Enhanced Iterative Methods Using Mamadu-Njoseh Polynomials for Solving the Heston Stochastic Partial Differential Equation

In this paper, a modified form of the Homotopy Perturbation Method (HPM) and the Variational Iteration Method (VIM), both developed by J.H. He, is presented using the newly constructed Mamadu-Njoseh orthogonal polynomial(MNPs)as modifier and basis function. The HPM combines principles from the field of topology and the usual perturbation techniques, while the goal of the VIM is to construct a correction functional for nonlinear systems. These two analytical techniques are modified through the orthogonal collocation method using the MNPs as basis function. The modified methods are employed to determine which approximates the Heston Stochastic Partial Differential Equation (HSPDE) faster to its exact solution. The Heston SPDE is a volatility model for determining the European bond and currency options as determined by stock pricing. While other methods exist in literature in determining the numerical solution to the HSPDE, our numerical schemes, the MHPM and the MVIM being presented as a new technique by the presence of the MNPs is noticed by comparison, to possess a faster approximation to the exact solution. In this work, we observe that the Modified VIM approximates faster to the exact solution of the HSPDE than the modified HPM due to its highly effective use of orthogonal polynomials, greater chances of handling of nonlinearities, superiority in the convergence properties, and having to adapt better to the boundary and initial conditions of the problem. This new approach is highly beneficial in handling large and complex nonlinear PDEs due to the presence of the MNPs and the iterative nature of the VIM.