• Fractional pennes' bioheat equation: Theoretical and numerical studies

      Ferras, Luis L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; Nobrega, Joao M.; University of Minho & University of Chester, University of Chester, UTAD, UNL Lisboa, University of Minho (de Gruyter, 2015-08-04)
      In 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 temperature-dependent 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.
    • Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method

      Morgado, Maria L.; Rebelo, Magda S.; Ferras, Luis L.; Ford, Neville J.; Universidade de Tras-os-Montes e Alto Douro; Universidade NOVA de Lisboa; University of Minho; University of Chester (Elsevier, 2016-11-09)
      In this work we present a new numerical method for the solution of the distributed order time fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed.
    • Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.

      Ferras, Luis L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; McKinley, Gareth H.; Nobrega, Joao M.; University of Chester, University of Minho, UTAD, Universidade Nova de Lisboa (Elsevier, 2018-07-12)
      In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular ow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for the fractional diffusion-wave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics 56 (2006) 193-209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional step-strain, we observe the different rates of stress relaxation obtained with different values of \alpha and \beta (the fractional order exponents that regulate the viscoelastic response of the complex fluids).