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A class of cubic and quintic spline modified collocation methods for the solution of two-point boundary value problems
This paper is concerned with the study of a class of methods for solving second and fourth-order two-point boundary-value problems. The methods under
consideration are modifications of the standard cubic and quintic spline
collocation techniques, and are derived by making use of recent results con- cerning the a posteriori correction of cubic and quintic interpolating spline
A multiscale collocation method for fractional differential problems
We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale
collocation method are proved and some numerical results are shown.We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional
derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage
of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale
collocation method are proved and some numerical results are shown
A fractional B-spline collocation method for the numerical solution of fractional predator-prey models
We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems: a computational study
An alternating direction implicit (ADI) orthogonal spline collocation (OSC)
method is described for the approximate solution of a class of nonlinear
reaction-diffusion systems. Its efficacy is demonstrated on the solution of
well-known examples of such systems, specifically the Brusselator, Gray-Scott,
Gierer-Meinhardt and Schnakenberg models, and comparisons are made with other
numerical techniques considered in the literature. The new ADI method is based
on an extrapolated Crank-Nicolson OSC method and is algebraically linear. It is
efficient, requiring at each time level only operations where
is the number of unknowns. Moreover,it is shown to produce
approximations which are of optimal global accuracy in various norms, and to
possess superconvergence properties
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