Sinan Deniz

Comparison of Adomian Decomposition Method and Taylor Matrix Method in Solving Different Kinds of Partial Differential Equations

In this paper, we will present a comparison between the Adomian Decomposition Method (ADM) and Taylor Matrix Method by solving some well-known partial differential equations (PDEs). In order to illustrate the analysis we examined the Telegraph equation, which is considered one of the most significant partial differential equations, describe wave propagation of electric signals in a cable transmission line and Klein-Gordon equation which is encountered in several applied physics fields such as, quantum field theory , fluid dynamics , optoelectronic devices design and numerical analysis. Our study shows that the decomposition method is faster and easy to use from a computational viewpoint.

Implementation of taylor collocation and adomian decomposition method for systems of ordinary differential equations

The importance of ordinary differential equation and also systems of these equations in scientific world is a crystal-clear fact. Many problems in chemistry, physics, ecology, biology can be modeled by systems of ordinary differential equations. In solving these systems numerical methods are very important because most realistic systems of these equations do not have analytic solutions in applied sciences In this study, we apply Taylor collocation method and Adomian decomposition method to solve the systems of ordinary differential equations. In these both scheme, the solution takes the form of a convergent power series with easily computable components. So, we will be able to make a comparison between Adomian decomposition and Taylor collocation methods after getting these power series.

MODIFIED ADOMIAN DECOMPOSITION METHOD FOR SOLVING RICCATI DIFFERENTIAL EQUATIONS

In this study, we solve Riccati differential equations by modified Adomian decomposition method which is constructed by different orthogonal polynomials. Here, Chebyshev polynomials are used instead of Taylor polynomials to expand the source function. We see the benefits of using these expansions to get better results.

Modification of perturbation-iteration method to solve different types of nonlinear differential equations

Perturbation iteration method has been recently constructed by Pakdemirli and co-workers. It has been also proven that this technique is very effective and applicable for solving some nonlinear differential equations. In this study we suggest a modification to expedite the solution process of perturbation-iteration algorithms. This work might greatly improve the computational efficiency of the perturbation iteration method and also its Mathematica package to solve nonlinear equations. Numerical illustrations are also given to show how modified method eliminates cumbersome computational work needed by perturbation iteration method.

Optimal perturbation iteration method for Bratu-type problems

In this paper, we introduce the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. The proposed method is illustrated by studying Bratu-type equations. Our results show that only a few terms are required to obtain an approximate solution which is more accurate and efficient than many other methods in the literature.

Applications of optimal perturbation iteration method for solving nonlinear differential equations

Perturbation iteration method has been recently constructed and it has been also proven that this technique is very effective for solving some nonlinear differential equations. In this study, we develop the new optimal perturbation iteration method based on the perturbation iteration algorithms for the approximate solutions of nonlinear differential equations of many types. This work will greatly improve the computational efficiency of the perturbation iteration method. Applications also show that only a few terms are required to get an approximate solution which is more accurate and efficient than many other methods in literature.

Application of adomian decomposition method for singularly perturbed fourth order boundary value problems

In this paper, we use Adomian Decomposition Method (ADM) to solve the singularly perturbed fourth order boundary value problem. In order to make the calculation process easier, first the given problem is transformed into a system of two second order ODEs, with suitable boundary conditions. Numerical illustrations are given to prove the effectiveness and applicability of this method in solving these kinds of problems. Obtained results shows that this technique provides a sequence of functions which converges rapidly to the accurate solution of the problems.

New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods

Abstract In this paper, we implement the optimal homotopy asymptotic method to find the approximate solutions of the Poisson–Boltzmann equation. We also use the results of the conjugate gradient method for comparison with those of the optimal homotopy asymptotic method. Our study reveals that the optimal homotopy asymptotic method gives more effective results than conjugate gradient algorithms for the considered problems.