In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo's type to Riemann-Liouville's type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.

The role of fractional differential equations in the advancement of science and technology cannot be overemphasized. The time fractional telegraph equation (TFTE) is a hyperbolic partial differential equation (HPDE) with applications in frequency transmission lines such as the telegraph wire, radio frequency, wire radio antenna, telephone lines, and among others. Consequently, numerical procedures (such as finite element method, H 1 -Galerkin mixed finite element method, finite difference method, and among others) have become essential tools for obtaining approximate solutions for these HPDEs. It is also essential for these numerical techniques to converge to a given analytic solution to certain rate. The Ritz projection is often used in the analysis of stability, error estimation, convergence and superconvergence of many mathematical procedures. Hence, this paper offers a rigorous and comprehensive analysis of convergence of the space discretized time-fractional telegraph equation. To this effect, we define a temporal mesh on with a finite element space in Mamadu-Njoseh polynomial space, , of degree An interpolation operator (also of a polynomial space) was introduced along the fractional Ritz projection to prove the convergence theorem. Basically, we have employed both the fractional Ritz projection and interpolation technique as superclose estimate in -norm between them to avoid a difficult Ritz operator construction to achieve the convergence of the method.

A model for both stochastic jumps and volatility for equity returns in the area of option pricing is the stochastic volatility process with jumps (SVPJ). A major advantage of this model lies in the area of mean reversion and volatility clustering between returns and volatility with uphill movements in price asserts. Thus, in this article, we propose to solve the SVPJ model numerically through a discretized variational iteration method (DVIM) to obtain sample paths for the state variable and variance process at various timesteps and replications in order to estimate the expected jump times at various iterates resulting from executing the DVIM as n increases. These jumps help in estimating the degree of randomness in the financial market. It was observed that the average computed expected jump times for the state variable and variance process is moderated by the parameters  (variance process through mean reversion), Θ (long-run mean of the variance process), σ (volatility variance process) and λ (constant intensity of the Poisson process) at each iterate.