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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.
- A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculusPublication . Ferreira, M.; Kraußhar, R. S.; Rodrigues, M. M.; Vieira, N.In this paper, we develop a fractional integro-differential operator calculus for Clifford-algebra valued functions. To do that we introduce fractional analogs of the Teodorescu and Cauchy-Bitsadze operators and we investigate some of their mapping properties. As a main result, we prove a fractional Borel-Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge-type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann-Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.
- Fractional gradient methods via ψ-Hilfer derivativePublication . Vieira, N.; Rodrigues, M. M.; Ferreira, M.Motivated by the increasing of practical applications in fractional calculus, we study the classical gradient method under the perspective of the ψ-Hilfer derivative. This allows us to cover in our study several definitions of fractional derivatives that are found in the literature. The convergence of the ψ-Hilfer continuous fractional gradient method is studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we develop an algorithm for the ψ-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and optimizing the step size, the ψ-Hilfer fractional gradient method shows better results in terms of speed and accuracy. Our results generalize previous works in the literature.
- Some representations for the eigenfunctions of the time-fractional wave operatorPublication . Rodrigues, M.M.; Ferreira, M.; Vieira, NelsonIn this work we present some new representations for the eigenfunctions of the time-fractional wave operator with the time-fractional derivative in the Caputo sense.
- A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator CalculusPublication . Ferreira, M.; Rodrigues, M. M.; Vieira, N.In this paper, we develop a time-fractional operator calculus in fractional Clifford analysis. Initially, we study the $L_p$-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogs of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.
- 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.
- 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.
- First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac operatorsPublication . Ferreira, Milton; Rodrigues, Manuela M.; Vieira, NelsonIn this work we obtain the first and second fundamental solutions (FS) of the multidimensional time-fractional equation with Laplace or Dirac operators, where the two time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] are in the Caputo sense. We obtain representations of the FS in terms of Hankel transform, double Mellin- Barnes integrals, and H-functions of two variables. As an application, the FS are used to solve Cauchy problems of Laplace and Dirac type.
- Psi-Hilfer fractional relaxation-oscillation equationPublication . Vieira, N.; Ferreira, M.; Rodrigues, M.M.In this work, we solve the ψ-Hilfer fractional relaxation-oscillation equation with a force term, where the time-fractional derivatives are in the ψ-Hilfer sense. The solution of the equation is presented in terms of bivariate Mittag-Leffler functions. An asymptotic analysis of the solution of the associated homogeneous equation is performed.