Browsing by Author "Rodrigues, M. Manuela"
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- Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph EquationPublication . Ferreira, M.; Rodrigues, M. Manuela; Vieira, NelsonIn this paper, we consider a non-homogeneous time-space-fractional telegraph equation in n-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm-Liouville operator defined in terms of right and left fractional Riemann-Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper, some illustrative examples are presented.
- First and second fundamental solutions of the time-fractional telegraph equation of order 2αPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, NelsonIn this work we obtain the first and second fundamental solutions of the multidimensional time-fractional equation of order 2α, α ∈]0, 1], where the two time-fractional derivatives are in the Caputo sense. We obtain representations of the fundamental solutions in terms of Hankel transform, double Mellin-Barnes integral, and H-functions of two variables. As an application, the fundamental solutions are used to solve a Cauchy problem and to study telegraph process with Brownian time.
- A Fractional Analysis in Higher Dimensions for the Sturm-Liouville ProblemPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, NelsonIn this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.
- Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph EquationPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, NelsonIn this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β.
- Fundamental solution of the time-fractional telegraph Dirac operatorPublication . Ferreira, Milton; Rodrigues, M. Manuela; Vieira, Nelsonn this work, we obtain the fundamental solution (FS) of the multidimensionaltime-fractional telegraph Dirac operator where the 2 time-fractional derivatives oforders𝛼∈]0,1]and𝛽∈]1,2]are in the Caputo sense. Explicit integral and seriesrepresentation of the FS are obtained for any dimension. We present and discusssome plots of the FS for some particular values of the dimension and of the frac-tional parameters𝛼and𝛽. Finally, using the FS, we study some Poisson and Cauchyproblems
- On a regular Ψ-fractional Sturm-Liouville problemPublication . Ferreira, M.; Rodrigues, M. Manuela; Vieira, NelsonIn this short paper, we consider a $\psi$-fractional Sturm-Liouville eigenvalue problem by using left $\psi$-Caputo and right $\psi$-Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem.
- Time-fractional diffusion equation with psi-Hilfer derivativePublication . Vieira, Nelson; Rodrigues, M. Manuela; Ferreira, MiltonIn this work, we consider the multidimensional time-fractional diffusion equation with the $\psi$-Hilfer derivative. This fractional derivative enables the interpolation between Riemann-Liouville and Caputo fractional derivatives and its kernel depends on an arbitrary positive monotone increasing function $\psi$ thus encompassing several fractional derivatives in the literature. This allows us to obtain general results for different families of problems that depend on the function $\psi$ selected. By employing techniques of Fourier, $\psi$-Laplace, and Mellin transforms, we obtain a solution representation in terms of convolutions involving Fox H-functions for the Cauchy problem associated with our equation. Series representations of the first fundamental solution are explicitly obtained for any dimension as well as the fractional moments of arbitrary positive order. For the one-dimensional case, we show that the series representation reduces to a Wright function, and we prove that it corresponds to a probability density function for any admissible $\psi$. Finally, some plots of the fundamental solution are presented for particular choices of the function $\psi$ and the order of differentiation.
- Time-fractional telegraph equation of distributed order in higher dimensionsPublication . Vieira, Nelson; Rodrigues, M. Manuela; Ferreira, MiltonIn this work, the Cauchy problem for the time-fractional telegraph equation of distributed order in Rn is considered. By employing the technique of the Fourier, Laplace, and Mellin transforms, a representation of the fundamental solution of this equation in terms of convolutions involving the Fox H-function is obtained. Some particular choices of the density functions in the form of elementary functions are studied. Fractional moments of the fundamental solution are computed in the Laplace domain. Finally, by application of the Tauberian theorems we study the asymptotic behaviour of the second-order moment (variance) in the time domain.
- Time-fractional telegraph equation of distributed order in higher dimensions with Hilfer fractional derivativesPublication . Vieira, Nelson; Rodrigues, M. Manuela; Ferreira, MiltonIn this paper, we consider time-fractional telegraph equations of distributed order in higher spatial dimensions, where the time derivatives are in the sense of Hilfer, thus interpolating between the Riemann-Liouville and the Caputo fractional derivatives. By employing the techniques of the Fourier, Laplace, and Mellin transforms, we obtain a representation of the solution of the Cauchy problem associated with the equation in terms of convolutions involving functions that are Laplace integrals of Fox H-functions. Fractional moments of the first fundamental solution are computed and for the special case of double-order distributed it is analyzed in detail the asymptotic behavior of the second-order moment, by application of the Tauberian Theorem. Finally, we exhibit plots of the variance showing its behavior for short and long times, and for different choices of the parameters along small dimensions.
- Time-fractional telegraph equation with ψ-Hilfer derivativesPublication . Vieira, Nelson; Ferreira, Milton; Rodrigues, M. ManuelaThis paper deals with the investigation of the solution of the time-fractional telegraph equation in higher dimensions with $\psi$-Hilfer fractional derivatives. By application of the Fourier and $\psi$-Laplace transforms the solution is derived in closed form in terms of bivariate Mittag-Leffler functions in the Fourier domain and in terms of convolution integrals involving Fox H-functions of two-variables in the space-time domain. A double series representation of the first fundamental solution is deduced for the case of odd dimension. The results derived here are of general nature since our fractional derivatives allow to interpolate between Riemann-Liouville and Caputo fractional derivatives and the use of an arbitrary positive monotone increasing function $\psi$ in the kernel allows to encompass most of the fractional derivatives in the literature. In the one dimensional case, we prove the conditions under which the first fundamental solution of our equation can be interpreted as a spatial probability density function evolving in time, generalizing the results of Orsingher and Beghin (2004). Some plots of the fundamental solutions for different fractional derivatives are presented and analysed, and particular cases are addressed to show the consistency of our results.