This paper considered the notion of European option which is geared towards solving analytical and numerical solutions. In particular, we examined Black-Scholes closed form solution modified (MBS) partial differential equation using Crank-Nicolson finite difference method. These equations were approximated to obtain Call Put prices. The explicit price both options found accordingly. solutions c...

In this paper, a numerical scheme named alternating segment Crank-Nikolson is used for solving heat equation. This scheme can be used directly on parallel computations. Truncation error and stability of the presented method is analyzed. Comparison in accuracy with the fully implicit Crank-Nikolson scheme is presented in numerical experiment.

Results from applying a Crank-Nicholson parabolic equation method (CN-PE) are presented in situations with a thin screen on a hard ground in a turbulent atmosphere, and with the acoustic source at ground level. The results are evaluated by comparison with G. A. Daigle’s model, which uses the sound scattering cross-section by V. I. Tatarskii together with diffraction theory. The results show a f...

The numerical solution of a Rayleigh-Taylor instability problem where an inviscid liquid of finite depth is accelerated into a gas of semi-infinite extent is obtained by transforming the irregular flow domain into a rectangular domain by a coordinate transformation. The free surface equation is solved by a Crank-Nicolson procedure. The boundary condition at the free surface for the velocity pot...

In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank–Nicolson schemes can be reused for higher–order fully implicit Runge–Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order–optimal for A–stable and irreducible Runge–Kutta schemes with invertible coefficient matrices. The theoretical inv...

In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, darkand Peregrine-type soliton solutio...

In this work we study Crank-Nicolson Leap-Frog (CNLF) methods with fast slow wave splittings for Navier-Stokes equation plus a Coriolis force term, which is a simplification of geophysical flows. We present a new stabilized CNLF method where the added stabilization completely removes the method’s CFL time step condition. We give a comprehensive stability and error analysis. We prove that for Os...

The preconditioned Crank-Nicolson (pCN) method is a MCMC algorithm for implementing the Bayesian inferences in function spaces. A remarkable feature of the algorithm is that, unlike many usual MCMC algorithms, which become arbitrary slow under the mesh refinement, the efficiency of the algorithm is dimension independent. In this work we develop an adaptive version of the pCN algorithm, where th...

In this study, a new algorithm is introduced for the numerical solution of equal width (EW) equation. This created by using collocation finite element method based on decic B-spline functions space discretization EW equation and Crank-Nicolson time his The obtained results are compared with previous ones to see efficiency accuracy proposed method.

In this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou andW. Deng, A class of second order difference approximation...