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Yayın Analytical solution of the heat conduction equation in one-dimensional spherical coordinates at nanoscale(Maltepe Üniversitesi, 2009) Mohammadi-Fakhar, V.; Momeni-Masuleh, S. H.Heat conduction equation at microscale has been widely applied to thermal analysis of thin metal films. The microscopic heat flux equation developed from physical and mathematical reasoning is different from the traditional heat equation. Here a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time will appear in the heat equation. An approximate analytical solution to the non-Fourier heat conduction equation in one-dimensional spherical coordinates based on the dual-phase-lag framework is obtained by employing the Adomian decomposition method (ADM). The application of ADM to partial differential equations, when the exact solution is not reached or existed, demands the use of truncated series. The major reduction in computational effort associated with the ADM is the main factor behind their popularity while other numerical methods require extensive computation. The ADM does not discretize variables and gives an analytical solution in the form of truncated series. If there are nonlinear factors in an equation, ADM gives the analytical solution without any need for linearization. In this presentation, the reliability and efficiency of the solution were verified using the ADM.Yayın Application Of generalized purcell method for real eigenvalue problems(Maltepe Üniversitesi, 2009) Rahmani, M.; Momeni-Masuleh, S. H.In numerical linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal and image processing, image compression and statistics. A new method based on generalized Purcell method for real eigenvalue problem and QR decomposition of an arbitrary matrix is proposed. The method in comparison to the inverse power method generates better results and has less computational cost. In addition, the method obtains directly the rank of a matrix and gives linearlly independent eigenvectors corresponding to an eigenvalue.Yayın Fully spectral methods for the solution of high order differential equations(Maltepe Üniversitesi, 2009) Vaissmoradi, N.; Malek, A.; Momeni-Masuleh, S. H.In the recent years spectral methods are used for solving stiff and non-stiff partial differential equations and ordinary differential equations. Various types of spectral methods for steady and unsteady problems are proposed to solve stiff and non-stiff partial differential equations effciently. In this article some schemes for solving stiff partial differential equations are derived. There are twofold: first method is based on Chebyshev polynomials for solving high-order boundary value problems. Second methods are based on Fourier-Galerkin and collocation spectral methods in space and Runge-Kutta, exponential time differencing, Taylor expansion and contour integral in time for solving stiff PDEs. Numerical results show the effciency of proposed schemes.Yayın Fully spectral methods for the solution of high order differential equations(Maltepe Üniversitesi, 2009) Vaissmoradi, N.; Malek, A.; Momeni-Masuleh, S. H.In the recent years spectral methods are used for solving stiff and non-stiff partial differential equations and ordinary differential equations. Various types of spectral methods for steady and unsteady problems are proposed to solve stiff and non-stiff partial differential equations efficiently. In this article some schemes for solving stiff partial differential equations are derived. There are twofold: first method is based on Chebyshev polynomials for solving high-order boundary value problems. Second methods are based on Fourier-Galerkin and collocation spectral methods in space and Runge-Kutta, exponential time differencing, Taylor expansion and contour integral in time for solving stiff PDEs. Numerical results show the efficiency of proposed schemes.Yayın Heat conduction equation at micro and nano scale: approximation methods(Maltepe Üniversitesi, 2009) Momeni-Masuleh, S. H.In the classical theory of diffusion, Fourier law of heat conduction, it is assumed that the heat flux vector and temperature gradient across a material volume occur at the same instant of time. It has shown that if the scale in one direction is at the microscale (of order 0.1 µm), then the heat flux and temperature gradient occur in this direction at different times. In the so-called non-Fourier heat conduction equation a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time will appear. Among the frameworks to study the non-Fourier heat conduction equation, the dual-phase-lag framework is employed. In this talk, some numerical approaches for solving the heat conduction equation in various domains are presented.
