Abstract
In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coeffcients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.
Highlights
Nonlocal multi-point boundary value problems arise in applied mathematics, physics, engineering and the various areas of mechanics such as theory of elasticity, theory of elastic stability and theory of plates and shells [14]
Finding solutions analytically to multi-point boundary value problems represented by linear ordinary differential equations with variable coefficients involving nonlocal boundary conditions can be based on determining of some fundamental solutions such as Green’s function or Green’s functional [10, 11, 12, 13, 15]
The presented method in [16] for solving the problem represented by an ordinary differential equation with only derivative conditions is implemented to obtain the approximate solution to linear fourth-order multi-point BVP governed by an ordinary differential equation with both derivative and integral conditions in this work
Summary
Nonlocal multi-point boundary value problems arise in applied mathematics, physics, engineering and the various areas of mechanics such as theory of elasticity, theory of elastic stability and theory of plates and shells [14]. Reproducing kernel theory has many potential applications in numerical analysis of differential equations [1, 2, 3, 4, 5, 6, 7, 9, 17] In this context, the presented method in [16] for solving the problem represented by an ordinary differential equation with only derivative conditions is implemented to obtain the approximate solution to linear fourth-order multi-point BVP governed by an ordinary differential equation with both derivative and integral conditions in this work. If {xi}∞ i=1 is dense on [0, 1] and the exact solution v(x) of problem (2.5)–(2.6) exists and is unique, it can be expanded in terms of a Fourier series about orthonormal system {ψi(x)}∞ i=1 as in (2.7) by noting that w(x), Rxi (x) W21 = w(xi) for each w(x) ∈ W21[0, 1]. Vn(x) is n-truncation of the Fourier series corresponding to v(x)
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