Abstract
Problem statement: The lacunary interpolation problem, which we had investigated in this study, consisted in finding the six degree spline S (x) of deficiency four, interpolating data given on the function value and third and fifth order in the int erval (0,1). Also, an extra initial condition was prescribed on the first derivative. Other purpose o f this construction was to solve the second order differential equations by two examples showed that the spline function being interpolated very well. The convergence analysis and the stability of the a pproximation solution were investigated and compared with the exact solution to demonstrate the prescribed lacunary spline (0, 3, 5) function interpolation. Approach: An approximation solution with spline interpolatio n functions of degree six and deficiency four was derived for solving initial value problems, with prescribed nonlinear endpoint conditions. Under suitable assumptions, the existen ces; uniqueness and the error bounds of the spline (0, 3, 5) function had been investigated; also the uppe r bounds of errors were obtained. Results: Numerical examples, showed that the presented spline function proved their effectiveness in solving the second order initial value problems. Also, we noted that, the better error bounds were obtained for a small s tep size h. Conclusion: In this study we treated for a first time a lacuna ry data (0,3,5) by constructing spline function of degree six which interpolated th e lacunary data (0,3,5) and the constructed spline function applied to solve the second order initial value problems.
Highlights
The initial value problems play an important role in mathematical physics, because many problems in science and technology are formulated mathematically in boundary value problems as in heat transfer and deflection in cables.Several numerical methods have been investigated for calculating the solutions of such problems
We present numerical results to demonstrate the convergence of the spline (0, 3, 5) function of degree six which constructed before to the second order initial value problem
Problem 1: we consider that the second order initial value problem y′′ = 1 (y′ + y) where x∈[0,1] and y(0) = y(0) = 1 with the exact solution y(x) = ex [6]
Summary
The initial value problems play an important role in mathematical physics, because many problems in science and technology are formulated mathematically in boundary value problems as in heat transfer and deflection in cables. Finite difference techniques and the shooting method play an important role. These methods provide the value of the unknown of some grid knots. The literature on the numerical solutions of initial value problems by using lacunary spline functions is not too much. Gyovari[2] solved Cauchy problem by sing modified lacunary spline function which interpolating the lacunary data (0, 2, 3) Saxena[7]. Saxena and Venturino[8] used two-point boundary value problem by using lacunary spline function which interpolates the lacunary data (0, 2). This study is organized as follows: First consider the spline function of degree six is presented which interpolates the lacunary data (0, 3, 5). To demonstrate the convergence of the prescribed lacunary spline function, numerical examples presented, we prescribe the conclusion and discussion of the result
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