Fractional Differential Equations

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

On the ψ-Hilfer fractional derivative

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total memory and work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t –1–α on the interval [Δt, T] with a uniform absolute error e. We give the theoretical analysis to show that the number of exponentials N exp needed is of order for T≫1 or for TH1 for fixed accuracy e. The resulting algorithm requires only storage and work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/AF5FDC74FD7A010ED0ACD291BB1A92B1/S1815240617000147a.pdf/div-class-title-fast-evaluation-of-the-caputo-fractional-derivative-and-its-applications-to-fractional-diffusion-equations-div.pdf

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

Existence of mild solution for evolution equation with Hilfer fractional derivative

We consider an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. We prove existence and uniqueness of global solutions in the space of weighted continuous functions. The stability of the solution for a weighted Cauchy-type problem is also analyzed.

Existence and uniqueness for a problem involving Hilfer fractional derivative

In this paper the nonlinear viscoelastic wave equation in canonical form |u t | ρ u tt - Δu - Δu tt + ∫ t 0 g(t - τ)Δu(τ) dτ = b|u| P-2 u with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set.

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

We consider the following nonlinear viscoelastic equation: | u t | ρ u t t − Δ u − Δ u t t + ∫ 0 t g ( t − s ) Δ u ( s ) d s = 0 , in a bounded domain Ω for ρ > 0  We show that the dissipation induced by the integral term is strong enough to stabilize the solution This result improves an earlier one given by Cavalcanti et al in [MM Cavalcanti, VND Cavalcanti, J Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math Methods Appl Sci 24 (2001) 1043–1053]

Exponential and polynomial decay for a quasilinear viscoelastic equation

Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

/pdf/on-a-differential-equation-involving-hilfer-hadamard-1mi2lez69s.pdf

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Papers

Existence and uniqueness for a problem involving Hilfer fractional derivative

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

Exponential and polynomial decay for a quasilinear viscoelastic equation

Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives

On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative