Therefore, the original fractional partial differential equation
can be converted into another partial differential equation
of integer order as follows:
Consider an kth-order system of partial differential equations
(pdes) of n independent variables [x.bar] = ([x.sub.1], [x.sub.2], ..., [x.sub.n]) and p dependent variables w = ([w.sub.1], [w.sub.2], ..., [w.sub.p]), namely,
The current study deals with mainly such type of implications of partial differential equations
. Finally, the numerical result obtained is compared with the exact solution and implementation of the method is verified as well.
Fractional partial differential equations
have been of great interest due to its wide applications in different branches such as physics , , , ,  and hydrology , .
Recently, the investigation of exact travelling wave solutions to nonlinear partial differential equations
plays an important role in the study of nonlinear modelling physical phenomena.
In Section 3, we formulate a stochastic volatility model and obtain an approximate price of the Parisian option in the form of a partial differential equation
. In Section 4, we compute the leading order price and a correction term by the finite difference method and find a stochastic volatility effect on the Black-Scholes price.
In the nonlinear sciences, it is well known that many nonlinear partial differential equations
are widely used to describe the complex phenomena in various fields.
The modeling software first reads the partial differential equation
system and problem domain definition in a natural, easy-to-learn language.
Definition 1 The system whose state is defined by (2)is said to be controllable if the observation ) (u z generates a dense (affine) subspace of the space of observations .H In the above setting, the equation (2) is typically a partial differential equation
, where the influence of u can take multiple different forms: typically, u can be an additional (force) term in the right-hand side of the equation, localized in a part of the domain; it can also appear in the boundary conditions; but other situations can clearly be envisaged (we will describe some of them).
A stable explicit method for the finite difference solution of a fourth order parabolic partial differential equation
The vector T = ([T.sup.1], ..., [T.sup.r]) is a conserved vector for the partial differential equation
and [T.sup.1], ..., [T.sup.r] are its components.
In this section we established an Algorithm using Laplace Decomposition method on the partial differential equations
which were nonlinear.