scholarly journals Partial Differential Equations and Quantum States in Curved Spacetimes

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1936
Author(s):  
Zhirayr Avetisyan ◽  
Matteo Capoferri
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Review Paper ◽  
Quantum States ◽  
Quantum Field ◽  
Partial Differential ◽  
Curved Spacetimes ◽  
Globally Hyperbolic Spacetimes ◽  
Hadamard States ◽  
Hyperbolic Spacetimes

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution.

scholarly journals ON SOME ASPECTS OF THE GEOMETRY OF DIFFERENTIAL EQUATIONS IN PHYSICS

2004 ◽  
Vol 01 (03) ◽  
pp. 265-284 ◽  
Author(s):  
XAVIER GRÀCIA ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
NARCISO ROMÁN-ROY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Control Theory ◽  
Review Paper ◽  
Applied Mathematics ◽  
Field Theories ◽  
Partial Differential ◽  
Singular Differential Equations ◽  
Systems Of Differential Equations ◽  
Hamilton Equations

In this review paper, we consider three kinds of systems of differential equations, which are relevant in physics, control theory and other applications in engineering and applied mathematics; namely: Hamilton equations, singular differential equations, and partial differential equations in field theories. The geometric structures underlying these systems are presented and commented on. The main results concerning these structures are stated and discussed, as well as their influence on the study of the differential equations with which they are related. In addition, research to be developed in these areas is also commented on.

scholarly journals ANNs-Based Method for Solving Partial Differential Equations : A Survey

2021 ◽  
Author(s):  
Danang Adi Pratama ◽  
Maharani Abu Bakar ◽  
Mustafa Man ◽  
M. Mashuri
Keyword(s):  
Machine Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Review Paper ◽  
Partial Differential ◽  
Poisson Equations ◽  
Finite Difference Approximations ◽  
Higher Dimensional ◽  
Set Up ◽  
Discretization Process

Conventionally, partial differential equations (PDE) problems are solved numerically through discretization process by using finite difference approximations. The algebraic systems generated by this process are then finalized by using an iterative method. Recently, scientists invented a short cut approach, without discretization process, to solve the PDE problems, namely by using machine learning (ML). This is potential to make scientific machine learning as a new sub-field of research. Thus, given the interest in developing ML for solving PDEs, it makes an abundance of an easy-to-use methods that allows researchers to quickly set up and solve problems. In this review paper, we discussed at least three methods for solving high dimensional of PDEs, namely PyDEns, NeuroDiffEq, and Nangs, which are all based on artificial neural networks (ANNs). ANN is one of the methods under ML which proven to be a universal estimator function. Comparison of numerical results presented in solving the classical PDEs such as heat, wave, and Poisson equations, to look at the accuracy and efficiency of the methods. The results showed that the NeuroDiffEq and Nangs algorithms performed better to solve higher dimensional of PDEs than the PyDEns.

scholarly journals Wick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations

2013 ◽  
Vol 469 (2158) ◽  
pp. 20130001 ◽  
Author(s):  
D. Venturi ◽  
X. Wan ◽  
R. Mikulevicius ◽  
B. L. Rozovskii ◽  
G. E. Karniadakis
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Stokes Equations ◽  
Wick Product ◽  
Navier Stokes ◽  
Quantum Field ◽  
Partial Differential ◽  
Stochastic Approximations ◽  
Navier Stokes Equations

Approximating nonlinearities in stochastic partial differential equations (SPDEs) via the Wick product has often been used in quantum field theory and stochastic analysis. The main benefit is simplification of the equations but at the expense of introducing modelling errors. In this paper, we study the accuracy and computational efficiency of Wick-type approximations to SPDEs and demonstrate that the Wick propagator, i.e. the system of equations for the coefficients of the polynomial chaos expansion of the solution, has a sparse lower triangular structure that is seemingly universal, i.e. independent of the type of noise. We also introduce new higher-order stochastic approximations via Wick–Malliavin series expansions for Gaussian and uniformly distributed noises, and demonstrate convergence as the number of expansion terms increases. Our results are for diffusion, Burgers and Navier–Stokes equations, but the same approach can be readily adopted for other nonlinear SPDEs and more general noises.

Partial Differential Equations

2020 ◽  
Author(s):  
A. K. Nandakumaran ◽  
P. S. Datti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential

Partial Differential Equations in Fluid Dynamics

2008 ◽  
Author(s):  
Isom H. Herron ◽  
Michael R. Foster
Keyword(s):  
Fluid Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential
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