scholarly journals Simple and accurate approach for solving of nonlinear heat convective-radiative equation in fin by using the collocation method and comparison with HPM and VIM

2014 ◽  
Vol 41 (3) ◽  
pp. 159-176
Author(s):  
Petroudi Rahimi ◽  
D.D. Ganji ◽  
Y. Rostamiyan ◽  
E. Rahimi ◽  
Nejad Khazayi
Keyword(s):  
Heat Transfer ◽  
Collocation Method ◽  
Analytical Techniques ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Mathematical Tool ◽  
Science And Engineering ◽  
Analytical Technique ◽  
Transfer Equations ◽  
Small Parameters

Collocation Method (CM) such as analytical technique, which does not need small parameters is here used to evaluate the analytical approximate solutions of the nonlinear heat transfer equation. The obtained results from Collocation Method are compared with other analytical techniques such as Homotopy Perturbation Method (HPM) and Variation Iteration Method (VIM). Also, boundary value problem (BVP) is applied as a numerical method for validation. The results reveal that the Collocation Method is very effective, simple and more accurate than other techniques. Also, it is found that this method is a powerful mathematical tool and can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering especially at some heat transfer equations.

APPLICATION OF HE'S PARAMETRIZED PERTURBATION METHOD FOR NONLINEAR HEAT DIFFUSION AND HEAT TRANSFER EQUATIONS

2009 ◽  
Vol 23 (23) ◽  
pp. 2791-2806 ◽  
Author(s):  
D. D. GANJI ◽  
N. JAMSHIDI ◽  
M. MOMENI
Keyword(s):  
Heat Transfer ◽  
Perturbation Method ◽  
Perturbation Methods ◽  
Real Life ◽  
Method Comparison ◽  
Nonlinear Problems ◽  
Heat Diffusion ◽  
Transfer Equations ◽  
Small Parameters ◽  
Comparison Of The Results

Perturbation methods depend on small parameters which are difficult to find for real-life nonlinear problems. He's parametrized perturbation method, which does not need small parameters, is one of the newest analytical methods to solve nonlinear equations. In this letter some nonlinear heat transfer equations are used as examples to illustrate the simple solution procedures of this method. Comparison of the results obtained by the new method with exact solutions reveals that the method is tremendously effective.

scholarly journals A variational iteration method integral transform technique for handling heat transfer problems

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 55-61 ◽  
Author(s):  
Yuejin Zhou ◽  
Shun Pang ◽  
Guo Chong ◽  
Xiaojun Yang ◽  
Xiaoding Xu ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Iteration Method ◽  
Integral Transform ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Excess Temperature ◽  
Variational Iteration ◽  
Transfer Equations ◽  
Transfer Problems ◽  
Integral Transform Technique

In this paper, we consider the heat transfer equations at the low excess temperature. The variational iteration method integral transform technique is used to find the approximate solutions for the problems. The used method is accurate and efficient.

scholarly journals Thin film flow of an Oldroyd 6-constant fluid over a moving belt: an analytic approximate solution

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 44-64 ◽  
Author(s):  
Remus-Daniel Ene ◽  
Vasile Marinca ◽  
Valentin Bogdan Marinca
Keyword(s):  
Thin Film ◽  
Differential Equation ◽  
Constant Velocity ◽  
Basic Equation ◽  
Film Flow ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Thin Film Flow ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method

AbstractIn this paper the thin film flow of an Oldroyd 6-constant fluid on a vertically moving belt is investigated. The basic equation of a non-Newtonian fluid in a container with a wide moving belt which passes through the container moving vertically upward with constant velocity, is reduced to an ordinary nonlinear differential equation. This equation is solved approximately by means of the Optimal Homotopy Asymptotic Method (OHAM). The solutions take into account the behavior of Newtonian and non-Newtonian fluids. Our procedure intended for solving nonlinear problems does not need small parameters in the equation and provides a convenient way to control the convergence of the approximate solutions.

scholarly journals Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Excellent Agreement ◽  
Analytical Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Volterra Equations ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Small Parameters

We apply an analytical method called the Optimal Parametric Iteration Method (OPIM) to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.

scholarly journals Analytical investigation of nonlinear model arising in heat transfer through the porous fin

2014 ◽  
Vol 18 (2) ◽  
pp. 409-417 ◽  
Author(s):  
Yasser Rostamiyan ◽  
Domiri Ganji ◽  
Rahimi Petroudi ◽  
Khazayi Nejad
Keyword(s):  
Heat Transfer ◽  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Analytical Techniques ◽  
Nonlinear Problems ◽  
Analytical Investigation ◽  
Homotopy Perturbation ◽  
Porous Fins ◽  
Variation Iteration Method

In this letter simple analytical methods called homotopy perturbation method(HPM), variation iteration method(VIM) and perturbation method(PM) are employed to approach temperature distribution of porous fins. also energy balance and Darcy's model used to formulate the heat transfer equation. To study the thermal performance, a type case considered is finite-length fin with insulated tip. The obtained results from variation iteration method (VIM) are compared with other analytical techniques proposed before. These methods are homotopy perturbation method and perturbation method (PM). Also BVP is applied as a numerical method for validation. The obtained results shows that the VIM is more accurate, stable and more appropriate than other techniques. Also it is found that this method is powerful mathematical tools and can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering specially some heat transfer equations.

A Software Tool for Determining Steady-State Temperature Distributions in a Square Domain by the Finite-Difference Method

2007 ◽  
Vol 35 (4) ◽  
pp. 305-315
Author(s):  
K. A. Oladejo ◽  
D. A. Adetan ◽  
O. A. Bamiro
Keyword(s):  
Heat Transfer ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Computer Aided Design ◽  
Linear Equations ◽  
Analytical Techniques ◽  
Software Tool ◽  
Transfer Equations ◽  
The Finite Difference Method

This paper presents the development of an interactive program (called SSTDD) to solve two-dimensional conduction heat transfer equations in a square domain using the finite-difference method. The development of the tool (based on a computer-aided design package), on a Visual BASIC 6.0 platform, involved the application of the heat transfer equations and the appropriate boundary conditions to a square domain. The finite-difference method was used to express the elliptic differential equation in a form suitable for numerical solution. The system of linear equations generated was solved by the Gauss–Seidel iterative technique. The SSTDD model was tested by using problems solved by conventional analytical techniques. The results generated by the model and the analytical method were in good agreement. Hence the model can be used to solve practical engineering problems, with good accuracy, and also as a demonstration tool to students in the area of design and heat transfer of mechanical engineering.

scholarly journals Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sunil Kumar ◽  
Amit Kumar ◽  
Shaher Momani ◽  
Mujahed Aldhaifallah ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Power Series ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Analytical Techniques ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Power Series Method ◽  
Analytical Technique ◽  
Residual Power Series ◽  
Residual Power

Abstract The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ L 2 and $L_{\infty }$ L ∞ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

Numerical Multistep Approach for Solving Fractional Partial Differential Equations

2017 ◽  
Vol 14 (03) ◽  
pp. 1750029 ◽  
Author(s):  
Mohammed Al-Smadi ◽  
Asad Freihat ◽  
Hammad Khalil ◽  
Shaher Momani ◽  
Rahmat Ali Khan
Keyword(s):  
Fractional Derivatives ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Time Frame ◽  
Differential Transformation Method ◽  
Heat Equations ◽  
One Dimensional ◽  
Analytical Technique ◽  
Differential Transformation

In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.

ANALYTICAL APPROACHES FOR SYSTEM OF NONLINEAR EQUATIONS ARISING IN MELTING OF A FINITE SLAB

2009 ◽  
Vol 23 (02) ◽  
pp. 209-222 ◽  
Author(s):  
S. M. VAREDI ◽  
D. D. GANJI ◽  
M. RAJABI
Keyword(s):  
Heat Transfer ◽  
Analytical Methods ◽  
Real Life ◽  
Nonlinear Problems ◽  
System Of Nonlinear Equations ◽  
Variational Theory ◽  
Nonlinear Heat Transfer ◽  
Finite Slab ◽  
Transfer Equations ◽  
Analytical Approaches

Most engineering problems, especially heat transfer equations are mostly nonlinear. Variational iteration method (VIM) and homotopy perturbation method (HPM) are useful numerical and analytical methods for solving nonlinear heat transfer equations. Perturbation methods depend on a small parameter, which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two newly powerful analytical methods are introduced to solve nonlinear heat transfer problems in our work; one is the VIM and the other one is the HPM. The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM makes a difficult problem much easier to be evaluated. In this research, these methods are used to solve nonlinear ordinary differential system for melting of a finite slab.

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