scholarly journals Different Approximations to the Solution of Upper-Convected Maxwell Fluid over a Porous Stretching Plate

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca ◽  
Romeo Negrea
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Maxwell Fluid ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Rapid Convergence ◽  
Similarity Transformations ◽  
Optimal Homotopy Asymptotic Method ◽  
Stretching Plate ◽  
Good Agreement

In the present paper, we consider an incompressible magnetohydrodynamic flow of two-dimensional upper-convected Maxwell fluid over a porous stretching plate with suction and injection. The nonlinear partial differential equations are reduced to an ordinary differential equation by the similarity transformations and taking into account the boundary layer approximations. This equation is solved approximately by means of the optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solutions. Different approximations to the solution are given, showing the exceptionally good agreement between the analytical and numerical solutions of the nonlinear problem. OHAM is very efficient in practice, ensuring a very rapid convergence of the solutions after only one iteration even though it does not need small or large parameters in the governing equation.

scholarly journals Analytic Approximate Solution for Falkner-Skan Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Boundary Layer Problem ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Good Agreement ◽  
Layer Problem

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

scholarly journals Dual Approximate Solutions of the Unsteady Viscous Flow over a Shrinking Cylinder with Optimal Homotopy Asymptotic Method

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene
Keyword(s):  
Viscous Flow ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Similarity Transformation ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Unsteady Viscous Flow ◽  
Optimal Homotopy Asymptotic Method ◽  
Shrinking Surface ◽  
Good Agreement

The unsteady viscous flow over a continuously shrinking surface with mass suction is investigated using the optimal homotopy asymptotic method (OHAM). The nonlinear differential equation is obtained by means of the similarity transformation. The dual solutions exist for a certain range of mass suction and unsteadiness parameters. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

Analytical approximate solutions to the Thomas-Fermi equation

Open Physics ◽  
2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene
Keyword(s):  
Excellent Agreement ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Thomas Fermi ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method

AbstractThe purpose of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to solve the nonlinear differential Thomas-Fermi equation. Our procedure 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 was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

scholarly journals A New Numerical Solution of Maxwell Fluid over a Shrinking Sheet in the Region of a Stagnation Point

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
S. S. Motsa ◽  
Y. Khan ◽  
S. Shateyi
Keyword(s):  
Differential Equations ◽  
Stagnation Point ◽  
Nonlinear Partial Differential Equations ◽  
Maxwell Fluid ◽  
Shrinking Sheet ◽  
Two Dimensional ◽  
Similarity Transformations ◽  
The Mathematical Model ◽  
Successive Linearisation Method ◽  
Fluid Parameters

The mathematical model for the incompressible two-dimensional stagnation flow of a Maxwell fluid towards a shrinking sheet is proposed. The developed equations are used to discuss the problem of being two dimensional in the region of stagnation point over a shrinking sheet. The nonlinear partial differential equations are transformed to ordinary differential equations by first-taking boundary-layer approximations into account and then using the similarity transformations. The obtained equations are then solved by using a successive linearisation method. The influence of the pertinent fluid parameters on the velocity is discussed through the help of graphs.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Analytic approximate solutions to the chemically reactive solute transfer problem with partial slip in the flow of a viscous fluid over an exponentially stretching sheet with suction/blowing

2018 ◽  
Vol 8 (1) ◽  
pp. 227-242
Author(s):  
Remus-Daniel Ene ◽  
Ilare Bordeaşu ◽  
Romeo Iosif Negrea
Keyword(s):  
Viscous Fluid ◽  
Transfer Problem ◽  
Approximate Solutions ◽  
Partial Slip ◽  
Solute Transfer ◽  
Similarity Transformations ◽  
Optimal Homotopy Asymptotic Method ◽  
Exponentially Stretching ◽  
First Time ◽  
Good Agreement

Abstract This paper analytically investigates the flow as well as the chemically reactive solute transfer problem in a viscous fluid. The motion equations are reduced to a nonlinear ordinary differential equations system using the similarity transformations. The obtained nonlinear differential system is for the first time approximately solved by means of the Optimal Homotopy Asymptotic Method (OHAM). The effects of the partial slip and suction/blowing parameters are analytically analyzed. Some examples are given; the obtained results provides us with a good agreement with the numerical results and reveal that our procedure is effective, accurate and easy to use.

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.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Comparison of Numerical and Perturbation Solutions of Two-Dimensional Nonlinear Water-Wave Problems

1976 ◽  
Vol 20 (03) ◽  
pp. 160-170
Author(s):  
Nils Salvesen ◽  
C. von Kerczek
Keyword(s):  
Perturbation Theory ◽  
Numerical Solutions ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Wave System ◽  
Two Dimensional ◽  
Third Order ◽  
Flow Past ◽  
Wave Problems ◽  
Good Agreement

Numerical solutions of the nonlinear problem of the steady two-dimensional potential flow past a submerged line vortex are obtained using the finite-difference iterative technique previously presented by the authors. These solutions are compared in detail with third-order perturbation theory solutions. It is found that very good agreement is obtained for cases of positive circulation of the vortex with strength large enough to produce downstream waves whose steepness is within 15 percent of the maximum possible steepness of irrotational free waves. These computed waves are as steep as the steepest waves obtained in a certain experiment involving the flow past a two-dimensional hydrofoil. For negative circulation, there is substantial difference between the numerical results and third-order perturbation theory. The failure of the perturbation theory is discussed. Details of the far-downstream wave system obtained by the numerical method are compared with other numerical solutions and very high-order perturbation theory solutions of the free-wave problem. Very good agreement is obtained in most cases.

scholarly journals Application of the optimal homotopy asymptotic method to nonlinear Bingham fluid dampers

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 620-626 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Liviu Bereteu
Keyword(s):  
Asymptotic Method ◽  
Approximate Solutions ◽  
Bingham Fluid ◽  
Rapid Convergence ◽  
Bingham Model ◽  
Mr Dampers ◽  
Approximate Analytical Solutions ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Fluid Dampers

AbstractDynamic response time is an important feature for determining the performance of magnetorheological (MR) dampers in practical civil engineering applications. The objective of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to give approximate analytical solutions of the nonlinear differential equation of a modified Bingham model with non-viscous exponential damping. Our procedure does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. OHAM is very efficient in practice for ensuring very rapid convergence of the solution after only one iteration and with a small number of steps.

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