**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**7

# exact solutions Related Publications

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

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

**Abstract:**

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

##### 6 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:**
exact solutions,
Eigenfunctions,
Helmholtz equation,
Nikiforov-Uvarov method

##### 5 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:**
exact solutions,
Klein-Gordon-Schödinger Equation,
Lie Classical Method

##### 4 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:**
exact solutions,
Shallow water wave equation,
(G'/G) expansion method

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

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

**Abstract:**

**Keywords:**
exact solutions,
Gravitational Fields,
Lie Classical Method

##### 2 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:**
Navier-Stokes equations,
exact solutions,
Fluid of variable viscosity,
Steady plane flows

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

**Authors:**
Imran Siddique

**Abstract:**

**Keywords:**
exact solutions,
velocity field,
shear stress,
Newtonian fluids,
Cylindrical domains