Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

On the numerical solution and dynamical laws of nonlinear fractional Schrödinger/Gross-Pitaevskii equations

International audienceThe purpose of this paper is to discuss some recent developments concerning the numerical simulation of space and time fractional Schrödinger and Gross-Pitaevskii equations. In particular, we address some questions related to the discretization of the models (order of accuracy and fast implementation) and clarify some of their dynamical properties. Some numerical simulations illustrate these points

Numerical solution methods for fractional partial differential equations

Fractional partial differential equations have been developed in many different fields such as physics, finance, fluid mechanics, viscoelasticity, engineering and biology. These models are used to describe anomalous diffusion. The main feature of these equations is their nonlocal property, due to the fractional derivative, which makes their solution challenging. However, analytic solutions of the fractional partial differential equations either do not exist or involve special functions, such as the Fox (H-function) function (Mathai & Saxena 1978) and the Mittag-Leffler function (Podlubny 1998) which are diffcult to evaluate. Consequently, numerical techniques are required to find the solution of fractional partial differential equations. This thesis can be considered as two parts, the first part considers the approximation of the Riemann-Liouville fractional derivative and the second part develops numerical techniques for the solution of linear and nonlinear fractional partial differential equations where the fractional derivative is defied as a Riemann-Liouville derivative. In the first part we modify the L1 scheme, developed initially by Oldham & Spanier (1974), to develop the three schemes which will be defined as the C1, C2 and C3 schemes. The accuracy of each method is considered. Then the memory effect of the fractional derivative due to nonlocal property is discussed. Methods of reduction of the computation L1 scheme are proposed using regression approximations. In the second part of this study, we consider numerical solution schemes for linear fractional partial differential equations. Here the numerical approximation schemes are developed using an approximation of the fractional derivative and a spatial discretization scheme. In this thesis the L1, C1, C2, C3 fractional derivative approximation schemes, developed in the first part of the thesis, are used in conjunction with either the Centred-finite difference scheme, the Dufort-Frankel scheme or the Keller Box scheme. The stability of these numerical schemes are investigated via the technique of the Fourier analysis (Von Neumann stability analysis). The convergence of each the numerical schemes is also discussed. Numerical tests were used to conform the accuracy and stability of each proposed method. In the last part of the thesis numerical schemes are developed to handle nonlinear partial differential equations and systems of nonlinear fractional partial differential equations. We considered two models of a reversible reaction in the presence of anomalous subdiffusion. The Centred-finite difference scheme and the Keller Box methods are used to spatially discretise the spatial domain in these schemes. Here the L1 scheme and a modification of the L1 scheme are used to approximate the fractional derivative. The accuracy of the methods are discussed and the convergence of the scheme are demonstrated by numerical experiments. We also give numerical examples to illustrate the e�ciency of the proposed scheme

An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

In this paper, we construct and analyze a linearized finite difference/Galerkin-Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0<β0<β1<β2<⋯<βm<1. The problem is first approximated by the L1 difference method on the temporal direction, and then, the Galerkin-Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2-βm in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results. © 2021 A. K. Omran et al.A. K. Omran is funded by a scholarship under the joint executive program between the Arab Republic of Egypt and Russian Federation. M. A. Zaky wishes to acknowledge the support of the Nazarbayev University Program (091019CRP2120). M. A. Zaky wishes also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”)

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems. © 2022 by the authors.King Saud University, KSUM. A. Zaky and A. Aldraiweesh extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia)

Узагальнення рівнянь фільтрації та тепломасоперенесення на випадок суфозійних процесів

Запропоновано методологію виведення рівнянь фільтрації в гетерогенних пористих середовищах з урахуванням довільної скінченної кількості факторів впливу. Методологія полягає в записі рівнянь нерозривності для фаз пористого середовища через повні похідні в часі. В якості конкретизації факторів впливу вибрано теплосолеперенесення, хімічну та механічну суфозії, що не зменшує її (методології) загальності. Зроблено огляд залежностей параметрів фаз ґрунту (густина, коефіцієнт фільтрації) від впливу теплосолеперенесення та концентрації суфозійних частинок. Сформовано математичну модель і визначено напрямки подальших досліджень.A methodology of analytical derivation of filtration equation for fluid equation in heterogeneous porous media taking into account finite number of the influence factors has been proposed. The methodology consists in the record of the continuity equations of liquid and solid phases of the porous medium with using a material derivative with time. For example, heat and mass transfer, chemical and mechanical suffusion have been chosen as impact factors. A survey of experimental and theoretical dependences of the densities of the soil phases and hydraulic conductivity of the soil upon these factors has been performed. The review phase dependences of parameters of soil (density, hydraulic conductivity) has been made. The mathematical model has been formed and areas for further research has been identified

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

An Accurate and Robust Numerical Scheme for Transport Equations

En esta tesis se presenta una nueva técnica de discretización para ecuaciones de transporte en problemas de convección-difusión para el rango completo de números de Péclet. La discretización emplea el flujo exacto de una ecuación de transporte unidimensional en estado estacionario para deducir una ecuación discreta de tres puntos en problemas unidimensionales y cinco puntos en problemas bidimensionales. Con "flujo exacto" se entiende que se puede obtener la solución exacta en función de integrales de algunos parámetros del fluido y flujo, incluso si estos parámetros son vari- ables en un volumen de control. Las cuadraturas de alto orden se utilizan para lograr resultados numéricos cercanos a la precisión de la máquina, incluso con mallas bastas.Como la discretización es esencialmente unidimensional, no está garantizada una solución con precisión de máquina para problemas multidimensionales, incluso en los casos en que las integrales a lo largo de cada coordenada cartesiana tienen una primitiva. En este sentido, la contribución principal de esta tesis consiste en una forma simple y elegante de obtener soluciones en problemas multidimensionales sin dejar de utilizar la formulación unidimensional. Además, si el problema es tal que la solución tiene precisión de máquina en el problema unidimensional a lo largo de las líneas coordenadas, también la tendrá para el dominio multidimensional.In this thesis, we present a novel discretization technique for transport equations in convection-diffusion problems across the whole range of Péclet numbers. The discretization employs the exact flux of a steady-state one-dimensional transport equation to derive a discrete equation with a three-point stencil in one-dimensional problems and a five-point stencil in two-dimensional ones. With "exact flux" it is meant that the exact solution can be obtained as a function of integrals of some fluid and flow parameters, even if these parameters are variable across a control volume. High-order quadratures are used to achieve numerical results close to machine- accuracy even with coarse grids. As the discretization is essentially one-dimensional, getting the machine- accurate solution of multidimensional problems is not guaranteed even in cases where the integrals along each Cartesian coordinate have a primitive. In this regard, the main contribution of this thesis consists in a simple and elegant way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain.<br /