Abstract
A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order two-point boundary value problems (BVPs) and initial value problems (IVPs) in ordinary differential equations. The second kind Chebyshev wavelets operational matrix of integration is derived and used to transform the problem to a system of algebraic equations. The uniform convergence analysis and error estimation for the proposed method are given. Accuracy and efficiency of the suggested method are established through comparing with the existing quintic B-spline collocation method, homotopy asymptotic method, and modified decomposition method. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literatures.
Highlights
Y(4) (a) = δ5, y(5) (a) = δ6, y(6) (a) = δ7, y(7) (a) = δ8, where y and f are continuous functions defined on the interval [a, b], and f ∈ C8[a, b] is real and αi, βi, γi, and δi, i = 1, 2, . . . , 8, are finite real numbers
Different types of wavelets and approximating functions have been used in the numerical solution of boundary value problems [14]
Chebyshev wavelets are widely used in solving nonlinear integrodifferential equations and partial differential equations [15,16,17,18]
Summary
Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. 2k−1, m is the degree of the second kind Chebyshev polynomials, and t is the normalized time They are defined on the interval [0, 1) as ψn,m (t). Um(t) are the second kind Chebyshev polynomials of degree m which are orthogonal with respect to the weight function ω(t) = √1 − t2 on the interval [−1, 1] and satisfy the following recursive formula: U0 (t) = 1, U1 (t) = 2t,. Note that when dealing with the second kind Chebyshev wavelets the weight function has to be dilated and translated as ωn (t) = ω (2kt − 2n + 1). Let f(x) be a second-order derivative squareintegrable function defined on [0, 1) with bounded secondorder derivative; say |f(x)| ≤ B for some constant B; (i) f(x) can be expanded as an infinite sum of the second kind Chebyshev wavelets and the series converges to f(x) uniformly; that is,
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