In this paper, we present an efficient and accurate method for calculating the Black-Scholes differential equations and solve the Black-Scholes equations using Jacoby and Airfoil orthogonal bases, with the collocation method. The BlackScholes equation is a partial differential equation, which describes the price of choice in terms of time and the collocation method is a method of deter-mining coefficients. Then we show the computational results and examine the performance of the method for the two options, the price of basic assets and its issues. These results show that the Jacoby method is more efficient in solving the Black Scholes equation, and the method error is less and the convergence rate is higher. In this paper, by applying numerical methods to the desired equation, nonlinear devices can be solved by nonlinear solution methods, such as Newton's iterative method. The existence, uniqueness of the solution, and convergence of the methods are examined, and we will show in an example that by repeating then |𝑢, 𝑛, +1−, 𝑢, 𝑛, | |𝑢, 𝑛, | < ε, can be reached and this indicates the accuracy of the response to In this paper, we present an efficient and accurate method for calculating the Black-Scholes differential equations and solve the Black-Scholes equations using Jacoby and Airfoil orthogonal bases, with the collocation method. The BlackScholes equation is a partial differential equation, which describes the price of choice in terms of time and the collocation method is a method of deter-mining coefficients. Then we show the computational results and examine the performance of the method for the two options, the price of basic assets and its issues. These results show that the Jacoby method is more efficient in solving the Black Scholes equation, and the method error is less and the convergence rate is higher. In this paper, by applying numerical methods to the desired equation, nonlinear devices can be solved by nonlinear solution methods, such as Newton's iterative method.

Our aim in this paper is to prove an analog of Younis's Theorem on the image under the JACOBI transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space Lp ( ;  )(R+), (1 < p  2). It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lipschitz condition in Lp.

In 2009 B. Zheng et al. proposed two modified Gauss-Seidel (MGS) methods for linear system with M-matrices. In this paper, we use the preconditioners introduced by B. Zheng et al. For modified JACOBI method. The comparison theorems and numerical examples show that the proposed methods are superior to the classical JACOBI method.

The aim of this paper is to introduce two new extensions of the JACOBI and Laguerre POLYNOMIALs as the eigenfunctions of two non-classical Sturm-Liouville problems. We prove some important properties of these operators such as: These sets of functions are orthogonal with respect to a positive definite inner product defined over the compact intervals [-1; 1] and [0,¥), respectively and also these sequences form two new orthogonal bases for the corresponding Hilbert spaces. Finally, the spectral and Rayleigh-Ritz methods are carry out using these basis functions to solve some examples. Our numerical results are compared with other existing results to confirm the efficiency and accuracy of our method.