scholarly journals The Wright Functions of the Second Kind in Mathematical Physics

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 884 ◽  
Author(s):  
Francesco Mainardi ◽  
Armando Consiglio
Keyword(s):  
Wave Equations ◽  
Diffusion Processes ◽  
Mathematical Physics ◽  
Fundamental Solutions ◽  
Point Of View ◽  
Diffusion Wave ◽  
Analytical Properties ◽  
Accessible Point ◽  
Non Gaussian ◽  
Gaussian Stochastic Processes

In this review paper, we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics. We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices, we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

scholarly journals Competing risks joint models using R-INLA

2021 ◽  
Vol 21 (1-2) ◽  
pp. 56-71
Author(s):  
Janet van Niekerk ◽  
Haakon Bakka ◽  
Håvard Rue
Keyword(s):  
Competing Risks ◽  
Count Data ◽  
Joint Model ◽  
Point Of View ◽  
Spatial Structures ◽  
Joint Models ◽  
Hazard Functions ◽  
Longitudinal Count Data ◽  
Non Gaussian ◽  
Made In

The methodological advancements made in the field of joint models are numerous. None the less, the case of competing risks joint models has largely been neglected, especially from a practitioner's point of view. In the relevant works on competing risks joint models, the assumptions of a Gaussian linear longitudinal series and proportional cause-specific hazard functions, amongst others, have remained unchallenged. In this article, we provide a framework based on R-INLA to apply competing risks joint models in a unifying way such that non-Gaussian longitudinal data, spatial structures, times-dependent splines and various latent association structures, to mention a few, are all embraced in our approach. Our motivation stems from the SANAD trial which exhibits non-linear longitudinal trajectories and competing risks for failure of treatment. We also present a discrete competing risks joint model for longitudinal count data as well as a spatial competing risks joint model as specific examples.

scholarly journals A Brief Review of Generalized Entropies

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 813 ◽  
Author(s):  
José Amigó ◽  
Sámuel Balogh ◽  
Sergio Hernández
Keyword(s):  
Diffusion Processes ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Point Of View ◽  
Scaling Exponent ◽  
Scaling Exponents ◽  
Physical Systems ◽  
New Phenomena ◽  
Generalized Entropies

Entropy appears in many contexts (thermodynamics, statistical mechanics, information theory, measure-preserving dynamical systems, topological dynamics, etc.) as a measure of different properties (energy that cannot produce work, disorder, uncertainty, randomness, complexity, etc.). In this review, we focus on the so-called generalized entropies, which from a mathematical point of view are nonnegative functions defined on probability distributions that satisfy the first three Shannon–Khinchin axioms: continuity, maximality and expansibility. While these three axioms are expected to be satisfied by all macroscopic physical systems, the fourth axiom (separability or strong additivity) is in general violated by non-ergodic systems with long range forces, this having been the main reason for exploring weaker axiomatic settings. Currently, non-additive generalized entropies are being used also to study new phenomena in complex dynamics (multifractality), quantum systems (entanglement), soft sciences, and more. Besides going through the axiomatic framework, we review the characterization of generalized entropies via two scaling exponents introduced by Hanel and Thurner. In turn, the first of these exponents is related to the diffusion scaling exponent of diffusion processes, as we also discuss. Applications are addressed as the description of the main generalized entropies advances.

Quantification of Model-Form Uncertainty in Drift-Diffusion Simulation Using Fractional Derivatives

2015 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Optimization Procedure ◽  
Physical World ◽  
Drift Diffusion ◽  
Long Range Correlations ◽  
Modeling Process ◽  
Model Form ◽  
Generic Description ◽  
Non Gaussian

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest such as rare events difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker-Planck equation (fFPE) is proposed to describe the drift-diffusion processes under long-range correlations and memory effects. A new model calibration approach based on the maximum accumulative mutual information is also proposed to reduce model-form uncertainty, where an optimization procedure is taken.

Bernhard Riemann’s legacy of 1859

2009 ◽  
Vol 93 (528) ◽  
pp. 468-475
Author(s):  
Graham Hoare
Keyword(s):  
Number Theory ◽  
Mathematical Physics ◽  
Point Of View ◽  
German Version ◽  
Profound Impact ◽  
Almost All ◽  
University Of Göttingen ◽  
The University ◽  
Modern Mathematics ◽  
Mathematics And Physics

The German version of Riemann’s Collected Works is confined to a single volume of 690 pages. Even so, this volume has had an abiding and profound impact on modern mathematics and physics, as we shall see. In fifteen years of activity, from 1851, when he gained his doctorate at the University of Göttingen, to his death in 1866, two months short of his fortieth birthday, Riemann contributed to almost all areas of mathematics. He perceived mathematics from the analytic point of view and used analysis to illuminate subjects as diverse as number theory and geometry. Although regarded principally as a mathematician Riemann had an abiding interest in physics and researched significantly in the methods of mathematical physics, particularly in the area of partial differential equations.

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