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Caputo derivative (6)

Adams method (2)Finite difference method (2)Fractional differential equation (2)Fractional differential equations (2)bioheat equation, (1)convergence (1)Distributed order differential equations (1)distributed order equations (1)Error estimates (1)View MoreJournalApplied Mathematics and Computation (1)BIT Numerical Mathematics (1)Computers and mathematics with applications (1)Electronic Transactions on Numerical Analysis (1)Fractional Calculus and Applied Analysis -
Fract. Calc. Appl. Anal (1)View MoreAuthorsFord, Neville J. (5)Morgado, Maria L. (3)Rebelo, Magda S. (2)Yan, Yubin (2)Diethelm, Kai (1)Ferras, Luis L. (1)Liu, Yanzhi (1)Nobrega, João M. (1)Pal, Kamal (1)Roberts, Jason (1)Types
Article (6)

Preprint (1)

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

Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S. (Kent State University/Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences, 2015-06-10)

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.

Higher order numerical methods for solving fractional differential equations

Yan, Yubin; Pal, Kamal; Ford, Neville J. (Springer, 2013-10-05)

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 < α < 1. The order of convergence of the numerical method is O(h^(3−α)). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α >0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Detailed error analysis for a fractional adams method with graded meshes

Liu, Yanzhi; Roberts, Jason; Yan, Yubin (Springer, 2017-09-21)

We consider a fractional Adams method for solving the nonlinear fractional differential equation $\, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0$, equipped with the initial conditions $y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots, \lceil \alpha \rceil -1$. Here $\alpha$ may be an arbitrary positive number and $ \lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ and the differential operator is the Caputo derivative. Under the assumption $\, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T]$, Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ and proved that this method has the optimal convergence order uniformly in $t_{n}$, that is $O(N^{-2})$ if $\alpha > 1$ and $O(N^{-1-\alpha})$ if $\alpha \leq 1$. They also showed that if $\, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T]$, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $ 0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if $\, ^{C}_{0}D^{\alpha}_{t} y$ behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Fractional pennes' bioheat equation: Theoretical and numerical studies

Ferras, Luis L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda S.; Nobrega, João M. (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.

Distributed order equations as boundary value problems

Ford, Neville J.; Morgado, Maria L. (Elsevier, 2012-01-20)

This preprint discusses the existence and uniqueness of solutions and proposes 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.

Multi-order fractional differential equations and their numerical solution

Diethelm, Kai; Ford, Neville J. (Elsevier, 2004)

This article considers the numerical solution of (possibly nonlinear) fractional differential equations of the form y(α)(t)=f(t,y(t),y(β1)(t),y(β2)(t),…,y(βn)(t)) with α>βn>βn−1>>β1 and α−βn1, βj−βj−11, 0

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