Runge-Kutta-Chebyshev Method For Stiff Volterra Integral Equations Of The Second Kind And Time-Fractional Reaction-diffusion Equations

In this paper, we construct explicit Pouzet-Runge-Kutta-Chebyshev (PRKC) method for nonlinear stiff Volterra integral equations of the second kind, and explicit Abel-Runge-Kutta-Chebyshev (ARKC) method for the initial-boundary value problem of time-fractional reaction-subdiffusion equations with Caputo derivative.Runge-Kutta (RK) is a classical one-step method for ordinary differential equa-tions. Explicit Runge-Kutta-Chebyshev (RKC) method is a stabilized RK method of order one and two, of which the length of absolute stability region along the negative real axis is proportional to s2 (the number of stages). Therefore explicit RKC method can be used to solve huge nonlinear stiff ordinary differential system. The first Cheby-shev polynomial is exploited to construct the stability function in order to extend the stability region of the RKC method.Adopting spatial discretization the initial-boundary value problem of time-fractional reaction-subdiffusion equations with Caputo derivative can be transformed into nonlin-ear stiff and weakly singular Volterra integral equations of the second kind. So the same characteristics of two problems is nonlinear and stiff. Because it is difficult to imple-ment implicit solver, we utilize the idea of RKC method to construct numerical methods for two problems.This paper includes four chapters. The conclusion are involved in the last chapter, and the other chapters are organized in the following way.In first chapter we introduce background and theory foundation of our research, i.e. Volterra integral equations of the second kind, time-fractional diffusion equations, RK method and explicit RKC method for ordinary differential equation, RK method for nonsingular and weakly singular Volterra integral equations of the second kind.In second chapter we investigate PRKC method for nonlinear stiff Volterra inte- gral equations of the second kind. Because there are the same absolute stability re-gion and order between Volterra-Runge-Kutta method of Pouzet type (Pouzet-Volterra-Runge-Kutta, PVRK) method for Volterra integral equations of the second kind and corresponding RK method for ordinary differential equations, we choose the PVRK method as our purpose. We analysis the stability of PRKC method with respect to the basic test equation. And we built the stability function of PVRK method based the first Chebyshev polynomial so that we derive explicit second-order PRKC method for stiff Volterra integral equation of the second kind. We investigate theoretically and numeri-cally the discuss the stability based on convolution test equation, and obtain the relation between the stability region and the value of coefficients ε and s. The experiments show that PRKC method can efficiently solve nonlinear stiff Volterra integral equations of the second kind, and the absolute stability region is related with the value of coefficients ε and s.In third chapter we investigate explicit ARKC method for the initial-boundary val-ue problem of time-fractional reaction-diffusion equation with Caputo derivative. This method combine explicit PRKC method and RK method for weakly singular Volter-ra integral equations of the second kind (Abel-Runge-Kutta, ARK). Adopting spatial discretization the initial-boundary value problem of time-fractional reaction-diffusion equations with Caputo derivative can be transformed into a nonlinear stiff and weakly singular Volterra integral equations of the second kind, then we investigate actually a nonlinear stiff and weakly singular Volterra integral equations of the second kind. First-ly, we find the relation of coefficients between ARK method and Volterra-Runge-Kutta (VRK) method. According to this relation we can construct coefficients of a first-order ARK method based on those of a VRK method. Secondly, We construct explicit ARKC method based on the relation of coefficients between ARK method and VRK method and explicit PRKC method. We analysis numerically stability with respect to the test equation, and derive the relation between coefficients s, ε and λ,α. The experiments show that explicit ARKC method can efficiently solve nonlinear stiff and weakly sin-gular Volterra integral equation of the second kind.1.Explicit PRKC method for stiff Volterra integral equations of the second kindWe are interested in the stiff Volterra integral equations of the second kind where (?)K/(?)y and (?)2K/(?)t(?)y are negative real of which absolute value are large.Let in= nh, n= 0,1,..., N, Nh= T. The explicit s-stage PRKC method for the stiff Volterra integral equations of the second kind (1) has the following formula where Yni and yn+i are approximations of y(tn+ci/h) and y(tn+h) respectively, and satisfy which is an estimation ofThe PRKC method has the following coefficients2.Explicit ARKC method for time-fractional reaction-diffusion equations with Caputo derivativeThe time-fractional reaction-diflusion equations with Caputo derivative can be transformed into nonlinear stiff and weakly singular Volterra integral equations of the second kindThe explicit Abel-Runge-Kutta-Cheby shev (ARKC) method for the nonlinear stiff and weakly singular Volterra integral equation of the second kind (3) has the following formThe ARKC method has the following coefficientswhere aij bj and ci are coefficients of s-stage explicit PRKC method (2).