scholarly journals A Homotopy-Analysis Approach for Nonlinear Wave-Like Equations with Variable Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Afgan Aslanov
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Equations ◽  
Nonlinear Wave ◽  
Analytical Solutions ◽  
Wave Operator ◽  
Series Solution ◽  
Variable Coefficients ◽  
Analysis Approach ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis

We are interested in the approximate analytical solutions of the wave-like nonlinear equations with variable coefficients. We use a wave operator, which provides a convenient way of controlling all initial and boundary conditions. The proposed choice of the auxiliary operator helps to find the approximate series solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method.

scholarly journals Homotopy Perturbation Method for Solving Wave-Like Nonlinear Equations with Initial-Boundary Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Boundary Conditions ◽  
Perturbation Method ◽  
Nonlinear Equations ◽  
Analytical Solutions ◽  
Homotopy Perturbation Method ◽  
Initial Boundary ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions

The homotopy perturbation method is employed to obtain approximate analytical solutions of the wave-like nonlinear equations with initial-boundary conditions. An efficient way of choosing the auxiliary operator is presented. The results demonstrate reliability and efficiency of the method.

Analytical solutions of linear and nonlinear Klein-Fock-Gordon equation

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Sajida Rasheed
Keyword(s):  
Analytical Solutions ◽  
Gordon Equation ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Relativistic Version ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

AbstractIn this paper, we deal with some linear and nonlinear Klein-Fock-Gordon (KFG) equations, which is a relativistic version of the Schrödinger equation. The approximate analytical solutions are obtained by using the homotopy analysis method (HAM). The efficiency of the HAM is that it provides a practical way to control the convergence region of series solutions by introducing an auxiliary parameter }. Analytical results presented are in agreement with the existing results in open literature, which confirm the effectiveness of this method.

scholarly journals The scattering of sound waves by a vortex: numerical simulations and analytical solutions

1994 ◽  
Vol 260 ◽  
pp. 271-298 ◽  
Author(s):  
Tim Colonius ◽  
Sanjiva K. Lele ◽  
Parviz Moin
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Acoustic Waves ◽  
Stokes Equations ◽  
Spatial Differentiation ◽  
Navier Stokes ◽  
Sound Waves ◽  
Navier Stokes Equations ◽  
Refraction Effect ◽  
Approximate Analytical Solutions

The scattering of plane sound waves by a vortex is investigated by solving the compressible Navier–-Stokes equations numerically, and analytically with asymptotic expansions. Numerical errors associated with discretization and boundary conditions are made small by using high-order-accurate spatial differentiation and time marching schemes along with accurate non-reflecting boundary conditions. The accuracy of computations of flow fields with acoustic waves of amplitude five orders of magnitude smaller than the hydrodynamic fluctuations is directly verified. The properties of the scattered field are examined in detail. The results reveal inadequacies in previous vortex scattering theories when the circulation of the vortex is non-zero and refraction by the slowly decaying vortex flow field is important. Approximate analytical solutions that account for the refraction effect are developed and found to be in good agreement with the computations and experiments.

scholarly journals Semi- analytic numerical method for solution of time-space fractional heat and wave type equations with variable coefficients

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 74-86 ◽  
Author(s):  
Rishi Kumar Pandey ◽  
Hradyesh Kumar Mishra
Keyword(s):  
Fractional Derivative ◽  
Series Solution ◽  
Variable Coefficients ◽  
Variable Order ◽  
Wave Type ◽  
Transform Method ◽  
Time Space ◽  
Homotopy Analysis ◽  
Sumudu Transform ◽  
Variable Order Derivative

AbstractThe time and space fractional wave and heat type equations with variable coefficients are considered, and the variable order derivative in He‘s fractional derivative sense are taken. The utility of the homotopy analysis fractional sumudu transform method is shown in the form of a series solution for these generalized fractional order equations. Some discussion with examples are presented to explain the accuracy and ease of the method.

Approximate Analytical Solutions for CO2 Injectivity Into Saline Formations

2013 ◽  
Vol 16 (02) ◽  
pp. 123-133 ◽  
Author(s):  
Ehsan Azizi ◽  
Yildiray Cinar
Keyword(s):  
Boundary Conditions ◽  
Relative Permeability ◽  
Analytical Solutions ◽  
Outer Boundary ◽  
Analytical Models ◽  
Co2 Injection ◽  
Approximate Analytical Solutions ◽  
New Models ◽  
Saline Formation ◽  
Carbon Dioxide Co2

Summary This paper presents new analytical models to estimate the bottomhole pressure (BHP) of a vertical carbon dioxide (CO2) injection well in a radial, homogeneous, horizontal saline formation. The new models include the effects of multiphase flow, CO2 dissolution in formation brine, and near-well drying out on the BHP. CO2 is injected into the formation at a constant rate. The analytical solutions are presented for three types of formation outer boundary conditions: closed boundary, constant-pressure boundary, and infinite-acting formation. The sensitivity of BHP computations to gas relative permeability, retardation factors, and CO2 compressibility is examined. The predictive capability of the analytical models is tested by use of numerical reservoir simulations. The results show a good agreement between the analytical and numerical computations for all three boundary conditions. Variations in gas compressibility, retardation factors, and gas relative permeability in the drying-out zone are found to have moderate effects on BHP computations. It is demonstrated for several hypothetical but realistic cases that the new models can estimate CO2 injectivity reliably.

scholarly journals Construction of Approximate Analytical Solutions to Strongly Nonlinear Coupled van der Pol Oscillators

2014 ◽  
Vol 6 ◽  
pp. 817570
Author(s):  
Y. H. Qian ◽  
W. K. Liu ◽  
S. M. Chen
Keyword(s):  
Analytical Solutions ◽  
Degrees Of Freedom ◽  
Van Der Pol Oscillator ◽  
Practical Significance ◽  
Engineering System ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Practical Engineering ◽  
Van Der Pol Oscillators

Using nonlinear theory to research vibration model of engineering system has important theoretical and practical significance. Multi-degree-of-freedom (MDOF) coupled van der Pol oscillator is a typical model in the nonlinear vibration; many complex dynamic problems in practical engineering can be simplified as this model to be solved in the end. This paper discusses a class of two-degrees-of-freedom (2-DOF) coupled van der Pol oscillator, which was divided into three parameters of different situations α1≠α2, β1≠β2, and γ1≠γ2 to discuss. Employing symbolic software such as Mathematica for those problems, the explicit analytical solutions of frequency ω and displacements x1( t) and x2( t) are well formulated. Results showed that the homotopy analysis method (HAM) can effectively deal with this kind of parameter of different coupled vibrators, just request the values of some parameters are not too big. Finally, we got four important theorems to simplify the solution of the nonlinear system.

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