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numerical methods (16)

fractional differential equations (6)bifurcations (3)integral equations (3)stochastic delay equations (3)multi-term equations (2)qualitative behaviour (2)Bagley-Torvik equation (1)Capute derivative (1)Caputo derivative (1)View MoreJournalJournal of Computational and Applied Mathematics (2)Computers and Fluids (1)Computers and mathematics with applications (1)Fractional calculus and applied analysis (1)AuthorsFord, Neville J. (13)Diogo, Teresa (3)Lima, Pedro M. (3)Morgado, Maria L. (3)Norton, Stewart J. (3)Ford, Judith M. (2)Simpson, A. Charles (2)Connolly, Joseph A. (1)Diethelm, Kai (1)Edwards, John T. (1)View MoreTypesArticle (7)Technical Report (4)Doctoral (2)PhD (2)Preprint (2)View More

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A high order numerical method for solving nonlinear fractional differential equation with non-uniform meshes

Fan, Lili; Yan, Yubin (Springer Link, 2019-01-18)

We introduce a high-order numerical method for solving nonlinear fractional differential equation with non-uniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N-1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j-1}, t_{j}, t_{j+1}$. A high-order numerical method is obtained. Then we apply this numerical method with the non-uniform meshes with the step size $\tau_{j}= t_{j+1}- t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the non-uniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same non-uniform meshes.

The numerical solution of linear multi-term fractional differential equations: Systems of equations

Edwards, John T.; Ford, Neville J.; Simpson, A. Charles (Elsevier, 2002-11)

This article discusses how the numerical approximation of a linear multi-term fractional differential equation can be calculated by the reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity.

Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Ford, Neville J.; Diogo, Teresa; Ford, Judith M.; Lima, Pedro M. (Elsevier, 2007-08)

Noise-induced changes to the behaviour of semi-implicit Euler methods for stochastic delay differential equations undergoing bifurcation

Ford, Neville J.; Norton, Stewart J. (Elsevier, 2009-07-15)

This article discusses estimating parameter values at which bifurcations occur in stochastic delay differential equations. After a brief review of bifurcation, we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation. In this paper we provide a formulaic relationship between the estimated bifurcation value, the level of noise, the choice of numerical scheme and the step length. We are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise.

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.

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.

Ferras, L. L.; Ford, Neville J.; Morgado, Maria L.; Rebelo, Magda; McKinley, Gareth H.; Nobrega, Joao M. (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).

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