This paper presents a new numerical method for the solution of ordinary differential equations (ODE). The linear second-order equations considered herein are solved using operational matrices of Wang-Ball Polynomials. By the improvement of the operational matrix, the singularity of the ODE is removed, hence ensuring that a solution is obtained. In order to show the employability of the method, several problems were considered. The results indicate that the method is suitable to obtain accurate solutions.

In this work, linear and nonlinear optical properties of two types of Iraqi heavy crude oil extracted from fields in southern Iraq were determined. The nonlinear optical properties were measured utilizing Z-scan technology with He-Ne laser at 632.8 nm. It was found that nonlinear refractive index (NLR) values for the Basra and Kut heavy crude oil samples are 6.34381×10^-4  and 8.25108×10^-4  cm²/mW, respectively, while those for the nonlinear absorption coefficient (NLA) are 2.68942×10^-5  and 2.58874×10^-5 , respectively. These results showed that the two samples with linear and nonlinear optical properties can be used in optics field applications as

In this paper, the homotopy perturbation method is presented for solving the second kind linear mixed Volterra-Fredholm integral equations. Then, Aitken method is used to accelerate the convergence. In this method, a series will be constructed whose sum is the solution of the considered integral equation. Convergence of the constructed series is discussed, and its proof is given; the error estimation is also obtained. For more illustration, the method is applied on several examples and programs, which are written in MATLAB (R2015a) to compute the results. The absolute errors are computed to clarify the efficiency of the method.

The aim of this paper is to employ the fractional shifted Legendre polynomials (FSLPs) in the matrix form to approximate the fractional derivatives and find the numerical solutions of the one-dimensional space-fractional bioheat equation (SFBHE). The Caputo formula was utilized to approximate the fractional derivative. The proposed methodology applied for two examples showed its usefulness and efficiency. The numerical results showed that the utilized technique is very efficacious with high accuracy and good convergence.

The aims of the paper are to present a modified symmetric fuzzy approach to find the best workable compromise solution for quadratic fractional programming problems (QFPP) with fuzzy crisp in both the objective functions and the constraints. We introduced a modified symmetric fuzzy by proposing a procedure, that starts first by converting the quadratic fractional programming problems that exist in the objective functions to crisp numbers and then converts the linear function that exists in the constraints to crisp numbers. After that, we applied the fuzzy approach to determine the optimal solution for our quadratic fractional programming problem which is supported theoretically and practically. The computer application for the algo