Journal of Difference Equations and Applications, Sep 1, 2012
ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete ... more ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.
This presentation deals with the numerical solution of a reaction-diffusion problems, where the t... more This presentation deals with the numerical solution of a reaction-diffusion problems, where the time derivative is of fractional order. Since the fractional derivative of a function depends on its past history, these systems can successfully model evolutionary problems with memory, as for example electrochemical processes, porous or fractured media, viscoelastic materials, bioengineering applications. On the side of numerical simulation, the research mainly focused on suitable extensions of methods for PDE. This approach often produced low accuracy and/or high computational methods, due to the lack of smoothness of the analytical solution and to the longrange history dependence of the fractional derivative. Here we consider a finite difference scheme along space, to discretize the integer-order spatial derivatives, while we adopt a spectral collocation method through time. A suitable choice of the function basis produces an exponential convergence though time at a low computational cost, since the spectral method avoids the step-by-step method
The time fractional derivative of a function y(t) depends on the past history of the function y(t... more The time fractional derivative of a function y(t) depends on the past history of the function y(t), and so time fractional differential systems are naturally suitable to describe evolutionary processes with memory. Fractional models are increasingly used in many modelling situations including, for example, viscoelastic materials in mechanics, anomalous diffusion in transport dynamics of complex systems and some biological processes in rheology. Here we consider a time-fractional reaction diusion problem [2]. This is a non-local model and as the solution depends on all its past history, numerical step-by-step methods are computationally expensive. We propose a mixed method, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods and is exponentially convergent [3]. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given [1]. References [1] Burrage, K., Cardone, A., D'Ambrosio, R. and Paternoster, B. 2017 Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116 8294. [2] Gafiychuk, V., Datsko, B. and Meleshko, V. 2008 Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1-2) 215225. [3] Zayernouri, M. and Karniadakis, G. Em 2014 Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1) A40A62
The aim of our research is the construction of efficient and accurate numerical methods for the s... more The aim of our research is the construction of efficient and accurate numerical methods for the solution of Volterra Integro-Differential Equations (VIDEs). In order to increase the order of convergence of classical one-step collocation methods, we propose multistep collocation methods, which have been successfully introduced for Volterra integral equations in [1; 2]. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. In this talk we describe the derivation of multistep collocation methods for VIDEs and the analysis of convergence and stability properties. We show some examples of methods which compare favorably with respect to existing one-step methods. This is a joint work with B. Paternoster and D. Conte from University of Salerno. REFERENCES [1] D. Conte, Z. Jackiewicz, B. Paternoster. Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput., 204 :839{853, 2008. [2] D. Conte, B. Paternoster. Multistep collocation methods for Volterra Integral Equations. Appl. Math. Comput., 59 :1721-1736, 2009
Journal of Difference Equations and Applications, Sep 1, 2012
ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete ... more ABSTRACT Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.
This presentation deals with the numerical solution of a reaction-diffusion problems, where the t... more This presentation deals with the numerical solution of a reaction-diffusion problems, where the time derivative is of fractional order. Since the fractional derivative of a function depends on its past history, these systems can successfully model evolutionary problems with memory, as for example electrochemical processes, porous or fractured media, viscoelastic materials, bioengineering applications. On the side of numerical simulation, the research mainly focused on suitable extensions of methods for PDE. This approach often produced low accuracy and/or high computational methods, due to the lack of smoothness of the analytical solution and to the longrange history dependence of the fractional derivative. Here we consider a finite difference scheme along space, to discretize the integer-order spatial derivatives, while we adopt a spectral collocation method through time. A suitable choice of the function basis produces an exponential convergence though time at a low computational cost, since the spectral method avoids the step-by-step method
The time fractional derivative of a function y(t) depends on the past history of the function y(t... more The time fractional derivative of a function y(t) depends on the past history of the function y(t), and so time fractional differential systems are naturally suitable to describe evolutionary processes with memory. Fractional models are increasingly used in many modelling situations including, for example, viscoelastic materials in mechanics, anomalous diffusion in transport dynamics of complex systems and some biological processes in rheology. Here we consider a time-fractional reaction diusion problem [2]. This is a non-local model and as the solution depends on all its past history, numerical step-by-step methods are computationally expensive. We propose a mixed method, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods and is exponentially convergent [3]. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given [1]. References [1] Burrage, K., Cardone, A., D'Ambrosio, R. and Paternoster, B. 2017 Numerical solution of time fractional diffusion systems. Appl. Numer. Math. 116 8294. [2] Gafiychuk, V., Datsko, B. and Meleshko, V. 2008 Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220(1-2) 215225. [3] Zayernouri, M. and Karniadakis, G. Em 2014 Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1) A40A62
The aim of our research is the construction of efficient and accurate numerical methods for the s... more The aim of our research is the construction of efficient and accurate numerical methods for the solution of Volterra Integro-Differential Equations (VIDEs). In order to increase the order of convergence of classical one-step collocation methods, we propose multistep collocation methods, which have been successfully introduced for Volterra integral equations in [1; 2]. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. In this talk we describe the derivation of multistep collocation methods for VIDEs and the analysis of convergence and stability properties. We show some examples of methods which compare favorably with respect to existing one-step methods. This is a joint work with B. Paternoster and D. Conte from University of Salerno. REFERENCES [1] D. Conte, Z. Jackiewicz, B. Paternoster. Two-step almost collocation methods for Volterra integral equations. Appl. Math. Comput., 204 :839{853, 2008. [2] D. Conte, B. Paternoster. Multistep collocation methods for Volterra Integral Equations. Appl. Math. Comput., 59 :1721-1736, 2009
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