Fractional Differential Equations

Recent work has shown how to obtain the Page curve of an evaporating black hole from holographic computations of entanglement entropy. We show how these computations can be justified using the replica trick, from geometries with a spacetime wormhole connecting the different replicas. In a simple model, we study the Page transition in detail by summing replica geometries with different topologies. We compute related quantities in less detail in more complicated models, including JT gravity coupled to conformal matter and the SYK model. Separately, we give a direct gravitational argument for entanglement wedge reconstruction using an explicit formula known as the Petz map; again, a spacetime wormhole plays an important role. We discuss an interpretation of the wormhole geometries as part of some ensemble average implicit in the gravity description.

Replica wormholes and the black hole interior

This book is devoted to the study of maximum principles in partial differential equations. I t contains a wealth of material much of which is presented for the first time in a book form. An attractive feature of the book is that it is completely elementary and thus accessible to a wide audience of readers. The book has four chapters. Chapter I deals with the one dimensional maximum principle. The discussion of this very simple model of a maximum principle forms a good introduction to the general theory. Various applications of the principle are given to show that it is a very useful tool even in the study of ordinary differential equations. As an example, it is shown that many oscillation and comparison results in the Sturm-Liouville theory could be deduced most easily by a maximum principle argument. The proper discussion of maximum principles in partial differential equations begins in Chapter II . This chapter, which is the backbone of the book, is devoted to elliptic equations. The material covered in this chapter includes the E. Hopf maximum principle and its generalizations; the Phragmèn-Lindelöf principle for solutions of elliptic equations; Serrin's version of the Harnack inequality for solutions of general elliptic equations in two variables (this is probably the most difficult result discussed in the book) ; various versions of the Hadamard three circles theorems for solutions of elliptic equations; applications of the maximum principle to nonlinear equations and to problems of fluid flow. Chapter III is devoted to parabolic equations. The plan of this chapter parallels that of Chapter II . The topics discussed include the L. Nirenberg strong maximum principle; a three curves theorem with an interesting application to the Tikhonov uniqueness theorem; a Phragmèn-Lindelöf principle for parabolic equations with applications to uniqueness results; nonlinear operators; a maximum principle for certain parabolic systems. The fourth and the last chapter is devoted to hyperbolic equations. The results in this chapter are somewhat special since a maximum principle in the proper sense does not hold for solutions of hyperbolic equations. Nevertheless, solutions of certain hyperbolic equations

Maximum Principles In Differential Equations

https://link.springer.com/content/pdf/10.1007/JHEP03(2022)205.pdf

A bstract Recent work has shown how to obtain the Page curve of an evaporating black hole from holographic computations of entanglement entropy. We show how these computations can be justified using the replica trick, from geometries with a spacetime wormhole connecting the different replicas. In a simple model, we study the Page transition in detail by summing replica geometries with different topologies. We compute related quantities in less detail in more complicated models, including JT gravity coupled to conformal matter and the SYK model. Separately, we give a direct gravitational argument for entanglement wedge reconstruction using an explicit formula known as the Petz map; again, a spacetime wormhole plays an important role. We discuss an interpretation of the wormhole geometries as part of some ensemble average implicit in the gravity description. 

/pdf/replica-wormholes-and-the-black-hole-interior-21kufowf.pdf

This work focuses on an investigation of the (n+1)-dimensional time-dependent fractional Schrodinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.

A numerical method for the fractional Schrödinger type equation of spatial dimension two

https://iconline.ipleiria.pt/bitstream/10400.8/3824/1/FS_Fract_Diffusion_Wave_Post_Print.pdf

Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $$\Delta _+^{(\alpha , \beta , \gamma )}:= D_{x_0^+}^{1+\alpha } +D_{y_0^+}^{1+\beta } +D_{z_0^+}^{1+\gamma },$$
 where $$(\alpha , \beta , \gamma ) \in \,]0,1]^3$$
 , and the fractional derivatives $$D_{x_0^+}^{1+\alpha }, D_{y_0^+}^{1+\beta }, D_{z_0^+}^{1+\gamma }$$
 are in the Riemann–Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $$\Delta _+^{(\alpha ,\beta ,\gamma )}$$
 in classes of functions admitting a summable fractional derivative. Making use of the Mittag–Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

/pdf/eigenfunctions-and-fundamental-solutions-of-the-fractional-3k4t5mp94i.pdf

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

Abstract In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β.

/pdf/fundamental-solution-of-the-multi-dimensional-time-2nzr3qen3s.pdf

Fundamental solution of the multi-dimensional time fractional telegraph equation

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers will be constructed

Nelson Vieira

Papers

A numerical method for the fractional Schrödinger type equation of spatial dimension two

Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

Fundamental solution of the multi-dimensional time fractional telegraph equation

Fractional Clifford Analysis