Numerical method for Volterra equation with a power-type nonlinearity

This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical computations challenging. Our strategy is to reduce the problem into a single one-dimensional Volterra integral equation for the selfsimilar solution and then to apply the discretization. The main difficulty arises due to the non-Lipschitzian behaviour of the equation's nonlinearity. By the analysis of the recurrence relation for the error we are able to prove that there exists a family of finite difference methods that is convergent for a large subset of the parameter space. We illustrate our results with a concrete example of a method based on the midpoint quadrature.for the main operator in the spatial diffusion, namely the fractional Laplacian, can be found in [20,11]. As for the nonlinear version of the anomalous diffusion, an interesting paper recently appeared in which the authors constructed a multi-grid waveform method of fast convergence [18]. Moreover, a similar equation to ours has been solved in [3] in the context of petroleum industry. Some other numerical approaches concern space-fractional nonlinear diffusion [17,14], nonlinear source terms [40,38,5] and variable order diffusion [26].Our approach is based on a transformation of the governing PDE to the equivalent nonlinear Volterra equation (for a recent survey of theory and numerical methods see [6]). In that case the classical convergence theory cannot be applied since the nonlinearity of the equation is non-Lipschitzian. According to our best knowledge, there is a scarce literature concerning similar problems. A very interesting paper is [7] where an iterative technique has been applied and convergence proofs given. Moreover, in [8] a short summary of the theoretical and numerical character has been published. Lastly, we mention our own work [31] from which the present considerations stem where we have given the convergence proof assuming the kernel's separation from zero. In the present discussion we relax this assumption.The paper is structured as follows. In the second section we formulate the problem in terms of the Volterra setting starting from the self-similar form of the time-fractional porous medium equation. We proceed to the main results in the third section where a construction of a convergent finite difference method is given. We end the paper with some numerical simulations illustrating our results.

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Numerical method for Volterra equation with a power-type nonlinearity

Cited by 5 publications

References 34 publications

A linear Galerkin numerical method for a quasilinear subdiffusion equation

A linear Galerkin numerical method for a quasilinear subdiffusion equation

Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

Numerical Method for the Time-Fractional Porous Medium Equation

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