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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 (6)Technical Report (4)Doctoral (2)PhD (2)Preprint (2)View More

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Theory and numerics for multi-term periodic delay differential equations, small solutions and their detection

Ford, Neville J.; Lumb, Patricia M. (University of Chester, 2006)

We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (†) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Diogo, Teresa; Ford, Judith M.; Ford, Neville J.; Lima, Pedro M. (University of Chester, 2006)

We consider the qualitative behaviour of solutions to linear integral equations of the form where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.

Fractional boundary value problems: Analysis and numerical methods

Ford, Neville J.; Morgado, Maria L. (Springer, 2011-07-28)

This journal article discusses nonlinear boundary value problems.

Numerical approaches to the solution of some fractional differential equations

Ford, Neville J.; Simpson, A. Charles (Lea Press, 2002)

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

Ford, Neville J.; Norton, Stewart J. (University of Chester, 2007)

We are concerned with 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 maybe 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.

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.

Solution of a singular integral equation by a split-interval method

Diogo, Teresa; Ford, Neville J.; Lima, Pedro M.; Thomas, Sophy M. (2007)

This article discusses a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method is discussed and examples demonstrate its effectiveness.

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.

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. (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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