This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonli

This study emphasizes the infinite-boundary integro-differential equation. To examine the approximate solution of the problem, two modified optimization algorithms are proposed based on generalized Laguerre functions. In the first technique, the proposed method is applied to the original problem by approximating the solution using the truncated generalized Laguerre polynomial of the unknown function, optimizing coefficients through error minimization, and transforming the integro-differential equation into an algebraic equation. In contrast, the second approach incorporates a penalty term into the objective function to effectively enforce boundary and integral constraints. This technique reduces the original problem to a mathematical optimi

In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R³ is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.

Mathematical Subject Classificat

This paper considers the nonlinear homogeneous fractional Burger's equation as a type of nonlinear fractional partial differential equations (FPDE). Our goal in this paper is to show that an initial value problem (IVP) can be modified with a second initial condition when (α ∈ ( 1,2 ]) as the velocity of the movement, and the obtained solution agrees with the nature of the wave with space and time for the problem. The Caputo fractional derivative is used in all the fractional derivatives. Also, the algorithm of the Laplace transform decomposition method (LTDM) for fractional PDEs is presented. The approximate solution converges to the exact solution in Theorem 1. Also, a numerical simulation is made to confirm the theoretical resu

Oscillation criteria are obtained for all solutions of the first-order linear delay differential equations with positive and negative coefficients where we established some sufficient conditions so that every solution of (1.1) oscillate. This paper generalized the results in [11]. Some examples are considered to illustrate our main results.

Abstract:The optimum design of the magnetic deflector with the lowest values of the radial and spiral distortion aberration coefficients was computed. The optimized calculations were made using three models, Glaser bell-shaped, Grivet-lenz and exponential models. By using the optimum axial field distribution, the pole pieces shape which gave rise to those field distributions was found by using the reconstruction method. The calculations show that the results of the three models coincide at the lower values of the excitation parameter. In general the Glaser- bell shaped model gives the optimum results at the whole range of the excitation parameter under investigation.The negative values of the spiral distortion aberration coefficient appears

In this paper Hermite interpolation method is used for solving linear and non-linear second order singular multi point boundary value problems with nonlocal condition. The approximate solution is found in the form of a rapidly convergent polynomial. We discuss behavior of the solution in the neighborhood of the singularity point which appears to perform satisfactorily for singular problems. The examples to demonstrate the applicability and efficiency of the method have been given.