Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration
ABSTRACT In this paper we present operational matrix of integration (OMI) of Chebyshev wavelets basis and the product operation matrix (POM) of it. Some comparative examples are included to demonstrate the superiority of operational matrix of Chebyshev wavelets to those of Legendre wavelets.
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ABSTRACT: Wavelet solution techniques have become the focus of interest among researchers in different disciplines of science and technology. In this paper, implementation of two different wavelet basis functions has been comparatively considered for dynamic analysis of structures. For this aim, computational technique is developed by using free scale of simple Haar wavelet, initially. Later, complex and continuous Chebyshev wavelet basis functions are presented to improve the time history analysis of structures. Free-scaled Chebyshev coefficient matrix and operation of integration are derived to directly approximate displacements of the corresponding system. In addition, stability of responses has been investigated for the proposed algorithm of discrete Haar wavelet compared against continuous Chebyshev wavelet. To demonstrate the validity of the wavelet-based algorithms, aforesaid schemes have been extended to the linear and nonlinear structural dynamics. The effectiveness of free-scaled Chebyshev wavelet has been compared with simple Haar wavelet and two common integration methods. It is deduced that either indirect method proposed for discrete Haar wavelet or direct approach for continuous Chebyshev wavelet is unconditionally stable. Finally, it is concluded that numerical solution is highly benefited by the least computation time involved and high accuracy of response, particularly using low scale of complex Chebyshev wavelet.Mathematical Problems in Engineering 02/2015; 2015. DOI:10.1155/2015/956793 · 1.08 Impact Factor
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ABSTRACT: Approximations of functions in terms of orthogonal polynomials have been used to develop and implement numerical approaches to solve spectrally initial and boundary value problems. The main idea behind these approaches is to express differential and integral operators by using matrices, and this, in turn, makes the numerical implementation easier to be expressed in computational algebraic languages. In this paper, the application of the methodology is enlarged by using Dirac’s formalism, combined with complex Fourier series.Communications in Nonlinear Science and Numerical Simulation 08/2014; 19(8):2614–2623. DOI:10.1016/j.cnsns.2014.01.001 · 2.57 Impact Factor
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ABSTRACT: a b s t r a c t In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one-and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work [1–3] from one-and two-dimensional boundary-value problems to one-and two-dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legen-dre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approx-imated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise per-formance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accu-racy in such cases, whereas HWCM produces a stable solution in such cases as well. Con-trary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann's boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numer-ical tests affirm better accuracy of the proposed methods for a range of benchmark problems.Applied Mathematical Modelling 04/2013; 37 (2013):9455-9481. DOI:10.1016/j.apm.2013.04.014 · 2.16 Impact Factor