1,721,234 research outputs found
Numerical solution of time fractional diffusion systems
Full text linkIn this paper a general class of diffusion problem is considered, where the standard time derivative is replaced by a fractional one. For the numerical solution, a mixed method is proposed, 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 to discretize the fractional derivative. 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
The African Institute for Mathematical Sciences
No full textIts mission is to promote Mathematics and Science in Africa and to provide a focal point for Mathematics university training in Africa. It offers scholarships for up to 50 students to come and study for a period of nine months. Of the 50 students, about 15 positions are reserved for females. In the 2006/2007 intake there were over 250 applicants.\ud
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The students are housed and fed and their return travel from their home town is fully funded. Lecturers also stay at AIMS and share their meals with the students, so that a rapport quickly develops. The students are away from their families and friends for nine months and are absolutely committed to the discipline of Mathematics. When they first arrive, some of them have little ability in English but since all tuition is in English they quickly learn. Some find the transitions difficult but they all support one another and at the end of their time their English skills are very good. The students do a series of subjects that last for about three weeks each, consisting of 30 contact hours, as well as a thesis/project. Each course has a number of assignments associated with it and these get evaluated. AIMS has seven or eight teaching assistants who help with the tutorials, marking, advice, and who are a vital component of AIMS
Numerical solution of time-fractional reaction-diffusion systems
No full textFractional differential systems model many dynamical phenomena all associated with memory aspects. These include anomalous diffusion in transport dynamics, the response of viscoelastic materials under mechanical stress, some biological processes in rheology and the kinetics of complex systems in spatially crowded domains. In recent years, considerable attention has been paid to fractional reaction-diffusion systems, where the fractional derivative index alpha produces new effects with respect to the classical model. For example, in the nonlinear model [1], when 0<alpha<1, a relaxation process arises, and when 1<alpha<2 periodic solutions may occur. In this talk we analyse the numerical treatment of time-fractional reaction-diffusion systems. As the solution depends on all its past history, numerical step-by-step methods are computationally expensive. On the other hand spectral methods can avoid the discretization of the ‘heavy tail’ and are exponentially convergent [2]. We propose a numerical scheme consisting of a spectral method through time, on a basis of functions suitably chosen for the problem, and a finite-difference method through space, whose coefficients are adapted according to the qualitative behaviour of the solution. Finally we illustrate preliminary numerical results on some significant test equations.
[1] V. Gafiychuk, B. Datsko, and V. Meleshko. Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math., 220(1-2):215–225, 2008.
[2] M. Zayernouri and G. Em Karniadakis. Fractional spectral collocation method. SIAM J. Sci. Comput., 36(1):A40–A62, 2014.
Keywords: reaction-diffusion systems, fractional differential equations, spectral methods, finite-difference schemes
Efficient simulation of stochastic chemical kinetics with the Stochastic Bulirsch-Stoer extrapolation method
Full text linkBackgroundBiochemical systems with relatively low numbers of components must be simulated stochastically in order to capture their inherent noise. Although there has recently been considerable work on discrete stochastic solvers, there is still a need for numerical methods that are both fast and accurate. The Bulirsch-Stoer method is an established method for solving ordinary differential equations that possesses both of these qualities.ResultsIn this paper, we present the Stochastic Bulirsch-Stoer method, a new numerical method for simulating discrete chemical reaction systems, inspired by its deterministic counterpart. It is able to achieve an excellent efficiency due to the fact that it is based on an approach with high deterministic order, allowing for larger stepsizes and leading to fast simulations. We compare it to the Euler ?-leap, as well as two more recent ?-leap methods, on a number of example problems, and find that as well as being very accurate, our method is the most robust, in terms of efficiency, of all the methods considered in this paper. The problems it is most suited for are those with increased populations that would be too slow to simulate using Gillespie’s stochastic simulation algorithm. For such problems, it is likely to achieve higher weak order in the moments.ConclusionsThe Stochastic Bulirsch-Stoer method is a novel stochastic solver that can be used for fast and accurate simulations. Crucially, compared to other similar methods, it better retains its high accuracy when the timesteps are increased. Thus the Stochastic Bulirsch-Stoer method is both computationally efficient and robust. These are key properties for any stochastic numerical method, as they must typically run many thousands of simulations
A mixed spectral method for time-fractional reaction-diffusion systems
No full textFractional differential systems arise in many fields, and are particularly suitable to model processes with memory. For example, the wall-friction through the fluid boundary layer exhibits some cumulative memory effects, and may be modeled by fractional partial derivatives in Navier–Stokes equations. Moreover reaction–diffusion phenomena with anomalous diffusion such as occurs in spatially inhomogeneous environments, are modeled by fractional reaction–diffusion equations. The numerical approximation to fractional differential systems is not trivial, since one has to take into account the non-local nature and the long the long-range history dependence of such problems. Here we consider the numerical solution of time-fractional reaction-diffusion equations. We propose a numerical scheme consisting of a spectral method through time, on a basis of functions suitably chosen for the problem, and a finite-difference method through space, whose coefficients are adapted according to the qualitative behaviour of the solution. The aim is to reduce the computational cost and to obtain an exponential superconvergence of the time discretization. On the other hand, a finite-difference scheme along space would simplify the implementation of the overall method, is enough accurate for most applications and is flexible, since its coefficients may be specially tuned on the problem
A higher-order numerical framework for stochastic simulation of chemical reaction systems
Full text linkBACKGROUND: In this paper, we present a framework for improving the accuracy of fixed-step methods for Monte Carlo simulation of discrete stochastic chemical kinetics. Stochasticity is ubiquitous in many areas of cell biology, for example in gene regulation, biochemical cascades and cell-cell interaction. However most discrete stochastic simulation techniques are slow. We apply Richardson extrapolation to the moments of three fixed-step methods, the Euler, midpoint and θ-trapezoidal τ-leap methods, to demonstrate the power of stochastic extrapolation. The extrapolation framework can increase the order of convergence of any fixed-step discrete stochastic solver and is very easy to implement; the only condition for its use is knowledge of the appropriate terms of the global error expansion of the solver in terms of its stepsize. In practical terms, a higher-order method with a larger stepsize can achieve the same level of accuracy as a lower-order method with a smaller one, potentially reducing the computational time of the system. RESULTS: By obtaining a global error expansion for a general weak first-order method, we prove that extrapolation can increase the weak order of convergence for the moments of the Euler and the midpoint τ-leap methods, from one to two. This is supported by numerical simulations of several chemical systems of biological importance using the Euler, midpoint and θ-trapezoidal τ-leap methods. In almost all cases, extrapolation results in an improvement of accuracy. As in the case of ordinary and stochastic differential equations, extrapolation can be repeated to obtain even higher-order approximations. CONCLUSIONS: Extrapolation is a general framework for increasing the order of accuracy of any fixed-step stochastic solver. This enables the simulation of complicated systems in less time, allowing for more realistic biochemical problems to be solved
Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise
No full textThere has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods
Effects of intrinsic and extrinsic noise can accelerate juxtacrine pattern formation
No full textEpithelial pattern formation is an important phenomenon that, for example, has roles in embryogenesis, development and wound-healing. The ligand Epithelial Growth Factor (EGF) and its receptor EGF-R, constitute a system that forms lateral induction patterns by juxtacrine signalling—binding of membrane-bound ligands to receptors on neighbouring cells. Owen et al. developed a generic ordinary differential equation model of juxtacrine lateral induction that exhibits stable patterning under some conditions. The model predicts relatively slow pattern formation. We examine here the effects of both intrinsic and extrinsic cellular noise arising from the stochastic treatment of this model, and show that this noise could have an accelerating effect on the patterning process
Krylov and steady-state techniques for the solution of the chemical master equation for the mitogen-activated protein kinase cascade
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