taylor series

In this paper, fractal differential equations are solved numerically. Here, the typical fractal equation is considered as follows:$$\frac{du(t)}{dt^{\alpha}}=f\left\{ t,u(t)\right\},~~~\alpha>0,$$ $f$ can be a nonlinear function and the main goal is to get $u(t)$. The continuous and discrete modes of this method have differences, so the subject must be carefully studied. How to solve fractal equations in their discrete form will be another goal of this research and also its generalization to higher dimensions than other aspects of this research.

The efficiency of the Adomian decomposition method in the solution of integro-differential equations cannot be overemphasized. However, improvement of the method is needed as its drawbacks have been analyzed and reported in recent literature. This present work develops a new modification of the method and its implementation on linear Fredholm type of integro-differential equations. The approach is based on the modification of the traditional Adomian decomposition method. The idea employs the Taylor series expansion of the source term whose resulting functions were combined in two terms for predicting the solution in each iteration. This approach yields a very high accuracy degree when compared to related methods in literature. The newly proposed method is said to accelerates and converges faster than the standard Adomian Decomposition Method. The procedure proves to be concise, effective and converges faster to the true solution of linear Fredholm Integro-differential problems.

In this paper, we propose a Taylor series expansion method for the second kind of Fredholm integral equation with smooth kernels. This method converts the integral equation into a linear equation system to reduce the amount of computation. We present convergence conditions. Finally, we show the efficiency of the method using some numerical examples.

This manuscript suggests an efficient scheme to find an approach for a class of differential equations arsing in the quantum calculus. The present scheme considers the solution in the form of a truncated Taylor series near zero with unknown coefficients. Then, by placing this approach into the problem and collocating the relation which is obtained at some nodes, a system of algebraic equations is achieved. The solution of this algebraic system is the unknown coefficients of the series. The ability of present method is examined by some examples.

As two-dimensional coupled system of nonlinear partial differential equations does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are most suitable and reliable. Therefore, modified numerical method based on Taylor series expansion and Haar wavelets is presented for solving coupled system of nonlinear partial differential equations. Efficiency and accuracy of the proposed method is depicted by comparing with classical methods.

In this paper second order explicit Galerkin finite element method based on cubic B-splines is constructed to compute numerical solutions of one dimensional nonlinear forced Burgers' equation. Taylor series expansion is used to obtain time discretization. Galerkin finite element method is set up for the constructed time discretized form. Stability of the corresponding linearized scheme is studied by using von Neumann analysis. The accuracy, efficiency, applicability and reliability of the present method is demonstrated by comparing numerical solutions of some test examples obtained by the proposed method with the exact and numerical solutions available in literature.

‎This paper presents the numerical solution for a class of fractional differential equations. The fractional derivatives are described in the Caputo \cite{1} sense. We developed a reproducing kernel method (RKM) to solve fractional differential equations in reproducing kernel Hilbert space. This method cannot be used directly to solve these equations, so an equivalent transformation is made by using Taylor series. Some numerical examples are studied to demonstrate the accuracy of the given ‎method.‎

In the present paper, we propose a method to solve a class of weakly singular Fredholm integral equations of the second kind in reproducing kernel spaces. The Taylor series of the unknown function is used to remove the singularity and bases of reproducing kernel spaces are used to solve this equation. Efficiency of the proposed method is investigated through various examples.

In this paper, the reduced di erential transform method is investigated for a nonlinear partial di erential equation modeling nematic liquid crystals, it is called the Hunter-Saxton equation. The main advantage of this method is that it can be applied directly to nonlinear di erential equations without requiring linearization, discretization, or perturbation. It is a semi analytical- numerical method that formulizes Taylor series in a very di erent manner. The numerical results denote that reduced di erential transform method is ecient and accurate for Hunter-Saxton equation.

In this paper, we integrate goal programming (GP), Taylor Series, Kuhn-Tucker conditions and Penalty Function approaches to solve linear fractional bi-level programming (LFBLP)problems. As we know, the Taylor Series is having the property of transforming fractional functions to a polynomial. In the present article by Taylor Series we obtain polynomial objective functions which are equivalent to fractional objective functions. Then on using the Kuhn-Tucker optimality condition of the lower level problem, we transform the linear bilevel programming problem into a corresponding single level programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty, that can be reduce to a single objective function. In the other words, suitable transformations can be applied to formulate FBLP problems. Finally a numerical example is given to illustrate the complexity of the procedure to the solution.

This paper presents a Taylor series approach for solving linear fractional de- centralized bi-level multi-objective decision-making (LFDBL-MODM) problems with a single decision maker at the upper level and multiple decision makers at the lower level. In the proposed approach, the membership functions associated with each objective(s) of the level(s) of LFDBL-MODM are transformed by using a Taylor series and then they are unified. On using the Kuhn-Tucker conditions, the problem is finally reduced to a single objective. Numerical example is given in order to illustrate the efficiency and superiority of the proposed approach.

Fuzzy integral equations play major roles in various areas, therefore a new method for finding a solution of the Fredholm fuzzy integral equation is presented. This method converts the fuzzy integral equation into linear system by using the Taylor series. For this scope, first the Taylor expansion of unknown function is substituted in parametric form of the given equation. Then we differentiate both sides of the resulting integral equation and also approximate the Taylor expansion by a suitable truncation limit. This work yields a linear system in crisp case. Now the solution of this system yields unknown Taylor coefficients of the solution functions. The proposed method is illustrated by several examples with computer simulations.

In this study, a Taylor method is developed for numerically solving the high-order most general nonlinear Fredholm integro-diff erential-difference equations in terms of Taylor expansions. The method is based on transferring the equation and conditions into the matrix equations which leads to solve a system of nonlinear algebraic equations with the unknown Taylor coefficients. Also, we test the method by numerical examples.

A simple mathematical model of steady state oxygen distribution subject to diffusive transport and non- linear uptake in a retinal cylinder has been developed. The approximate analytical solution to a reaction- diffusion equation are obtained by using series expansions. The computational results for the scaled variables are presented through graphs. The effect of the important parameters (1) diffusion coefficient (2) metabolic rate constant (3) retinal capillary concentration are examined and discussed.

In this study we produced a new method for solving regular differential equations with step size h and Taylor series. This method analyzes a regular differential equation with initial values and step size h. this types of equations include quadratic and cubic homogenous equations with constant coeffcients and cubic and second-level equations.

In this paper, we will compare a Homotopy perturbation algorithm and Taylor series expansin method for a system of second kind Fredholm integral equations. An application of He’s homotopy perturbation method is applied to solve the system of Fredholm integral equations. Taylor series expansin method reduce the system of integral equations to a linear system of ordinary differential equation.