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Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
01 Jan 1999-
TL;DR: In this article, the authors present a method for computing fractional derivatives of the Fractional Calculus using the Laplace Transform Method and the Fourier Transformer Transform of fractional Derivatives.
Abstract: Preface. Acknowledgments. Special Functions Of Preface. Acknowledgements. Special Functions of the Fractional Calculus. Gamma Function. Mittag-Leffler Function. Wright Function. Fractional Derivatives and Integrals. The Name of the Game. Grunwald-Letnikov Fractional Derivatives. Riemann-Liouville Fractional Derivatives. Some Other Approaches. Sequential Fractional Derivatives. Left and Right Fractional Derivatives. Properties of Fractional Derivatives. Laplace Transforms of Fractional Derivatives. Fourier Transforms of Fractional Derivatives. Mellin Transforms of Fractional Derivatives. Existence and Uniqueness Theorems. Linear Fractional Differential Equations. Fractional Differential Equation of a General Form. Existence and Uniqueness Theorem as a Method of Solution. Dependence of a Solution on Initial Conditions. The Laplace Transform Method. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. Fractional Green's Function. Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General Case: n-term Equation. Other Methods for the Solution of Fractional-order Equations. The Mellin Transform Method. Power Series Method. Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. Numerical Evaluation of Fractional Derivatives. Approximation of Fractional Derivatives. The "Short-Memory" Principle. Order of Approximation. Computation of Coefficients. Higher-order Approximations. Numerical Solution of Fractional Differential Equations. Initial Conditions: Which Problem to Solve? Numerical Solution. Examples of Numerical Solutions. The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations. Fractional-Order Systems and Controllers. Fractional-Order Systems and Fractional-Order Controllers. Example. On Viscoelasticity. Bode's Analysis of Feedback Amplifiers. Fractional Capacitor Theory. Electrical Circuits. Electroanalytical Chemistry. Electrode-Electrolyte Interface. Fractional Multipoles. Biology. Fractional Diffusion Equations. Control Theory. Fitting of Experimental Data. The "Fractional-Order" Physics? Bibliography. Tables of Fractional Derivatives. Index.
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TL;DR: In this article, a new paradigm of learning partial differential equations from small data is presented, which is essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equation, to extract patterns from high-dimensional data generated from experiments.
986 citations
Cites background from "Fractional differential equations :..."
...ut + (−∇α x )u = 0, (31) where α is the unknown parameter and (−∇α x ) is the fractional Laplacian operator [65]....
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...Here, kn,n−1(x, x′; θ, λ1, λ2) was obtained by taking the inverse Fourier transform [65] of [1 − tλ1(−iw ′)λ2 ]̂k(w, w ′; θ), where ̂k(w, w ′; θ) is the Fourier transform of the kernel k(x, x′; θ)....
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...In particular, λ2 is the fractional order of the operator D2 −∞,x that is defined in the Riemann–Liouville sense [65]....
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TL;DR: A new deep neural network called DeepONet can lean various mathematical operators with small generalization error and can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations.
Abstract: It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.
675 citations
TL;DR: In this article, a collection of numerical algorithms for the solution of various problems arising in fractional models is presented, which will give the engineer the necessary tools required to work with fractional model in an efficient way.
592 citations
TL;DR: It is found that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics, which is consistent with fractional order differentiation.
Abstract: Neural systems adapt to changes in stimulus statistics. The authors find that neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics, and that for individual neurons the firing is a fractional derivative of slowly varying stimulus parameters. Neural systems adapt to changes in stimulus statistics. However, it is not known how stimuli with complex temporal dynamics drive the dynamics of adaptation and the resulting firing rate. For single neurons, it has often been assumed that adaptation has a single time scale. We found that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics. This multiple time scale adaptation is consistent with fractional order differentiation, such that the neuron's firing rate is a fractional derivative of slowly varying stimulus parameters. Biophysically, even though neuronal fractional differentiation effectively yields adaptation with many time scales, we found that its implementation required only a few properly balanced known adaptive mechanisms. Fractional differentiation provides single neurons with a fundamental and general computation that can contribute to efficient information processing, stimulus anticipation and frequency-independent phase shifts of oscillatory neuronal firing.
588 citations
10 Sep 2008
TL;DR: In this article, the commutativity properties of the fractional sum and fractional difference operators are studied. But the particular goal is to define and solve well-defined discrete fractional differential equations.
Abstract: This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators Then a v-th (0 < v ≤ 1) order fractional difference equation is defined A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method
576 citations