A study of the hyperbolic heat conduction problem and laplace inversion using generalized haar wavelet operational matrix method / Suazlan bin Mt Aznam - Mt Aznam, Suazlan (2012)
Wavelets have been applied successfully in image and signal processing. Many attempts have been made in mathematics to use wavelet function as numerical computational tool. In this study, an orthogonal wavelet function namely Haar wavelet function is considered. We used the operational matrix based on Haar basis to solve hyperbolic heat conduction equation problem and Laplace inversion. It is remarkably known by many that one of the difficulties encountered in numerical method for non-Fourier heat conduction problem is the numerical oscillation within the vicinity of jump discontinuities at the wave front. We propose a new method of solving non-Fourier heat conduction equation problem which is also a hyperbolic partial differential equation. Our new method for solving partial differential equation of hyperbolic heat conduction equation is a hybrid of finite difference method and pseudo spectral method, where the former for time discretization and the latter for spatial discretization. The time discretization is performed prior to spatial discretization. In this sense, partial differential equation is reduced to ordinary differential equation and solved implicitly with Haar wavelet basis. For pseudo spectral method, Haar wavelet expansion has been used considering its advantage of the absence of the Gibbs phenomenon at the jump discontinuities. Furthermore, definition of Haar wavelet basis in this work allows a pleasant way in computing inverse of Haar wavelet matrix. We also derived generalized Haar wavelet operational matrix in the interval of [0,X). The propose method have been applied into one physical problem namely, thin surface layers. It is found that the proposed numerical ii results could suppress and eliminate the numerical oscillation in the jump vicinity at a certain value of discretization. We also present a numerical method for inversion of Laplace transform using the method of Haar wavelet operational matrix. We prove the method for the case of the transfer function using the extension of Riemann-Liouville fractional integral. The proposed method extends the work of Wu et al. to cover the whole of time domain as we used the generalized Haar wavelet operational matrix. Moreover, this method gives an alternative numerical way to find the solution for inversion of Laplace transform in a simple way. The use of numerical generalized Haar operational matrix method is much simpler than the conventional contour integration method and it can be easily coded. Examples in finding Laplace inversion for rational, irrational and exponential transfer function are illustrated.Furthermore, examples on solving differential equation by Laplace transform method are also included. Both of the proposed numerical methods are stable, convergent and easily coded. Numerical results also demonstrate good performance of the method in term of accuracy and competitiveness compared to other numerical methods. Additionally, few benefits come from its great features such as faster computation and attractiveness.