A nonpolynomial collocation method for fractional terminal value problems

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational and Applied Mathematics, 275, February 2015, doi: 10.1016/j.cam.2014.06.013In this paper we propose a non-polynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a non-polynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.The work was supported by an International Research Excellence Award funded through the Santander Universities scheme

An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time

In this paper we are concerned with the numerical solution of a diffusion equation in which the time order derivative is distributed over the interval [0,1]. An implicit numerical method is presented and its unconditional stability and convergence are proved. A numerical example is provided to illustrate the obtained theoretical results

A distributed order viscoelastic model for small deformations

In this work we discuss the connection between classical, fractional and dis- tributed order viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the dis- tributed order Maxwell model. We derive the storage and loss modulus functions for the distributed order viscoelastic model and perform a fitting to experimental data. The fitting results are compared with the Maxwell and Fractional Maxwell models.L.L. Ferr´as would also like to thank FCT for financial support through projects UIDB/ 00013/2020 and UIDP/00013/2020. M.L. Morgado aknowledges funding by FCT through project UID/Multi/04621/2019 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Su perior T´ecnico, University of Lisbon. This work was partially supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (Por tuguese Foundation for Science and Technology) through the project UIDB/00297/2020 (Centro de Matem´atica e Aplica¸c˜oes). The authors also acknowledge financial support from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology)

Stable and convergent finite difference schemes on nonuniformtime meshes for distributed-order diffusion equations

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.The authors acknowledge the support of the Center for Mathematics and Applications (CMA)—FCT-NOVA, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, and CMAT—Centre of Mathematics—University of Minho. The first author acknowledges Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) within Projects UIDB/04621/2020 and UIDP/04621/2020. The second author acknowledges the Fundação para a Ciência e a Tecnologia through Project UIDB/00297/2020 (Centro de Matemática e Aplicações). The third author acknowledges the funding by Fundação para a Ciência e Tecnologia through Projects UIDB/00013/2020 and UIDP/00013/202

High-order methods for systems of fractional ordinary differential equations and their application to time-fractional diffusion equations

Taking into account the regularity properties of the solutions of fractional differential equations, we develop a numerical method which is able to deal, with the same accuracy, with both smooth and nonsmooth solutions of systems of fractional ordinary differential equations of the Caputo-type. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples. Finally, we solve the time-fractional diffusion equation using a combination of the method of lines and the newly developed hybrid method.L.L. Ferras would like to thank FCT - Fundacao para a Ciencia e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) for financial support through the scholarship SFRH/BPD/100353/2014 and Project UID-MAT-00013/2013. M.L. Morgado aknowledges the financial support of FCT, through the Project UID/Multi/04621/2019 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior Tecnico, University of Lisbon. This work was also partially supported by FCT through the Project UID/MAT/00297/2019 (Centro de Matematica e Aplicacoes)

Fractional pennes' bioheat equation: Theoretical and numerical studies

Accepted for publication in Fractional calculus and applied analysisOriginally published in the journal Fract. Cal. Appl. Anal. Vol. 18 No. 4 / 2015 / pp.1080–1106 / DOI 10.1515/fca-2015-0062. The original publication is available at: http://www.degruyter.com/view/j/fca.2015.18.issue-4/fca-2015-0062/fca-2015-0062.xml?rskey=sWWcn0&result=1In 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.The authors L.L. Ferras and J. M. Nobrega acknowledge financial funding by FEDER through the COMPETE 2020 Programme and by FCT- Portuguese Foundation for Science and Technology under the projects UID/CTM/50025/2013 and EXPL/CTM-POL/1299/2013. L.L. Ferras acknowledges financial funding by the Portuguese Foundation for Science and Technology through the scholarship SFRH/BPD/100353/2014. M. Rebelo acknowledges financial funding by the Portuguese Foundation for Science and Technology through the project UID/MAT/00297/2013

Semi-analytical solutions for the poiseuille-couette flow of a generalised Phan-Thien-Tanner fluid

This work presents new analytical and semi-analytical solutions for the pure Couette and Poiseuille-Couette flows, described by the recently proposed (Ferras et al., A Generalised Phan-Thien-Tanner Model, JNNFM 2019) viscoelastic model, known as the generalised Phan-Thien-Tanner constitutive equation. This generalised version considers the Mittag-Leffler function instead of the classical linear or exponential functions of the trace of the stress tensor, and provides one or two new fitting constants in order to achieve additional fitting flexibility. The analytical solutions derived in this work allow a better understanding of the model, and therefore contribute to improve the modelling of complex materials, and will provide an interesting challenge to computational rheologists, to benchmarking and to code verification.This research was funded by FEDER through COMPETE2020-Programa Operacional Competitividade e Internacionalizacao (POCI) and by national funds through FCT-Fundacao para a Ciencia e a Tecnologia, I. P. through Projects PTDC/EMS-ENE/3362/2014, POCI-01-0145-FEDER-016665, UID-MAT-00013/2013, and UID/MAT/00297/2013 as well as grant number SFRH/BPD/100353/2014. This work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matematica e Aplicacoes)

Recent Advances in Complex Fluids Modeling

In this chapter, we present a brief description of existing viscoelastic models, starting with the classical differential and integral models, and then focusing our attention on new models that take advantage of the enhanced properties of the Mittag-Leffler function (a generalization of the exponential function). The generalized models considered in this work are the fractional Kaye-Bernstein, Kearsley, Zapas (K-BKZ) integral model and the differential generalized exponential Phan-Thien and Tanner (PTT) model recently proposed by our research group. The integral model makes use of the relaxation function obtained from a step-strain applied to the fractional Maxwell model, and the differential model generalizes the familiar exponential Phan-Thien and Tanner constitutive equation by substituting the exponential function of the trace of the stress tensor by the Mittag-Leffler function. Since the differential model is based on local operators, it reduces the computational time needed to predict the flow behavior, and, it also allows a simpler description of complex fluids. Therefore, we explore the rheometric properties of this model and its ability (or limitations) in describing complex flows

Platinum-Triggered Bond-Cleavage of Pentynoyl Amide and N-Propargyl Handles for Drug-Activation.

The ability to create ways to control drug activation at specific tissues while sparing healthy tissues remains a major challenge. The administration of exogenous target-specific triggers offers the potential for traceless release of active drugs on tumor sites from antibody-drug conjugates (ADCs) and caged prodrugs. We have developed a metal-mediated bond-cleavage reaction that uses platinum complexes [K2PtCl4 or Cisplatin (CisPt)] for drug activation. Key to the success of the reaction is a water-promoted activation process that triggers the reactivity of the platinum complexes. Under these conditions, the decaging of pentynoyl tertiary amides and N-propargyls occurs rapidly in aqueous systems. In cells, the protected analogues of cytotoxic drugs 5-fluorouracil (5-FU) and monomethyl auristatin E (MMAE) are partially activated by nontoxic amounts of platinum salts. Additionally, a noninternalizing ADC built with a pentynoyl traceless linker that features a tertiary amide protected MMAE was also decaged in the presence of platinum salts for extracellular drug release in cancer cells. Finally, CisPt-mediated prodrug activation of a propargyl derivative of 5-FU was shown in a colorectal zebrafish xenograft model that led to significant reductions in tumor size. Overall, our results reveal a new metal-based cleavable reaction that expands the application of platinum complexes beyond those in catalysis and cancer therapy.EPSRC studentship for Benjamin Stenton