In this paper, the Saul’yev finite difference scheme for a fully nonlinear partial differential equation with initial and boundary conditions is analyzed. The main advantage of this scheme is that it is unconditionally stable and explicit. Consistency and monotonicity of the scheme are discussed. Several finite difference schemes are used to compare the Saul’yev scheme with them. Numerical illustrations are given to demonstrate the efficiency and robustness of the scheme. In each case, it is found that the elapsed time for the Saul’yev scheme is shortest, and the solution by the Saul’yev scheme is nearest to the Crank–Nicolson method.

In this paper two different methods are presented to approximate the solution of the Black-Scholes equation for valuation of barrier option. These techniques can be applied directly for all types of differential equations, homogeneous or inhomogeneous. The use of these methods provides the solution of the problem in a closed form while the mesh point techniques provide the approximation at mesh points only. Also, the two schemes need less computational work in comparison with the traditional methods. These techniques can be employed for problems with initial condition. In this paper we use the variational iteration and homotopy perturbation methods for solving the Black-Scholes equation with terminal condition. Numerical results are compared with theoretical solutions in order to confirm the validity of the presented procedures.

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Functional perturbation method (FPM) is presented for the solution of differential equations with boundary conditions. Some properties of FPM are utilized to reduce the differential equation with variable coefficients to the equations with constant coefficients. The FPM can be applied directly for many types of differential equations. The exact solution is obtained by only the first term of the Frechet series for polynomial cases. Four examples are included to demonstrate the method.

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