scholarly journals Space-Time Discontinuous Petrov–Galerkin Methods for Linear Wave Equations in Heterogeneous Media

2019 ◽  
Vol 19 (3) ◽  
pp. 465-481 ◽  
Author(s):  
Johannes Ernesti ◽  
Christian Wieners
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Heterogeneous Media ◽  
Space Time ◽  
Linear Wave ◽  
Computational Domain ◽  
Partial Differential ◽  
Time Space ◽  
Appl Math

AbstractWe establish an abstract space-time DPG framework for the approximation of linear waves in heterogeneous media. The estimates are based on a suitable variational setting in the energy space. The analysis combines the approaches for acoustic waves of Gopalakrishnan–Sepulveda [J. Gopalakrishnan and P. Sepulveda, A space-time DPG method for the wave equation in multiple dimensions, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 129–154] and Ernesti–Wieners [J. Ernesti and C. Wieners, A space-time discontinuous Petrov–Galerkin method for acoustic waves, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 99–127] and is based on the abstract definition of traces on the skeleton of the time-space substructuring. The method is evaluated by large-scale parallel computations motivated from applications in seismic imaging, where the computational domain can be restricted substantially to a subset of the full space-time cylinder.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

Fitting partial differential equations to space-time dynamics

1999 ◽  
Vol 59 (1) ◽  
pp. 337-342 ◽  
Author(s):  
Markus Bär ◽  
Rainer Hegger ◽  
Holger Kantz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Space Time ◽  
Partial Differential ◽  
Time Dynamics

scholarly journals A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Ji Juan-Juan ◽  
Guo Ye-Cai ◽  
Zhang Lan-Fang ◽  
Zhang Chao-Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Space Time ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Table Lookup ◽  
Exact Analytical Solutions ◽  
Function Solution

A table lookup method for solving nonlinear fractional partial differential equations (fPDEs) is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1)-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.

scholarly journals The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

2021 ◽  
Vol 2090 (1) ◽  
pp. 012031
Author(s):  
E.A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Differential Operators ◽  
Space Time ◽  
Partial Differential ◽  
Stochastic Problems ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

scholarly journals Asymptotic Expansion of the Solutions to Time-Space Fractional Kuramoto-Sivashinsky Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Weishi Yin ◽  
Fei Xu ◽  
Weipeng Zhang ◽  
Yixian Gao
Keyword(s):  
Asymptotic Expansion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Power Series Method ◽  
Time Space ◽  
Residual Power Series ◽  
Asymptotic Expansion Of Solutions

This paper is devoted to finding the asymptotic expansion of solutions to fractional partial differential equations with initial conditions. A new method, the residual power series method, is proposed for time-space fractional partial differential equations, where the fractional integral and derivative are described in the sense of Riemann-Liouville integral and Caputo derivative. We apply the method to the linear and nonlinear time-space fractional Kuramoto-Sivashinsky equation with initial value and obtain asymptotic expansion of the solutions, which demonstrates the accuracy and efficiency of the method.

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