Nonlinear fractional-periodic boundary value problems with Hilfer fractional derivative: existence and numerical approximations of solutions

The Hilfer fractional derivative generalizes and interpolates between the commonly used Riemann-Liouville and Caputo fractional derivative. In general, solutions to Hilfer fractional derivative initial value problems are singular for $t \downarrow t_0$, providing a challenge in numerical approximation. In this work, we present a Bernstein splines method to obtain accurate approximations to solutions of nonlinear Hilfer fractional derivative initial value problems, including solutions with singular behavior. We provide explicit convergence requirements and asymptotic convergence rates to analytical solutions. Moreover, the method is efficiently implemented, using a vectorized approach and parallelization in polynomial order. Numerical experiments show empirical convergence corresponding to analytical results and a significantly higher accuracy compared to the commonly used fractional Adams-Bashforth-Moulton predictor-corrector method. Furthermore, our method is able to simulate nonlinea …

This work presents a new framework for approximating Caputo fractional derivatives (FDs) of any positive order using a shifted Gegenbauer pseudospectral (SGPS) method. By transforming the Caputo FD into a scaled integral of the $m$th-derivative of the Lagrange interpolating polynomial (with $m$ being the ceiling of the fractional order $α$), we mitigate the singularity near zero, improving stability and accuracy. The method links $m$th-derivatives of shifted Gegenbauer (SG) polynomials with SG polynomials of lower degrees, allowing for precise integration using SG quadratures. We employ orthogonal collocation and SG quadratures in barycentric form to obtain an accurate and efficient approach for solving fractional differential equations. We provide error analysis showing that the SGPS method is convergent in a semi-analytic framework and conditionally convergent with exponential rate for smooth functions in finite-precision arithmetic. This exponential convergence improves accuracy com …

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional Caputo derivative of order $α\in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $ρ$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $ρ\partial_t^αu-\textrm{div}(a \nabla u)+ q u=0$ on $(0,T)\times Ω$ are recovered simultaneously.

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(NM\log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semialgebraic mode analysis is also developed to the …

Twenty years ago the anti-reflective numerical boundary conditions (BCs) were introduced in a context of signal processing and imaging, for increasing the quality of the reconstruction of a blurred signal/image contaminated by noise and for reducing the overall complexity to that of few fast sine transforms i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of pixels. Here for quality of reconstruction we mean a better accuracy and the elimination of boundary artifacts, called ringing effects. Now we propose numerical anti-reflective BCs in the context of nonlocal problems of fractional differential type: the goals are the same i.e. a smaller approximation error and the reduction of boundary artifacts. In the latter setting, we compare various types of numerical BCs, including the anti-symmetric ones considered in the case of fractional differential problems for modeling reasons. More in detail, given important similarities between anti-symmetric and anti-reflecti …