scholarly journals Efficient Modification of the Decomposition Method for Solving a System of PDEs

2021 ◽  
pp. 3061-3070
Author(s):  
L. N. M. Tawfiq ◽  
Z. H. Kareem
Keyword(s):  
Decomposition Method ◽  
Series Solution ◽  
Initial Conditions ◽  
High Accuracy ◽  
Rapid Convergence ◽  
Adomian Polynomials ◽  
Convergence Of Series ◽  
Fast Approach ◽  
Systems Of Pdes ◽  
Non Linear System

     This paper presents an analysis solution for systems of partial differential equations using a new modification of the decomposition method to overcome the computational difficulties. Convergence of series solution was discussed with two illustrated examples, and the method showed a high-precision, being a fast approach to solve the non-linear system of PDEs with initial conditions. There is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. The method does not require any discretization or assumption for a small parameter to be present in the problem. The steps of the suggested method are easily implemented, with high accuracy and rapid convergence to the exact solution, compared with other methods that can be used to solve systems of PDEs.

Adomian Decomposition Method for Approximating the Solutions of the Bidirectional Sawada-Kotera Equation

2010 ◽  
Vol 65 (8-9) ◽  
pp. 658-664 ◽  
Author(s):  
Xian-Jing Lai ◽  
Xiao-Ou Cai
Keyword(s):  
Decomposition Method ◽  
Initial Conditions ◽  
Soliton Solutions ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Absolute Error ◽  
Solitary Wave Solutions ◽  
Adomian Polynomials ◽  
Wave Solutions ◽  
Adomian Decomposition

In this paper, the decomposition method is implemented for solving the bidirectional Sawada- Kotera (bSK) equation with two kinds of initial conditions. As a result, the Adomian polynomials have been calculated and the approximate and exact solutions of the bSK equation are obtained by means of Maple, such as solitary wave solutions, doubly-periodic solutions, two-soliton solutions. Moreover, we compare the approximate solution with the exact solution in a table and analyze the absolute error and the relative error. The results reported in this article provide further evidence of the usefulness of the Adomian decomposition method for obtaining solutions of nonlinear problems

scholarly journals Numeric-Analytic Solutions for Nonlinear Oscillators via the Modified Multi-Stage Decomposition Method

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 550 ◽  
Author(s):  
Emad A. Az-Zo’bi ◽  
Kamel Al-Khaled ◽  
Amer Darweesh
Keyword(s):  
Decomposition Method ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Analytic Solutions ◽  
Adomian Polynomials ◽  
Large Domain ◽  
Physical Problems ◽  
Runge Kutta Method ◽  
Multi Stage

This work deals with a new modified version of the Adomian-Rach decomposition method (MDM). The MDM is based on combining a series solution and decomposition method for solving nonlinear differential equations with Adomian polynomials for nonlinearities. With application to a class of nonlinear oscillators known as the Lienard-type equations, convergence and error analysis are discussed. Several physical problems modeled by Lienard-type equations are considered to illustrate the effectiveness, performance and reliability of the method. In comparison to the 4th Runge-Kutta method (RK4), highly accurate solutions on a large domain are obtained.

scholarly journals An accelerated solution for some classes of nonlinear partial differential equations

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
I. L. El-Kalla ◽  
E. M. Mohamed ◽  
H. A. A. El-Saka
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Adomian Polynomials ◽  
Numerical Examples ◽  
Adomian Decomposition

AbstractIn this paper, we apply an accelerated version of the Adomian decomposition method for solving a class of nonlinear partial differential equations. This version is a smart recursive technique in which no differentiation for computing the Adomian polynomials is needed. Convergence analysis of this version is discussed, and the error of the series solution is estimated. Some numerical examples were solved, and the numerical results illustrate the effectiveness of this version.

Numerical solution of large deflection beams by using the Laplace Adomian decomposition method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ming-Xian Lin ◽  
Chia-Hsiang Tseng ◽  
Chao Kuang Chen
Keyword(s):  
Decomposition Method ◽  
Large Deflection ◽  
Adomian Decomposition Method ◽  
Nonlinear Behavior ◽  
Bernoulli Beam ◽  
Rapid Convergence ◽  
Content Type ◽  
Adomian Decomposition ◽  
Euler Bernoulli Beam ◽  
Better Than

PurposeThis paper presents the problems using Laplace Adomian decomposition method (LADM) for investigating the deformation and nonlinear behavior of the large deflection problems on Euler-Bernoulli beam.Design/methodology/approachThe governing equations will be converted to characteristic equations based on the LADM. The validity of the LADM has been confirmed by comparing the numerical results to different methods.FindingsThe results of the LADM are found to be better than the results of Adomian decomposition method (ADM), due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms. LADM are presented for two examples for large deflection problems. The results obtained from example 1 shows the effects of the loading, horizontal parameters and moment parameters. Example 2 demonstrates the point loading and point angle influence on the Euler-Bernoulli beam.Originality/valueThe results of the LADM are found to be better than the results of ADM, due to this method's rapid convergence and accuracy to obtain the solutions by using fewer iterative terms.

SERIES SOLUTION OF TYPHOID FEVER MODEL USING DIFFERENTIAL TRANSFORM METHOD

2018 ◽  
Vol 3 (1) ◽  
pp. 67
Author(s):  
O J Peter ◽  
Oluwaseun B Akinduko ◽  
C Y Ishola ◽  
O A Afolabi ◽  
A B Ganiyu
Keyword(s):  
Typhoid Fever ◽  
Disease Transmission ◽  
Series Solution ◽  
Transformation Method ◽  
Type Model ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Model Equations ◽  
Non Linear System

This paper presents an analysis of PSIuIeTR type model, which are used to study the transmission dynamics of typhoid fever diseases in a population. Basic idea of typhoid fever disease transmission using compartmental modeling is discussed. Differential Transformation Method (DTM) is discussed in detail, which is used to compute the series solution of the non-linear system of differential equation governing the model equations. The validity of the (DTM) in solving the proposed model is established by classical fourth-order Runge-Kutta method which is implemented in Maple 18. Graphical results confirm that (DTM) is in good agreement with RK-4 and this produced correctly same behaviour of the model, thus validating the efficiency and accuracy of (DTM) in finding the series solution of an epidemic model.

Buckling of a viscous strut loaded eccentrically

1974 ◽  
Vol 9 (4) ◽  
pp. 227-229 ◽  
Author(s):  
J P Ellington
Keyword(s):  
Trigonometric Series ◽  
Series Solution ◽  
Maximum Deflection ◽  
Rapid Convergence ◽  
Poor Convergence

A solution to the problem of buckling of a viscous strut with eccentrically applied end loads has been found previously as an infinite trigonometric series having poor convergence for small values of time. An alternative series solution has been found, using powers rather than exponentials of time, which provides not only rapid convergence but simple closed forms for bounds on the maximum deflection.

scholarly journals Reliable Iterative Method for solving Volterra - Fredholm Integro Differential Equations

2021 ◽  
Vol 26 (2) ◽  
Author(s):  
Samaher Marez
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Iterative Method ◽  
Iterative Process ◽  
Initial Conditions ◽  
High Accuracy ◽  
Series Solutions ◽  
Math Program ◽  
The Right

  The aim of this paper, a reliable iterative method is presented for resolving many types of Volterra - Fredholm Integro - Differential Equations of the second kind with initial conditions. The series solutions of the problems under consideration are obtained by means of the iterative method.  Four various problems are resolved with high accuracy to make evident the enforcement of the iterative method on such type of integro differential equations. Results were compared with the exact solution which exhibit that this technique has compatible with the right solutions, simple, effective and easy for solving such problems. To evaluate the results in an iterative process the MATLAB is used as a math program for the calculations.

A robust technique based solution of time-fractional seventh-order Sawada–Kotera and Lax’s KdV equations

2021 ◽  
pp. 2150265
Author(s):  
Rajarama Mohan Jena ◽  
Snehashish Chakraverty ◽  
Dumitru Baleanu ◽  
Waleed Adel ◽  
Hadi Rezazadeh
Keyword(s):  
Convergence Analysis ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Absolute Error ◽  
Transform Method ◽  
Cpu Time ◽  
Kdv Equations ◽  
Differential Transform ◽  
Seventh Order

In this paper, the fractional reduced differential transform method (FRDTM) is used to obtain the series solution of time-fractional seventh-order Sawada–Kotera (SSK) and Lax’s KdV (LKdV) equations under initial conditions (ICs). Here, the fractional derivatives are considered in the Caputo sense. The results obtained are contrasted with other previous techniques for a specific case, [Formula: see text] revealing that the presented solutions agree with the existing solutions. Further, convergence analysis of the present results with an increasing number of terms of the solution and absolute error has also been studied. The behavior of the FRDTM solution and the effects on different values [Formula: see text] are illustrated graphically. Also, CPU-time taken to obtain the solutions of the title problems using FRDTM has been demonstrated.

Nonlinear Equations for High Accuracy X-Ray Crystal Orientation

1991 ◽  
Vol 35 (A) ◽  
pp. 681-685 ◽  
Author(s):  
Danut Dragoi
Keyword(s):  
Nonlinear Equations ◽  
Crystal Orientation ◽  
High Accuracy ◽  
Bragg Angle ◽  
Large Area ◽  
X Ray ◽  
Linear System Of Equations ◽  
Several Variables ◽  
Points Of View ◽  
Non Linear System

AbstractNonlinear equations are given for determining the crystallographic orientation of surfaces of single crystals. The equations are solved by an iterative method in several variables. The angle ϕ between the surface plane and the lattice plane in question is decomposed into two components α and β. These two components are obtained from the solution of a non-linear system of equations using two measurements and the Bragg angle. The diffractometric system considered is the well known θ/2θ with a sufficiently large area of x-ray detection and the capability of holding single crystal samples. The results obtained are discussed from experimental and theoretical points of view.

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