Abstract
In this paper, He's projected differential transform method (PDTM) has been used to obtain solution nonlinear coupled Burgers equation. This method involves less computational work and can, thus, be easily applied to initial value problems. (PDTM) is used to determine the exact solutions of some nonlinear time and space--fractional partial differential equations. A number of illustrative examples are provided and compared with the other methods. The numerical results obtained by these examples are found to be the same.
Highlights
Projected differential transform method is numerical solution technique that is based on Taylor series expansion which constructs an analytical solution in the form of a polynomial .; projected differential transform method obtained a polynomial series solution by means of an iterative procedure. [3], [10], [12]
Riemann-Lioville fractional derivative is mostly employed by mathematicians but this approach is not suitable for real world physical problems since it requires the definition of fractional order initial conditions, which has the advantage of defining integer order initial conditions which have no physically meaning full explanation as yet
We introduce the fractional projected differential transform method used in the sequent to obtain approximate analytical solutions for a fractional oscillator this method has been developed by Arikoglu and Ozkol as follows
Summary
Projected differential transform method is numerical solution technique that is based on Taylor series expansion which constructs an analytical solution in the form of a polynomial .; projected differential transform method obtained a polynomial series solution by means of an iterative procedure. [3], [10], [12]. Unlike the Riemann Liouville approach which derives its definition from repeated integration, the Grunwald Letnikow formulation approaches the problem from the derivative perspective. This approach is mostly used in numerical algorithms. E.g Laplace transform method, fractional Green’s function method, Mellin transform method and method of orthogonal polynomials Among these solution techniques, the power series method is the most transparent method of solution of fractional differential and integral equations. The two equations define Riemann Liouville and caputo fractional derivatives will of order , respectively. The Riemann – Liouville fractional derivative is computed in the reverse order we have chosen to use the Coputo fractional derivative because it allows traditional initial and boundary conditions to include in the formulation of the problem, but for homogeneous initial condition assumption, these two operators coincide [2], [7], [9], [11]
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