Browsing Mathematics by Publisher "de Gruyter"
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An algorithm for the numerical solution of twosided spacefractional partial differential equations.We introduce an algorithm for solving twosided spacefractional partial differential equations. The spacefractional derivatives we consider here are lefthanded and righthanded Riemann–Liouville fractional derivatives which are expressed by using Hadamard finitepart integrals. We approximate the Hadamard finitepart integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the spacefractional derivative with convergence order

Fractional pennes' bioheat equation: Theoretical and numerical studiesIn this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperaturedependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bio heat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.

Identification of the initial function for nonlinear delay differential equationsWe consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudoNewton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.