667 research outputs found
Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations
We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein
integral equations with non-Lipschitz nonlinearity. We present different kinds
of existence and uniqueness of nontrivial collocation solutions and we give
conditions for such existence and uniqueness in some cases. Finally we
illustrate these methods with an example of a collocation problem, and we give
some examples of collocation problems that do not fit in the cases studied
previously.Comment: 18 pages, 4 figure
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 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 spline collocation method for the fractional order logistic equation
We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method
A fractional wavelet Galerkin method for the fractional diffusion problem
The aim of this paper is to solve some fractional differential problems hav-
ing time fractional derivative by means of a wavelet Galerkin method that
uses the fractional scaling functions introduced in a previpous paper as approximating
functions. These refinable functions, which are a generalization of the
fractional B-splines, have many interesting approximation properties.
In particular, their fractional derivatives have a closed form that involves
just the fractional difference operator. This allows us to construct accurate
and efficient numerical methods to solve fractional differential problems.
Some numerical tests on a fractional diffusion problem will be given
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