Maria Luísa Morgado

Fractional boundary value problems: Analysis and numerical methods

In this paper we consider nonlinear boundary value problems for differential equations of fractional order α, 0 < α < 1. We study the existence and uniqueness of the solution and extend existing published results. In the last part of the paper we study a class of prototype methods to determine their numerical solution.

Numerical approximation of distributed order reaction-diffusion equations

In this paper an implicit scheme for the numerical approximation of the distributed order time-fractional reaction-diffusion equation with a nonlinear source term is presented. The stability and the convergence order of the numerical scheme are analysed and illustrated through some numerical examples.

Analysis and numerical methods for fractional differential equations with delay

In this paper, we consider fractional differential equations with delay. We focus on linear equations. We summarise existence and uniqueness theory based on the method of steps and we give a theorem on the propagation of derivative discontinuities. We discuss the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward difference method to the delay case.

Distributed order equations as boundary value problems

There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the solution depends, among other parameters, on Caputo-type initial conditions. In this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin.

Bubbles and droplets in nonlinear physics models: Analysis and numerical simulation of singular nonlinear boundary value problem

For a second-order nonlinear ordinary differential equation (ODE), a singular Boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble-type (droplet-type) solutions are sought. The equation, defined on a semi-infinite interval 0 < r < ∞, possesses a regular singular point as r→ 0 and an irregular one as r→ ∞. We give the restrictions to the parameters for a correct mathematical statement of the limit boundary conditions in singular points and their accurate transfer into the neighborhoods of these points using certain results for singular Cauchy problems and stable initial manifolds. The necessary and sufficient conditions for the existence of bubble-type (droplet-type) solutions are discussed (in the form of additional restrictions to the parameters) and some estimates are obtained. A priori detailed analysis of a singular nonlinear BVP leads to efficient shooting methods for solving it approximately. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

High Order Numerical Methods for Fractional Terminal Value Problems

Abstract. In this paper we present a shooting algorithm to solve fractional terminal (or boundary) value problems. We provide a convergence analysis of the numerical method, derived based upon properties of the equation being solved and without the need to impose smoothness conditions on the solution. The work is a sequel to our recent investigation where we constructed a nonpolynomial collocation method for the approximation of the solution to fractional initial value problems. Here we show that the method can be adapted for the effective approximation of the solution of terminal value problems. Moreover, we compare the efficiency of this numerical scheme against other existing methods.

Fractional pennes' bioheat equation: theoretical and numerical studies

Abstract 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 bioheat 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.

A nonpolynomial collocation method for fractional terminal value problems

In this paper we propose a nonpolynomial 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 nonpolynomial 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.

Analysis and numerical approximation of singular boundary value problems with the p-Laplacian in fluid mechanics

This paper studies a generalization of the Cahn-Hilliard continuum model for multi-phase fluids where the classical Laplacian has been replaced by a degenerate one (i.e., the so-called p-Laplacian). The solutions asymptotic behavior is analyzed at two singular points; namely, at the origin and at infinity. An efficient technique for treating such singular boundary value problems is presented, and results of numerical integration are discussed and compared with earlier computed data.

Density profile equation with p-Laplacian: Analysis and numerical simulation

Analytical properties of a nonlinear singular second order boundary value problem in ordinary differential equations posed on an unbounded domain for the density profile of the formation of microscopic bubbles in a nonhomogeneous fluid are discussed. Especially, sufficient conditions for the existence and uniqueness of solutions are derived. Two approximation methods are presented for the numerical solution of the problem, one of them utilizes the open domain Matlab code bvpsuite. The results of numerical simulations are presented and discussed.