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"It is a simple equation: time is brain," says Madhok.
Third equation: Time = space.
Sometimes the relationship between variables in a dynamic system cannot be represented in a single equation time series model.
To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error.
Numerical experiments show the integration procedure is stable for time steps much larger than the diffusion equation time step condition.
In a single equation time series framework, it is possible to study the effect of a shock contemporaneous or time dependent in an exogenous variable, using intervention or transfer function analysis, respectively.
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Roughly speaking, the fractional models can be classified into three principal kinds: space-fractional differential equation, time-fractional differential equation, and time-space fractional differential equation.
A mathematical model was established that describes two-dimensional groundwater flow in an unconfined aquifer near a river with a linearized Boussinesq equation, time-dependent sources and a sloped tile.
In the equation, time-exponent is proposed for expressing the decreasing diffusion with time, which contains a referential period and diffusion coefficient, and effect of mineral admixtures like Fly Ash (FA), Ground Granulated Blast Furnace Slag (GGBFS), and Silica Fume (SF) (Thomas and Bamforth 1999; Lingjie et al. 2017; Thomas and Bentz 2002).
We solve several FPDEs including time- and space-fractional advection-equation, time- and space-fractional advection diffusion equation, and finally the space-fractional Burgers' equation to demonstrate the performance of the method.
In the following equations, time subscripts are omitted to simplify the notation.
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