scholarly journals Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Nemat Dalir
Keyword(s):  
Fluid Mechanics ◽  
Analytical Solutions ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Differential Operators ◽  
Second Order ◽  
Nonlinear Pdes ◽  
Initial Value ◽  
Exact Analytical Solutions ◽  
First Order

Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.

The Formation of Implicit Second Order Backward Difference Adam’s Formulae for Solving Stiff Systems of First Order Initial Value Problems of Ordinary Differential Equations

2020 ◽  
pp. 21-29
Author(s):  
Sabo J. ◽  
Kyagya T. Y. ◽  
Ayinde A. M.
Keyword(s):  
Initial Value Problems ◽  
Absolute Stability ◽  
Multistep Method ◽  
Second Order ◽  
Stiff Systems ◽  
Initial Value ◽  
First Order ◽  
Systems Of Odes ◽  
Linear Odes ◽  
Region Of Absolute Stability

The formation of implicit second order backward difference Adam’s formulae for solving stiff systems of ODEs was study in this paper. We used interpolation and collocation in deriving backward differentiae Adam’s formulae. The basic properties of our method was analyzed, and it was found to be consistent, zero-stability and convergent, we further plotted the region of absolute stability and it was shown to be A-stable. Numerical evidences shows that the multistep method develop is very effective method for in handling linear ODEs either initial value problems or boundary value problems.

scholarly journals Adomian decomposition method for solving initial value problems in vibration models

2018 ◽  
Vol 159 ◽  
pp. 02007
Author(s):  
Sudi Mungkasi ◽  
I Made Wicaksana Ekaputra
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
External Source ◽  
Accurate Solution ◽  
Second Order ◽  
Initial Value ◽  
Adomian Decomposition

A number of engineering problems have second-order ordinary differential equations as their mathematical models. In practice, we may have a large scale problem with a large number of degrees of freedom, which must be solved accurately. Therefore, treating the mathematical model governing the problems correctly is required in order to get an accurate solution. In this work, we use Adomian decomposition method to solve vibration models in the forms of initial value problems of second-order ordinary differential equations. However, for problems involving an external source, the Adomian decomposition method may not lead to an accurate solution if the external source is not correctly treated. In this paper, we propose a strategy to treat the external source when we implement the Adomian decomposition method to solve initial value problems of second-order ordinary differential equations. Computational results show that our strategy is indeed effective. We obtain accurate solutions to the considered problems. Note that exact solutions are often not available, so they need to be approximated using some methods, such as the Adomian decomposition method.

scholarly journals Numerical algorithm for solving second order nonlinear fuzzy initial value problems

2020 ◽  
Vol 10 (6) ◽  
pp. 6497
Author(s):  
A. F. Jameel ◽  
N. R. Anakira ◽  
A. H. Shather ◽  
Azizan Saaban ◽  
A. K. Alomari
Keyword(s):  
Set Theory ◽  
Fuzzy Set Theory ◽  
Initial Value Problems ◽  
Second Order ◽  
Computational Technique ◽  
Solution Properties ◽  
Initial Value ◽  
First Order ◽  
Fuzzy Solution ◽  
Fifth Order

The purpose of this analysis would be to provide a computational technique for the numerical solution of second-order nonlinear fuzzy initial value (FIVPs). The idea is based on the reformulation of the fifth order Runge Kutta with six stages (RK56) from crisp domain to the fuzzy domain by using the definitions and properties of fuzzy set theory to be suitable to solve second order nonlinear FIVP numerically. It is shown that the second order nonlinear FIVP can be solved by RK56 by reducing the original nonlinear equation intoa system of couple first order nonlinear FIVP. The findings indicate that the technique is very efficient and simple to implement and satisfy the Fuzzy solution properties. The method’s potential is demonstrated by solving nonlinear second-order FIVP.

scholarly journals Solving nonlinear and non-autonomous ODEs systems by the ADM using a new several-variables Adomian polynomials

2021 ◽  
Author(s):  
Idriss Noureddine Zaouagui ◽  
Toufik Badredine
Keyword(s):  
Taylor Series ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Differential Operators ◽  
Adomian Decomposition Method ◽  
Initial Value ◽  
Adomian Polynomials ◽  
Solution Series ◽  
Several Variables ◽  
Adomian Decomposition

In this work, we adapted another time the Adomian decomposition method for solving nonlinear and non-autonomous ODEs systems. Therefore, our expressions of the Adomian polynomials are determined for a several-variable differential operators. The solution series is shown that it stay coincide with the Taylor series. Thus new conditions of convergence have been established, some systemes has been solved by ADM using Maple 2020. keywords Adomian decomposition method, Adomian polynomials, ODEs systems, initial value problems, several-variables differential operators. Classification 26B12, 34L30, 47E05, 65B10, 65L05, 65L80

Nonlinear initial value problems with a singular perturbation of hyperbolic type

1981 ◽  
Vol 89 (3-4) ◽  
pp. 333-345 ◽  
Author(s):  
R. Geel
Keyword(s):  
Initial Value Problems ◽  
A Priori ◽  
Second Order ◽  
Initial Value ◽  
Existence Of A Solution ◽  
Nonlinear Operators ◽  
First Order ◽  
Order Part ◽  
The Difference ◽  
Small Factor

SynopsisThis paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.

scholarly journals Exponentially Fitted Chebyshev Based Algorithm as Second Order Initial Value Solver

2020 ◽  
pp. 51-54
Author(s):  
Emmanuel Adeyefa ◽  
O. S. Esan
Keyword(s):  
Exponential Function ◽  
Numerical Algorithm ◽  
Analytical Solutions ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Research Work ◽  
Second Order ◽  
Initial Value ◽  
Numerical Integrator ◽  
Exponentially Fitted

In this research work, we focus on development of a numerical algorithm which is well suited as integrator of initial value problems of order two. Exponential function is fitted into the Chebyshev polynomials for the formulation of this new numerical integrator. The efficiency, ingenuity and computational reliability of any numerical integrator are determined by investigating the zero stability, consistency and convergence of the integrator. Findings reveal that this algorithm is convergent. On comparison, the solutions obtained through the algorithm compare favourably well with the analytical solutions.

scholarly journals Quasilinearization for the Boundary Value Problem of Second-Order Singular Differential System

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Peiguang Wang ◽  
Tiantian Kong
Keyword(s):  
Boundary Value Problem ◽  
Initial Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Quasilinearization Method ◽  
Initial Value ◽  
First Order ◽  
Monotone Iterative ◽  
Singular Integrodifferential Equations ◽  
The Given

We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

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