order

These fields were found to be best described using fractional differential equ- ations (FDEs) to model their processes and equations. One of the well-known methods for solving fractional differential equations is the Shifted Legendre operational matrix (LOM) method. In this article, I proposed a numerical method based on Shifted Legendre polynomials for solving a class of frac- tional differential equations. A fractional order operational matrix of Legen- dre polynomials is also derived where the fractional derivatives are described by the Caputo derivative sense. By using the operational matrix, the initial and boundary equations are transformed into the products of several matrix- es and by scattering the coefficients and the products of matrixes. I got a sys- tem of linear equations. Results obtained by using the proposed method (LOM) presented here show that the numerical method is very effective and appropriate for solving initial and boundary value problems of fractional or- dinary differential equations. Moreover, some numerical examples are pro- vided and the comparison is presented between the obtained results and those analytical results achieved that have proved the method’s validity.
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### The numerical solution of boundary value problems in partial differential equations

in the difference solution were sho?n to oe dependent on the pov.ors 01 thia aatr'x, and, therefore, to grow with the number of time steps* In chapte: 1 it was pointed out that a spectral radius of this order would give hounded errors In a closed region, 0 < t < T, but that the errors would become unbounded as t -» <*• Thus, problems of numerical instability in the solution of the third boundary value probl«a for the heat equation, which ari^e from the boundary conditions, arc important only for large values of the time. However, this type of instability aastnes a new
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### A problem-solving environment for the numerical solution of boundary value problems

Abstract Boundary value problems (BVPs) are systems of ordinary differential equations (ODEs) with boundary conditions imposed at two or more distinct points. Such problems arise within mathe- matical models in a wide variety of applications. Numerically solving BVPs for ODEs generally requires the use of a series of complex numerical algorithms. Fortunately, when users are required to solve a BVP, they have a variety of BVP software packages from which to choose. However, all BVP software packages currently available implement a specific set of numerical algorithms and therefore function quite differently from each other. Users must often try multiple software packages on a BVP to find the one that solves their problem most effectively. This creates two problems for users. First, they must learn how to specify the BVP for each software package. Second, because each package solves a BVP with specific numerical algorithms, it becomes difficult to determine why one BVP package outperforms another. With that in mind, this thesis offers two contributions. First, this thesis describes the development of the BVP component to the fully featured problem- solving environment (PSE) for the numerical solution of ODEs called pythODE. This software allows users to select between multiple numerical algorithms to solve BVPs. As a consequence, they are able to determine the numerical algorithms that are effective at each step of the solution process. Users are also able to easily add new numerical algorithms to the PSE. The effect of adding a new algorithm can be measured by making use of an automated test suite.
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### Mean-field games and two-point boundary value problems

Mean–field games and twopoint boundary value problems Thulasi Mylvaganam and Dario Bauso and Alessandro Astolfi Abstract— A large population of agents seeking to regulate their state to values characterized by a low density is consid- ered. The problem is posed as a mean-field game, for which solutions depend on two partial differential equations, namely the Hamilton-Jacobi-Bellman equation and the Fokker-Plank- Kolmogorov equation. The case in which the distribution of agents is a sum of polynomials and the value function is quadratic is considered. It is shown that a set of ordinary differ- ential equations, with two-point boundary value conditions, can be solved in place of the more complicated partial differential equations associated with the problem. The theory is illustrated by a numerical example.
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### On the Solvability of Discrete Nonlinear Two Point Boundary Value Problems

The first equation in 1.8 was recently analyzed by Kon´e et al. 12 and Ouaro 13 and generalized to a Radon measure data by Kon´e et al. 14 for an homogeneous Dirichlet boundary condition u 0 on ∂Ω. The study of 1.8 will be done in a forthcoming work. Problems like 1.8 have been intensively studied in the last decades since they can model various phenomena arising from the study of elastic mechanics see 15, 16, electrorheological fluids see 15, 17–19, and image restoration see 20. In 20, Chen et al. studied a functional with variable exponent 1 ≤ px ≤ 2 which provides a model for image denoising, enhancement, and restoration. Their paper created another interest for the study of problems with variable exponent.
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