Mojtaba Ranjbar

A numerical method for solving a fractional partial differential equation through converting it into an NLP problem

In this paper, we propose a new approach for solving fractional partial differential equations, which is very easy to use and can also be applied to equations of other types. The main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations. Using this approach, we convert a fractional partial differential equation into a nonlinear programming problem. Several numerical examples are used to demonstrate the effectiveness and accuracy of the method.

Numerical solution of homogeneous Smoluchowski's coagulation equation

Smoluchowskis equation is widely applied to describe the time evolution of the cluster-size distribution during aggregation processes. Analytical solutions for this equation, however, are known only for a very limited number of kernels. Therefore, numerical methods have to be used to describe the time evolution of the cluster-size distribution. A numerical technique is presented for the solution of the homogeneous Smoluchowskis coagulation equation with constant kernel. In this paper, we use Taylor polynomials and radial basis functions together to solve the equation. This method converts Smoluchowskis equation to a system of nonlinear equations that can be solved for unknown parameters. A numerical example with known solution is included to demonstrate the validity and applicability of the technique.

Collocation method based on shifted Chebyshev and radial basis functions with symmetric variable shape parameter for solving the parabolic inverse problem

ABSTRACT This work introduces a new numerical solution to the inverse parabolic problem with source control parameter that has important applications in large fields of applied science. We expand the approximate solution of the inverse problem in terms of shifted Chebyshev polynomials in time and radial basis functions with symmetric variable shape parameter in space, with unknown coefficients. Unknown coefficient matrix determined using the collocation technique. Sample results show that the proposed method is very accurate. Moreover, the proposed method is compared with two other methods, fourth-order compact difference scheme and method of lines. Finally, we examine the stability of our method for the case where there is additive noise in input data.

A computationally efficient numerical approach for multi-asset option pricing

ABSTRACT Numerical solution of the multi-dimensional partial differential equations arising in the modelling of option pricing is a challenging problem. Mesh-free methods using global radial basis functions (RBFs) have been successfully applied to several types of such problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we propose a numerical scheme to solve the Black–Scholes equation for valuation of options prices on several underlying assets. We use the derivatives of linear combinations of multiquadric RBFs to approximate the spatial derivatives and a straightforward finite difference to approximate the time derivative. The advantages of the scheme are that it does not require solving a full matrix at each time step and the algorithm is easy to implement. The accuracy of our scheme is demonstrated on a test problem.