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 A New Method for Inextensible Flows of Timelike Curves in Minkowski Space-Time E14

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Talat Körpinar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Minkowski Space ◽  
Space Time ◽  
New Method ◽  
Frenet Frame ◽  
Partial Differential ◽  
Minkowski Space Time ◽  
Inextensible Flows ◽  
The Given

We construct a new method for inextensible flows of timelike curves in Minkowski space-time E14. Using the Frenet frame of the given curve, we present partial differential equations. We give some characterizations for curvatures of a timelike curve in Minkowski space-time E14.

scholarly journals DISCRETE-TIME MODELLING OP DISPERSION IN ESTURARIES

1980 ◽  
Vol 1 (17) ◽  
pp. 182
Author(s):  
T. Wood
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
High Speed ◽  
Transport Theory ◽  
State Parameter ◽  
Space Time ◽  
Tidal River ◽  
Primary Concern ◽  
Transport Parameters ◽  
Partial Differential

This paper aims to put forward a case in favour of a simple discretetime model describing mixing in an estuary. The model derives from the remarkably simple concepts developed by Ketchum (1951 a,b) which describe mixing in terms of tidal prism exchanges between segments. The author's view is that Ketchum1s ideas were abandoned before they were fully explored. A major factor was the advent of the high-speed computer which opened up the possibility of using an approach based on the space-time formulation of the problem in terms of the partial differential equations of transport theory. Intrinsically this approach, based on a continuum description, is more attractive than a gross description based on relatively large segments: one obvious reason is the possibility of providing a comprehensive space-time prediction of the spread of a pollutant. In practice, though, significant problems arise in its use: in particular, the following can be mentioned - a) substantial computing costs relating to computer program development and machine time b) specification of transport parameters inherent in the partial differential equations of transport: for example, dispersion coefficients c) model validation and state/parameter estimation. The last of these is the primary concern of this paper. It is probably true to say that, to date, too little attention has been given to these topics, in the context of estuarine modelling. The point to be made is that there is small justification in using a sophisticated description of a system if the resulting predictions of the model cannot be effectively validated. The ideas used in this paper stem from those put forward by Beck and Young (1975) in studies on non-tidal river pollution. The subsequent discussion suggests an extension to estuarine systems-.

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