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In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L2 and L∞ norms. [ABSTRACT FROM AUTHOR]

In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the L 1 scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics. [ABSTRACT FROM AUTHOR]

In this study, we introduce a novel framework for exploring the dynamics of tumor growth and an evolution model for two-stage carcinogenic mutations in two-dimensions based on a system of reaction-diffusion equations. It is shown theoretically that the system is globally stable in the absence of both delay and diffusion. The inclusion of diffusion does not destabilize the system, while including delay does capture the key elements of how normal cells convert into cancer cells. To further validate these results, several numerical experiments are performed for different parameter values involved in the model equation. These parameter values are chosen in the sense that they have some biological meanings using the steady states of the equilibrium points. For the purpose of simulation, a stable Euler scheme is used for temporal discretization, while a Fourier spectral method is used for space variables, which is a natural choice due to the periodic boundary conditions in the model equation. The numerical simulation results further confirm our theoretical justification. [ABSTRACT FROM AUTHOR]

In this article, we use the applications of special functions in the form of Chebyshev polynomials to find the approximate solution of hyperbolic partial differential equations (PDEs) arising in the mathematical modeling of transmission line subject to appropriate symmetric Dirichlet and Neumann boundary conditions. The special part of the model equation is discretized using a Chebyshev differentiation matrix, which is centro-asymmetric using the symmetric collocation points as grid points, while the time derivative is discretized using the standard central finite difference scheme. One of the disadvantages of the Chebyshev differentiation matrix is that the resultant matrix, which is obtained after replacing the special coordinates with the derivative of Chebyshev polynomials, is dense and, therefore, needs more computational time to evaluate the resultant algebraic equation. To overcome this difficulty, an algorithm consisting of fast Fourier transformation is used. The main advantage of this transformation is that it significantly reduces the computational cost needed for N collocation points. It is shown that the proposed scheme converges exponentially, provided the data are smooth in the given equations. A number of numerical experiments are performed for different time steps and compared with the analytical solution, which further validates the accuracy of our proposed scheme. [ABSTRACT FROM AUTHOR]

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.compeleceng.2010.04.003 Byline: Nasro Min-Allah (a)(b), Ishtiaq Ali (c), Jiansheng Xing (d), Yongji Wang (b) Keywords: Real-time systems; Fixed-priority scheduling; Preemptive real-time scheduling; Feasibility analysis; Online schedulability Abstract: Due to polynomial time complexity, utilization based tests are desired for online feasibility analysis of periodic task systems. However, the associated disadvantage with these tests is that they propose a bound on system utilization, which trade processor utilization for performance. On the contrary, response time based tests share pseudo-polynomial time complexity, which are very expensive in terms of analysis time and therefore, impractical for analyzing feasibility of online systems. Realizing the advantage of utilization based tests over response time tests, attempts are being made to propose utilization based exact tests that achieve 100% CPU utilization for the system by modifying task parameters such as restricting task periods to be harmonic. We show that in systems where task deadlines are large, better results are obtained by making the task deadlines harmonic. The paper proposes a novel solution to feasibility problem of periodic task system under the assumption of composite deadline by providing a utilization based exact test with an upper bound of 1 and complexity O(n). Author Affiliation: (a) Department of Computer Science, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan (b) Laboratory for Internet Technologies, Institute of Software, Chinese Academy of Sciences, Beijing 100080, China (c) Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan (d) Department of Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA Article History: Received 9 October 2008; Revised 19 October 2009; Accepted 15 April 2010 Article Note: (footnote) [star] Reviews processed and proposed for publication to the Editor-in-Chief by Associate Editor Dr. F. Sahin.

In this article we present a semi-analytic method for solving elliptic partial differential equations. The technique is based on using a spectral method approximation for the second-order derivative in one of the spatial directions followed by solving the resulting system of second-order differential equations by an analytic method. That is, the system of second-order two-point boundary-value problems are solved analytically by casting them in first-order form and solving the resulting set of first-order equations by using the matrix exponential. An important aspect of our technique is that the solution obtained is semi-analytic, e.i., using analytic method in y and spectral method in x. The new method can be used for both linear and non-linear boundary conditions as well as for nonlinear elliptic partial differential equations. [ABSTRACT FROM AUTHOR]