x nJ+l matrix and r is an nJ+l-vector Employing Euler's method to approximate the differential equation lala, this formulation of the discrete problem becomes This example will be used t[r]

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We present in Section 2, the development and derivation of the **numerical** method for solving problem (1). Local truncation error and convergence is discussed in Section 3. The possibility of stability and computational performance of the method on model **problems** is discussed respectively in Sections 4 and 5. Conclusion and out view for future research are discussed in final Section 6.

In this article we have considered Fredholm integro-differential equation type second-order **boundary** **value** **problems** and proposed a rational difference method for **numerical** **solution** of the **problems**. The composite trapezoidal quadrature and non-standard difference method are used to convert Fredholm integro-differential equation into a system of equations. The **numerical** results in experiment on some model **problems** show the simplicity and efficiency of the method. Numer- ical results showed that the proposed method is convergent and at least second-order of accu- rate.

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where p, q ∈ C (a, b) and f ∈ L w [a, b] are suﬃciently regular given functions such that equation () satisﬁes the existence and uniqueness of the **solution**. α, β are ﬁnite constants. Without loss of generality, we can assume that the **boundary** conditions in equation () are homogeneous []. In this paper, based on reproducing kernel theory, reproducing ker- nels with polynomial form will be constructed and a computational method is described in order to obtain the accurate **numerical** **solution** with polynomial form of equation () in the reproducing kernel spaces spanned by the Chebyshev basis polynomials. The paper is organized as follows. In the following section, a proper closed form of the Chebyshev orthonormal basis polynomials which independently satisfy the homogeneous **boundary** conditions on [a, b] will be introduced. In addition, a reproducing kernel with polynomial form will be constructed. In Section , our method as a Chebyshev reproducing kernel method (C-RKM) is introduced. A convergence analysis and an error estimation for the present method in L

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[11] Y. N. Reddy and P. Pramod Chakravarthy, “Method of Reduction of Order for Solving Singularly Perturbed **Two**-**Point** **Boundary** **Value** **Problems**,” Applied Mathe- matics and Computation, Vol. 136, No. 1, 2003, pp. 27- 45. doi:10.1016/S0096-3003(02)00015-2

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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each of the regularization methods (R1)–(R3) has a stabilising/regularizing eﬀect on the **numerical** **solution** of the inverse **boundary** **value** problem (B), provided that an appropriate criterion is employed for the selection of the regularization parameter λ or the truncation number k. More precisely, both the LC and the DP are suitable criteria for the regularization methods (R1)–(R3), whilst the GCV fails to provide a good **value** for λ or k and hence a corresponding accurate **solution** for Example 1. The best combinations, in terms of the accuracy, for the **numerical** **solution** of Example 1, are TRM-LC, DSVD-LC and TSVD-DP, and these are displayed in Figs. 1–3, respectively. From these ﬁgures, as well as Table 1, one can conclude that the **numerical** solutions of Problem (B) for Example 1 obtained using these methods are all very accurate and stable with respect to decreasing the amount of noise in the data.

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The exact **solution** is y(x) = cos(π x/2) and the order of convergence, according to the scheme in [7], is O(hln h). **Numerical** results using the new scheme are shown in Table 4.1. The results show that the order of the convergence of the relative error is about 3.6.

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Many authors have studied the existence, nonexistence, and multiplicity of positive solu- tions for multipoint **boundary** **value** **problems** by using the ﬁxed-**point** theorem, the ﬁxed **point** index theory, and the lower and upper solutions method. We refer the readers to the references [–]. Recently, Hao, Liu and Wu [] studied the existence, nonexistence, and multiplicity of positive solutions for the following nonhomogeneous **boundary** **value** **problems**:

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For other results on the existence and multiplicity of positive solutions and nodal solutions for boundary value problems of ordinary di ﬀ erential equations based on bifurcation techniq[r]

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The purpose of this paper is to develop the method of upper and lower solutions and the method of quasilinearization 22–26. Under suitable conditions on f, we obtain a monotone sequence of solutions of linear **problems**. We show that the sequence of approximants converges uniformly and quadratically to a unique **solution** of the problem.

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This note is concerned with an iterative method for the **solution** of singular **boundary** **value** prob- lems. It can be considered as a predictor-corrector method. Sufficient conditions for the conver- gence of the method are introduced. A number of **numerical** examples are used to study the appli- cability of the method.

From table 1 and 2, it was observed that the Standard Collocation and Perturbed Collocation Method were both accurate methods (i.e Numerically) of solving higher order **boundary** **value** **problems** with the perturbed Collocation having slightly greater degree of accuracy than Standard Collocation Method, but Perturbed Collocation Method involving more tedious work when compared to the Standard Collocation Method.

3 . (1.2) 2. A technical treatment of (1.1). We observe that in (1.1), equation (1.1a) is a third order equation with an unknown constant β, while the **boundary** **value** condition (1.1b) contains four equalities. Hence following [4], we make the following technical treat- ment of (1.1).

grid points are clustered in regions of rapid variation of the **solution** and relatively few grid points are placed in regions where the **solution** is smooth. Most codes attempt to control the error in the **solution** either by estimating the discrete error at the mesh points [7] or else by controlling a residual [28]. The important **point** to realise is that in both cases the codes attempt to control the local error on the assumption that the problem is well conditioned so that if the local error is small then the global error will also be small. However in the case where the problem is ill-conditioned a standard backward error analysis shows that it is possible for an accepted **solution** to have a small local error but to have an unacceptably large global error. There is the additional problem that if a monitor function ( [1], p. 363) takes no account of conditioning of a problem then the grid choosing algorithm may become very inefficient. This is manifested by a sort of cycling where points are added into the grid on conditioning considerations and are then removed in the next remeshing due to accuracy considerations. To illustrate these ideas we present a test problem which we solve by using the code TWPBVPC [11] both with the standard mesh selection strategy based on the estimation of the local error and with a mesh selection strategy which considers both conditioning and local error estimation. We note that to change from the code TWPBVPC, which takes account of conditioning, to TWPBVP, which uses a conventional mesh choosing strategy, we need to change just one input parameter of the code TWPBVPC.

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The applications of **boundary** **value** **problems** (BVPs) are almost unlimited, and they play an important role in all the branches of science, engineering, and technology. They are ap- plied to model many systems in several ﬁelds of science and engineering. In recent years, there has been signiﬁcant advancement in solving **problems** related to a system of linear and nonlinear partial and ordinary diﬀerential equations concerning **boundary** conditions (BC). **Two** **point** nonlinear BVPs often cannot be solved by analytical techniques. With cumulative interest in ﬁnding solutions to linear/nonlinear BVPs has come an increas- ing requirement for **solution** techniques. In the present paper, we will study the algebraic results of the following linear 6th order BVP:

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Abstract: In this article, we present a novel finite difference method for the **numerical** **solution** of the eighth order **boundary** **value** **problems** in ordinary differential equations. We have discretized the problem by using the **boundary** conditions in a natural way to obtain a system of equations. Then we have solved system of equations to obtain a **numerical** **solution** of the problem. Also we obtained **numerical** values of derivatives of **solution** as a byproduct of the method. The **numerical** experiments show that proposed method is efficient and fourth order accurate.

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are presented in Tables and Figures in comparison with the exact solutions. HWM solutions are quite satisfactory in comparison with the existing **numerical** solutions available in the literature [1, 3, 4, 12, 13, 24, 30]. This scheme is easy to implement with computer programs and it can be extend for higher order with slight modiﬁca- tion. Proposed Theorem 3.1 reveals that the present method will contribute exact **solution** for diﬀerential equations, whose solutions are in the form of polynomials of ﬁnite degree. This is important for the development of new research in the ﬁeld of **numerical** analysis and beneﬁcial for new researchers. Also, Theorems 3.2 and 3.3 on uniform convergence and error analysis.

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