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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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

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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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 ].

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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.

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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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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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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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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**
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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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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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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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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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.

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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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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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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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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**
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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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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