**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1752

# Search results for: Exact solutions

##### 1752 Exploring Solutions in Extended Horava-Lifshitz Gravity

**Authors:**
Aziza Altaibayeva,
Ertan Gudekli,
Ratbay Myrzakulov

**Abstract:**

In this letter, we explore exact solutions for the Horava-Lifshitz gravity. We use of an extension of this theory with first order dynamical lapse function. The equations of motion have been derived in a fully consistent scenario. We assume that there are some spherically symmetric families of exact solutions of this extended theory of gravity. We obtain exact solutions and investigate the singularity structures of these solutions. Specially, an exact solution with the regular horizon is found.

**Keywords:**
Quantum gravity,
Horava-Lifshitz gravity,
black hole,
spherically symmetric space times.

##### 1751 Exact Solutions of the Helmholtz equation via the Nikiforov-Uvarov Method

**Authors:**
Said Laachir,
Aziz Laaribi

**Abstract:**

The Helmholtz equation often arises in the study of physical problems involving partial differential equation. Many researchers have proposed numerous methods to find the analytic or approximate solutions for the proposed problems. In this work, the exact analytical solutions of the Helmholtz equation in spherical polar coordinates are presented using the Nikiforov-Uvarov (NU) method. It is found that the solution of the angular eigenfunction can be expressed by the associated-Legendre polynomial and radial eigenfunctions are obtained in terms of the Laguerre polynomials. The special case for k=0, which corresponds to the Laplace equation is also presented.

**Keywords:**
Helmholtz equation,
Nikiforov-Uvarov method,
exact solutions,
eigenfunctions.

##### 1750 Exact Solutions of Steady Plane Flows of an Incompressible Fluid of Variable Viscosity Using (ξ, ψ)- Or (η, ψ)- Coordinates

**Authors:**
Rana Khalid Naeem,
Asif Mansoor,
Waseem Ahmed Khan,
Aurangzaib

**Abstract:**

The exact solutions of the equations describing the steady plane motion of an incompressible fluid of variable viscosity for an arbitrary state equation are determined in the (ξ,ψ) − or (η,ψ )- coordinates where ψ(x,y) is the stream function, ξ and η are the parts of the analytic function, ϖ =ξ( x,y )+iη( x,y ). Most of the solutions involve arbitrary function/ functions indicating that the flow equations possess an infinite set of solutions.

**Keywords:**
Exact solutions,
Fluid of variable viscosity,
Navier-Stokes equations,
Steady plane flows

##### 1749 Exact Pfaffian and N-Soliton Solutions to a (3+1)-Dimensional Generalized Integrable Nonlinear Partial Differential Equations

**Authors:**
Magdy G. Asaad

**Abstract:**

**Keywords:**
Bilinear operator,
G-BKP equation,
Integrable nonlinear PDEs,
Jimbo-Miwa equation,
Ma-Fan equation,
N-soliton solutions,
Pfaffian solutions.

##### 1748 Constructing Distinct Kinds of Solutions for the Time-Dependent Coefficients Coupled Klein-Gordon-Schrödinger Equation

**Authors:**
Anupma Bansal

**Abstract:**

We seek exact solutions of the coupled Klein-Gordon-Schrödinger equation with variable coefficients with the aid of Lie classical approach. By using the Lie classical method, we are able to derive symmetries that are used for reducing the coupled system of partial differential equations into ordinary differential equations. From reduced differential equations we have derived some new exact solutions of coupled Klein-Gordon-Schrödinger equations involving some special functions such as Airy wave functions, Bessel functions, Mathieu functions etc.

**Keywords:**
Klein-Gordon-Schödinger Equation,
Lie Classical Method,
Exact Solutions

##### 1747 Exact Solution of Some Helical Flows of Newtonian Fluids

**Authors:**
Imran Siddique

**Abstract:**

**Keywords:**
Newtonian fluids,
Velocity field,
Exact solutions,
Shear stress,
Cylindrical domains.

##### 1746 A Spectral Decomposition Method for Ordinary Differential Equation Systems with Constant or Linear Right Hand Sides

**Authors:**
R. B. Ogunrinde,
C. C. Jibunoh

**Abstract:**

**Keywords:**
Spectral decomposition,
eigenvalues of the Jacobian,
linear RHS,
homogeneous linear systems.

##### 1745 On Symmetry Analysis and Exact Wave Solutions of New Modified Novikov Equation

**Authors:**
Anupma Bansal,
R. K. Gupta

**Abstract:**

In this paper, we study a new modified Novikov equation for its classical and nonclassical symmetries and use the symmetries to reduce it to a nonlinear ordinary differential equation (ODE). With the aid of solutions of the nonlinear ODE by using the modified (G/G)-expansion method proposed recently, multiple exact traveling wave solutions are obtained and the traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions and rational functions.

**Keywords:**
New Modified Novikov Equation,
Lie Classical Method,
Nonclassical Method,
Modified (G'/G)-Expansion Method,
Traveling Wave Solutions.

##### 1744 On Symmetries and Exact Solutions of Einstein Vacuum Equations for Axially Symmetric Gravitational Fields

**Authors:**
Nisha Goyal,
R.K. Gupta

**Abstract:**

**Keywords:**
Gravitational fields,
Lie Classical method,
Exact solutions.

##### 1743 Group Invariant Solutions of Nonlinear Time-Fractional Hyperbolic Partial Differential Equation

**Authors:**
Anupma Bansal,
Rajeev Budhiraja,
Manoj Pandey

**Abstract:**

**Keywords:**
Nonlinear time-fractional hyperbolic PDE,
Lie
Classical method,
exact solutions.

##### 1742 Constructing Approximate and Exact Solutions for Boussinesq Equations using Homotopy Perturbation Padé Technique

**Authors:**
Mohamed M. Mousa,
Aidarkhan Kaltayev

**Abstract:**

**Keywords:**
Homotopy perturbation method,
Padé approximants,
cubic Boussinesq equation,
modified Boussinesq equation.

##### 1741 Traveling Wave Solutions for the (3+1)-Dimensional Breaking Soliton Equation by (G'/G)- Expansion Method and Modified F-Expansion Method

**Authors:**
Mohammad Taghi Darvishi,
Maliheh Najafi,
Mohammad Najafi

**Abstract:**

In this paper, using (G/G )-expansion method and modified F-expansion method, we give some explicit formulas of exact traveling wave solutions for the (3+1)-dimensional breaking soliton equation. A modified F-expansion method is proposed by taking full advantages of F-expansion method and Riccati equation in seeking exact solutions of the equation.

**Keywords:**
Exact solution,
The (3+1)-dimensional breaking soliton equation,
( G G )-expansion method,
Riccati equation,
Modified Fexpansion method.

##### 1740 New Exact Three-Wave Solutions for the (2+1)-Dimensional Asymmetric Nizhnik-Novikov-Veselov System

**Authors:**
Fadi Awawdeh,
O. Alsayyed

**Abstract:**

New exact three-wave solutions including periodic two-solitary solutions and doubly periodic solitary solutions for the (2+1)-dimensional asymmetric Nizhnik-Novikov- Veselov (ANNV) system are obtained using Hirota's bilinear form and generalized three-wave type of ansatz approach. It is shown that the generalized three-wave method, with the help of symbolic computation, provides an e¤ective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics.

**Keywords:**
Soliton Solution,
Hirota Bilinear Method,
ANNV System.

##### 1739 Approximated Solutions of Two-Point Nonlinear Boundary Problem by a Combination of Taylor Series Expansion and Newton Raphson Method

**Authors:**
Chinwendu. B. Eleje,
Udechukwu P. Egbuhuzor

**Abstract:**

One of the difficulties encountered in solving nonlinear Boundary Value Problems (BVP) by many researchers is finding approximated solutions with minimum deviations from the exact solutions without so much rigor and complications. In this paper, we propose an approach to solve a two point BVP which involves a combination of Taylor series expansion method and Newton Raphson method. Furthermore, the fourth and sixth order approximated solutions are obtained and we compare their relative error and rate of convergence to the exact solution. Finally, some numerical simulations are presented to show the behavior of the solution and its derivatives.

**Keywords:**
Newton Raphson method,
non-linear boundary value problem,
Taylor series approximation,
Michaelis-Menten equation.

##### 1738 Some Exact Solutions of the (2+1)-Dimensional Breaking Soliton Equation using the Three-wave Method

**Authors:**
Mohammad Taghi Darvishi,
Mohammad Najafi

**Abstract:**

This paper considers the (2+1)-dimensional breaking soliton equation in its bilinear form. Some exact solutions to this equation are explicitly derived by the idea of three-wave solution method with the assistance of Maple. We can see that the new idea is very simple and straightforward.

**Keywords:**
Soliton solution,
computerized symbolic computation,
painleve analysis,
(2+1)-dimensional breaking soliton equation,
Hirota's bilinear form.

##### 1737 Some Solitary Wave Solutions of Generalized Pochhammer-Chree Equation via Exp-function Method

**Authors:**
Kourosh Parand,
Jamal Amani Rad

**Abstract:**

In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.

**Keywords:**
Exp-function method,
generalized Pochhammer- Chree equation,
solitary wave solution,
ODE's.

##### 1736 On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations

**Authors:**
H. D. Ibrahim,
H. C. Chinwenyi,
H. N. Ude

**Abstract:**

In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.

**Keywords:**
conjugate gradient,
linear equations,
symmetric and positive definite matrix,
Gauss-Seidel,
Jacobi,
algorithm

##### 1735 Traveling Wave Solutions for Shallow Water Wave Equation by (G'/G)-Expansion Method

**Authors:**
Anjali Verma,
Ram Jiwari,
Jitender Kumar

**Abstract:**

This paper presents a new function expansion method for finding traveling wave solution of a non-linear equation and calls it the (G'/G)-expansion method. The shallow water wave equation is reduced to a non linear ordinary differential equation by using a simple transformation. As a result the traveling wave solutions of shallow water wave equation are expressed in three forms: hyperbolic solutions, trigonometric solutions and rational solutions.

**Keywords:**
Shallow water wave equation,
Exact solutions,
(G'/G) expansion method.

##### 1734 Laplace Decomposition Approximation Solution for a System of Multi-Pantograph Equations

**Authors:**
M. A. Koroma,
C. Zhan,
A. F. Kamara,
A. B. Sesay

**Abstract:**

In this work we adopt a combination of Laplace transform and the decomposition method to find numerical solutions of a system of multi-pantograph equations. The procedure leads to a rapid convergence of the series to the exact solution after computing a few terms. The effectiveness of the method is demonstrated in some examples by obtaining the exact solution and in others by computing the absolute error which decreases as the number of terms of the series increases.

**Keywords:**
Laplace decomposition,
pantograph equations,
exact
solution,
numerical solution,
approximate solution.

##### 1733 Multiple Soliton Solutions of (2+1)-dimensional Potential Kadomtsev-Petviashvili Equation

**Authors:**
Mohammad Najafi,
Ali Jamshidi

**Abstract:**

We employ the idea of Hirota-s bilinear method, to obtain some new exact soliton solutions for high nonlinear form of (2+1)-dimensional potential Kadomtsev-Petviashvili equation. Multiple singular soliton solutions were obtained by this method. Moreover, multiple singular soliton solutions were also derived.

**Keywords:**
Hirota bilinear method,
potential Kadomtsev-Petviashvili equation,
multiple soliton solutions,
multiple singular soliton solutions.

##### 1732 Some Rotational Flows of an Incompressible Fluid of Variable Viscosity

**Authors:**
Rana Khalid Naeem,
Waseem Ahmed Khan,
Muhammad Akhtar,
Asif Mansoor

**Abstract:**

The Navier Stokes Equations (NSE) for an incompressible fluid of variable viscosity in the presence of an unknown external force in Von-Mises system x,\ are transformed, and some new exact solutions for a class of flows characterized by equation y f x a\b for an arbitrary state equation are determined, where f x is a function, \ the stream function, a z 0 and b are the arbitrary constants. In three, out of four cases, the function f x is arbitrary, and the solutions are the solutions of the flow equations for all the flows characterized by the equationy f x a\b. Streamline patterns for some forms of f x in unbounded and bounded regions are given.

**Keywords:**
Bounded and unbounded region,
Exact solution,
Navier Stokes equations,
Streamline pattern,
Variable viscosity,
Von- Mises system

##### 1731 Solution of Two-Point Nonlinear Boundary Problems Using Taylor Series Approximation and the Ying Buzu Shu Algorithm

**Authors:**
U. C. Amadi,
N. A. Udoh

**Abstract:**

One of the major challenges faced in solving initial and boundary problems is how to find approximate solutions with minimal deviation from the exact solution without so much rigor and complications. The Taylor series method provides a simple way of obtaining an infinite series which converges to the exact solution for initial value problems and this method of solution is somewhat limited for a two point boundary problem since the infinite series has to be truncated to include the boundary conditions. In this paper, the Ying Buzu Shu algorithm is used to solve a two point boundary nonlinear diffusion problem for the fourth and sixth order solution and compare their relative error and rate of convergence to the exact solution.

**Keywords:**
Ying Buzu Shu,
nonlinear boundary problem,
Taylor series algorithm,
infinite series.

##### 1730 Solving Inhomogeneous Wave Equation Cauchy Problems using Homotopy Perturbation Method

**Authors:**
Mohamed M. Mousa,
Aidarkhan Kaltayev

**Abstract:**

In this paper, He-s homotopy perturbation method (HPM) is applied to spatial one and three spatial dimensional inhomogeneous wave equation Cauchy problems for obtaining exact solutions. HPM is used for analytic handling of these equations. The results reveal that the HPM is a very effective, convenient and quite accurate to such types of partial differential equations (PDEs).

**Keywords:**
Homotopy perturbation method,
Exact solution,
Cauchy problem,
inhomogeneous wave equation

##### 1729 New Exact Solutions for the (3+1)-Dimensional Breaking Soliton Equation

**Authors:**
Mohammad Taghi Darvishi,
Maliheh Najafi,
Mohammad Najafi

**Abstract:**

In this work, we obtain some analytic solutions for the (3+1)-dimensional breaking soliton after obtaining its Hirota-s bilinear form. Our calculations show that, three-wave method is very easy and straightforward to solve nonlinear partial differential equations.

**Keywords:**
(3+1)-dimensional breaking soliton equation,
Hirota'sbilinear form.

##### 1728 Periodic Solutions for Some Strongly Nonlinear Oscillators by He's Energy Balance Method

**Abstract:**

In this paper, applying He-s energy balance method to determine frequency formulation relations of nonlinear oscillators with discontinuous term or fractional potential. By calculation and computer simulations, compared with the exact solutions show that the results obtained are of high accuracy.

**Keywords:**
He's energy balance method,
periodic solution,
nonlinear oscillator,
discontinuous,
fractional potential.

##### 1727 Cubic B-spline Collocation Method for Numerical Solution of the Benjamin-Bona-Mahony-Burgers Equation

**Authors:**
M. Zarebnia,
R. Parvaz

**Abstract:**

In this paper, numerical solutions of the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation are obtained by a method based on collocation of cubic B-splines. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. The method is applied on some test examples, and the numerical results have been compared with the exact solutions. The L∞ and L2 in the solutions show the efficiency of the method computationally.

**Keywords:**
Benjamin-Bona-Mahony-Burgers equation,
Cubic Bspline,
Collocation method,
Finite difference.

##### 1726 Control of the Thermal Evaporation of Organic Semiconductors via Exact Linearization

**Authors:**
Martin Steinberger,
Martin Horn

**Abstract:**

In this article, a high vacuum system for the evaporation of organic semiconductors is introduced and a mathematical model is given. Based on the exact input output linearization a deposition rate controller is designed and tested with different evaporation materials.

**Keywords:**
Effusion cell,
organic semiconductors,
deposition rate,
exact linearization.

##### 1725 The Direct Ansaz Method for Finding Exact Multi-Wave Solutions to the (2+1)-Dimensional Extension of the Korteweg de-Vries Equation

**Authors:**
Chuanjian Wang,
Changfu Liu,
Zhengde Dai

**Abstract:**

In this paper, the direct AnsAz method is used for constructing the multi-wave solutions to the (2+1)-dimensional extension of the Korteweg de-Vries (shortly EKdV) equation. A new breather type of three-wave solutions including periodic breather type soliton solution, breather type of two-solitary solution are obtained. Some cases with specific values of the involved parameters are plotted for each of the three-wave solutions. Mechanical features of resonance interaction among the multi-wave are discussed. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.

**Keywords:**
EKdV equation,
Breather,
Soliton,
Bilinear form,
The
direct AnsAz method.

##### 1724 Analytical Based Truncation Principle of Higher-Order Solution for a x1/3 Force Nonlinear Oscillator

**Authors:**
Md. Alal Hosen

**Abstract:**

In this paper, a modified harmonic balance method based an analytical technique has been developed to determine higher-order approximate periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x^{1/3}. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. In this article, different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. We find a modified harmonic balance method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the x^{1/3} force nonlinear oscillator but it is also useful for many other nonlinear problems.

**Keywords:**
Approximate solutions,
Harmonic balance method,
Nonlinear oscillator,
Perturbation.

##### 1723 Extend Three-wave Method for the (3+1)-Dimensional Soliton Equation

**Authors:**
Somayeh Arbabi Mohammad-Abadi,
Maliheh Najafi

**Abstract:**

In this paper, we study (3+1)-dimensional Soliton equation. We employ the Hirota-s bilinear method to obtain the bilinear form of (3+1)-dimensional Soliton equation. Then by the idea of extended three-wave method, some exact soliton solutions including breather type solutions are presented.

**Keywords:**
Three-wave method,
(3+1)-dimensional Soliton equation,
Hirota's bilinear form.