**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**81

# Search results for: Exact solution

##### 81 Aerodynamic Design Optimization of High-Speed Hatchback Cars for Lucrative Commercial Applications

**Authors:**
A. Aravind,
M. Vetrivel,
P. Abhimanyu,
C. A. Akaash Emmanuel Raj,
K. Sundararaj,
V. R. S. Kumar

**Abstract:**

The choice of high-speed, low budget hatchback car with diversified options is increasing for meeting the new generation buyers trend. This paper is aimed to augment the current speed of the hatchback cars through the aerodynamic drag reduction technique. The inverted airfoils are facilitated at the bottom of the car for generating the downward force for negating the lift while increasing the current speed range for achieving a better road performance. The numerical simulations have been carried out using a 2D steady pressure-based k-ɛ realizable model with enhanced wall treatment. In our numerical studies, Reynolds-averaged Navier-Stokes model and its code of solution are used. The code is calibrated and validated using the exact solution of the 2D boundary layer displacement thickness at the *Sanal flow choking* condition for adiabatic flows. We observed through the parametric analytical studies that the inverted airfoil integrated with the bottom surface at various predesigned locations of Hatchback cars can improve its overall aerodynamic efficiency through drag reduction, which obviously decreases the fuel consumption significantly and ensure an optimum road performance lucratively with maximum permissible speed within the framework of the manufactures constraints.

**Keywords:**
aerodynamics of commercial cars,
downward force,
hatchback car,
inverted airfoil

##### 80 An Efficient Collocation Method for Solving the Variable-Order Time-Fractional Partial Differential Equations Arising from the Physical Phenomenon

**Authors:**
Haniye Dehestani,
Yadollah Ordokhani

**Abstract:**

**Keywords:**
collocation method,
legendre-laguerre functions,
fractional partial differential
equations,
pseudo-operational matrix
of integration

##### 79 Free Vibration of Axially Functionally Graded Simply Supported Beams Using Differential Transformation Method

**Authors:**
A. Selmi

**Abstract:**

Free vibration analysis of homogenous and axially functionally graded simply supported beams within the context of Euler-Bernoulli beam theory is presented in this paper. The material properties of the beams are assumed to obey the linear law distribution. The effective elastic modulus of the composite was predicted by using the rule of mixture. Here, the complexities which appear in solving differential equation of transverse vibration of composite beams which limit the analytical solution to some special cases are overcome using a relatively new approach called the Differential Transformation Method. This technique is applied for solving differential equation of transverse vibration of axially functionally graded beams. Natural frequencies and corresponding normalized mode shapes are calculated for different Young’s modulus ratios. MATLAB code is designed to solve the transformed differential equation of the beam. Comparison of the present results with the exact solutions proves the effectiveness, the accuracy, the simplicity, and computational stability of the differential transformation method. The effect of the Young’s modulus ratio on the normalized natural frequencies and mode shapes is found to be very important.

**Keywords:**
functionally graded material,
natural frequency,
mode shape,
differential transformation method

##### 78 Unsteady Temperature Distribution in a Finite Functionally Graded Cylinder

**Authors:**
A. Amiri Delouei

**Abstract:**

In the current study, two-dimensional unsteady heat conduction in a functionally graded cylinder is studied analytically. The temperature distribution is in radial and longitudinal directions. Heat conduction coefficients are considered a power function of radius both in radial and longitudinal directions. The proposed solution can exactly satisfy the boundary conditions. Analytical unsteady temperature distribution for different parameters of functionally graded cylinder is investigated. The achieved exact solution is useful for thermal stress analysis of functionally graded cylinders. Regarding the analytical approach, this solution can be used to understand the concepts of heat conduction in functionally graded materials.

**Keywords:**
temperature distribution,
functionally graded materials,
cylinder,
unsteady heat conduction

##### 77 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

##### 76 An Approximate Lateral-Torsional Buckling Mode Function for Cantilever I-Beams

**Authors:**
H. Ozbasaran

**Abstract:**

**Keywords:**
cantilever,
buckling mode,
I-beam,
lateral-torsional
buckling

##### 75 Numerical Computation of Sturm-Liouville Problem with Robin Boundary Condition

**Authors:**
Theddeus T. Akano,
Omotayo A. Fakinlede

**Abstract:**

**Keywords:**
Eigenvalue Problems,
Finite Element Method,
Sturm-Liouville problem,
Robin boundary condition

##### 74 Numerical Investigation of Unsteady MHD Flow of Second Order Fluid in a Tube of Elliptical Cross-Section on the Porous Boundary

**Authors:**
S. B. Kulkarni,
Hasim A. Chikte,
V. Murali Mohan

**Abstract:**

**Keywords:**
Numerical Simulation,
Porous Media,
Elastico-viscous fluid,
Elliptic cross-section,
Magnetic parameter

##### 73 A Numerical Solution Based On Operational Matrix of Differentiation of Shifted Second Kind Chebyshev Wavelets for a Stefan Problem

**Authors:**
Rajeev,
N. K. Raigar

**Abstract:**

**Keywords:**
operational matrix of differentiation,
shifted second kind chebyshev wavelets,
Similarity
transformation,
Stefan
problem

##### 72 Application of Residual Correction Method on Hyperbolic Thermoelastic Response of Hollow Spherical Medium in Rapid Transient Heat Conduction

**Authors:**
Po-Jen Su,
Huann-Ming Chou

**Abstract:**

**Keywords:**
maximum principle,
non-Fourier heat conduction,
residual correction method,
thermo-elastic response

##### 71 Unsteady MHD Flow of an Incompressible Elastico-Viscous Fluid in a Tube of Spherical Cross Section on a Porous Boundary

**Authors:**
Sanjay Baburao Kulkarni

**Abstract:**

Exact solution of an unsteady MHD flow of elasticoviscous fluid through a porous media in a tube of spherical cross section under the influence of magnetic field and constant pressure gradient has been obtained in this paper. Initially, the flow is generated by a constant pressure gradient. After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of spherical cross section by taking into account of the porosity factor and magnetic parameter of the bounding surface is investigated. The problem is solved in two-stages the first stage is a steady motion in tube under the influence of a constant pressure gradient, the second stage concern with an unsteady motion. The problem is solved employing separation of variables technique. The results are expressed in terms of a non-dimensional porosity parameter (K), magnetic parameter (m) and elasticoviscosity parameter (β), which depends on the Non-Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter and magnetic parameter tends to zero and porosity tends to infinity. It is seen that the effect of elastico-viscosity parameter, porosity parameter and magnetic parameter of the bounding surface has significant effect on the velocity parameter.

**Keywords:**
Porous Media,
Elastico-viscous fluid,
Second order
fluids,
Spherical cross-section,
Magnetic parameter

##### 70 Unsteady Poiseuille Flow of an Incompressible Elastico-Viscous Fluid in a Tube of Spherical Cross Section on a Porous Boundary

**Authors:**
Sanjay Baburao Kulkarni

**Abstract:**

Exact solution of an unsteady flow of elastico-viscous fluid through a porous media in a tube of spherical cross section under the influence of constant pressure gradient has been obtained in this paper. Initially, the flow is generated by a constant pressure gradient. After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of spherical cross section by taking into account of the porosity factor of the bounding surface is investigated. The problem is solved in twostages the first stage is a steady motion in tube under the influence of a constant pressure gradient, the second stage concern with an unsteady motion. The problem is solved employing separation of variables technique. The results are expressed in terms of a nondimensional porosity parameter (K) and elastico-viscosity parameter (β), which depends on the Non-Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter tends to zero and porosity tends to infinity. It is seen that the effect of elastico-viscosity parameter, porosity parameter of the bounding surface has significant effect on the velocity parameter.

**Keywords:**
Porous Media,
Elastico-viscous fluid,
Second order
fluids,
Spherical cross-section

##### 69 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:**
black hole,
quantum gravity,
Horava-Lifshitz gravity,
spherically symmetric space times

##### 68 Unsteady Flow of an Incompressible Elastico-Viscous Fluid of Second order Type in Tube of Ellipsoidal Cross Section on a Porous Boundary

**Authors:**
Sanjay Baburao Kulkarni

**Abstract:**

Exact solution of an unsteady flow of elastico-viscous fluid through a porous media in a tube of ellipsoidal cross section under the influence of constant pressure gradient has been obtained in this paper. Initially, the flow is generated by a constant pressure gradient. After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of ellipsoidal cross section by taking into account of the porosity factor of the bounding surface is investigated. The problem is solved in twostages the first stage is a steady motion in tube under the influence of a constant pressure gradient, the second stage concern with an unsteady motion. The problem is solved employing separation of variables technique. The results are expressed in terms of a nondimensional porosity parameter (K) and elastico-viscosity parameter (β), which depends on the Non-Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter tends to zero and porosity tends to infinity. It is seen that the effect of elastico-viscosity parameter and the porosity parameter of the bounding surface has significant effect on the velocity parameter.

**Keywords:**
Porous Media,
Elastico-viscous fluid,
Second order fluids,
Ellipsoidal cross-section

##### 67 On a New Inverse Polynomial Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations

**Authors:**
R. B. Ogunrinde

**Abstract:**

**Keywords:**
Differential Equations,
Numerical,
Initial value
problem,
Polynomials

##### 66 Theoretical Modal Analysis of Freely and Simply Supported RC Slabs

**Authors:**
M. S. Ahmed,
F. A. Mohammad

**Abstract:**

This paper focuses on the dynamic behavior of reinforced concrete (RC) slabs. Therefore, the theoretical modal analysis was performed using two different types of boundary conditions. Modal analysis method is the most important dynamic analyses. The analysis would be modal case when there is no external force on the structure. By using this method in this paper, the effects of freely and simply supported boundary conditions on the frequencies and mode shapes of RC square slabs are studied. ANSYS software was employed to derive the finite element model to determine the natural frequencies and mode shapes of the slabs. Then, the obtained results through numerical analysis (finite element analysis) would be compared with the exact solution. The main goal of the research study is to predict how the boundary conditions change the behavior of the slab structures prior to performing experimental modal analysis. Based on the results, it is concluded that simply support boundary condition has obvious influence to increase the natural frequencies and change the shape of the mode when it is compared with freely supported boundary condition of slabs. This means that such support conditions have the direct influence on the dynamic behavior of the slabs. Thus, it is suggested to use free-free boundary condition in experimental modal analysis to precisely reflect the properties of the structure. By using free-free boundary conditions, the influence of poorly defined supports is interrupted.

**Keywords:**
modal analysis,
ANSYS software,
natural frequencies,
mode shapes,
RC slabs

##### 65 Unsteady Flow of an Incompressible Viscous Electrically Conducting Fluid in Tube of Elliptical Cross Section under the Influence of Magnetic Field

**Authors:**
Sanjay Baburao Kulkarni

**Abstract:**

Exact solution of an unsteady flow of elastico-viscous electrically conducting fluid through a porous media in a tube of elliptical cross section under the influence of constant pressure gradient and magnetic field has been obtained in this paper. Initially, the flow is generated by a constant pressure gradient. After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of elliptical cross section by taking into account of the transverse magnetic field and porosity factor of the bounding surface is investigated. The problem is solved in twostages the first stage is a steady motion in tube under the influence of a constant pressure gradient, the second stage concern with an unsteady motion. The problem is solved employing separation of variables technique. The results are expressed in terms of a nondimensional porosity parameter (K), magnetic parameter (m) and elastico-viscosity parameter (β), which depends on the Non- Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter and magnetic parameter tends to zero and porosity tends to infinity. It is seen that the effect of elastico-viscosity parameter, magnetic parameter and the porosity parameter of the bounding surface has significant effect on the velocity parameter.

**Keywords:**
Porous Media,
Elastico-viscous fluid,
Elliptic cross-section,
Second order fluids

##### 64 On Bianchi Type Cosmological Models in Lyra’s Geometry

**Authors:**
R. K. Dubey

**Abstract:**

**Keywords:**
Bianchi type-I cosmological model,
variable
gravitational coupling (G) and Cosmological Constant term (β)

##### 63 The Application of Hybrid Orthonomal Bernstein and Block-Pulse Functions in Finding Numerical Solution of Fredholm Fuzzy Integral Equations

**Authors:**
Mahmoud Zarrini,
Sanaz Torkaman

**Abstract:**

In this paper, we have proposed a numerical method for solving fuzzy Fredholm integral equation of the second kind. In this method a combination of orthonormal Bernstein and Block-Pulse functions are used. In most cases, the proposed method leads to the exact solution. The advantages of this method are shown by an example and calculate the error analysis.

**Keywords:**
Fuzzy Fredholm Integral Equation,
Bernstein,
Block-Pulse,
Orthonormal

##### 62 Exp-Function Method for Finding Some Exact Solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries Equations

**Authors:**
Ehsan Mahdavi

**Abstract:**

In this paper, we apply the Exp-function method to Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara equation is the combination of the Rosenau and standard Kawahara equations and Rosenau-KdV equation is the combination of the Rosenau and standard KdV equations. These equations are nonlinear partial differential equations (NPDE) which play an important role in mathematical physics. Exp-function method is easy, succinct and powerful to implement to nonlinear partial differential equations arising in mathematical physics. We mainly try to present an application of Exp-function method and offer solutions for common errors wich occur during some of the recent works.

**Keywords:**
Exp-function method,
Rosenau Kawahara equation,
Rosenau Korteweg-de Vries equation,
nonlinear partial differential
equation

##### 61 MHD Unsteady Free Convection of Heat and Mass Transfer Flow through Porous Medium with Time Dependent Suction and Constant Heat Source/Sink

**Authors:**
Praveen Saraswat,
Rudraman Singh

**Abstract:**

In this paper, we have investigated the free convection MHD flow due to heat and mass transfer through porous medium bounded by an infinite vertical non-conducting porous plate with time dependent suction under the influence of uniform transverse magnetic field of strength H_{0}. When Temperature (T) and Concentration (C) at the plate is oscillatory with time about a constant non-zero mean. The velocity distribution, the temperature distribution, co-efficient of skin friction and role of heat transfer is investigated. Here the partial differential equations are involved. Exact solution is not possible so approximate solution is obtained and various graphs are plotted.

**Keywords:**
MHD,
Convection,
Porous,
Time Dependent Suction

##### 60 Numerical Treatment of Block Method for the Solution of Ordinary Differential Equations

**Authors:**
A. M. Sagir

**Abstract:**

Discrete linear multistep block method of uniform order for the solution of first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution.

**Keywords:**
Hybrid,
block method,
first order ordinary differential equations,
Self starting

##### 59 On the Approximate Solution of Continuous Coefficients for Solving Third Order Ordinary Differential Equations

**Authors:**
A. M. Sagir

**Abstract:**

This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y''' = f(x, y, y', y''), y(α)=y_{0}, y'(α)=β, y''(α)=η with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non – stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution.

**Keywords:**
Hybrid,
block method,
linear multistep,
third order ordinary differential equations,
Self starting

##### 58 Unsteady Transient Free Convective Flow of an Incompressible Viscous Fluid under Influence of Uniform Transverse Magnetic Field

**Authors:**
Praveen Saraswat,
Vipin Kumar Verma,
Rudraman Singh

**Abstract:**

The unsteady transient free convection flow of an incompressible dissipative viscous fluid between parallel plates at different distances have been investigated under porous medium. Due to presence of heat flux under the influence of uniform transverse magnetic field the velocity distribution and the temperature distribution, is shown graphically. Since exact solution is not possible so we find parametrical solution by perturbation technique. The result is shown in graph for different parameters. We notice that heat generation effects fluid velocity keeping in which of free convection which cools.

**Keywords:**
MHD,
Convection,
Porous,
transient,
viscous

##### 57 Transient Population Dynamics of Phase Singularities in 2D Beeler-Reuter Model

**Authors:**
Hidetoshi Konno,
Akio Suzuki

**Abstract:**

The paper presented a transient population dynamics of phase singularities in 2D Beeler-Reuter model. Two stochastic modelings are examined: (i) the Master equation approach with the transition rate (i.e., λ(n, t) = λ(t)n and μ(n, t) = μ(t)n) and (ii) the nonlinear Langevin equation approach with a multiplicative noise. The exact general solution of the Master equation with arbitrary time-dependent transition rate is given. Then, the exact solution of the mean field equation for the nonlinear Langevin equation is also given. It is demonstrated that transient population dynamics is successfully identified by the generalized Logistic equation with fractional higher order nonlinear term. It is also demonstrated the necessity of introducing time-dependent transition rate in the master equation approach to incorporate the effect of nonlinearity.

**Keywords:**
Birth-death process,
Transient population dynamics,
Phase singularity,
Non-stationary Master equation,
nonlinear Langevin equation,
generalized Logistic equation

##### 56 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:**
Numerical Solution,
approximate solution,
pantograph equations,
Laplace decomposition,
exact
solution

##### 55 A Family of Zero Stable Block Integrator for the Solutions of Ordinary Differential Equations

**Authors:**
A. M. Sagir

**Abstract:**

**Keywords:**
Hybrid,
block method,
Linear Multistep Method,
Self – starting,
Special Second Order

##### 54 Maximum Norm Analysis of a Nonmatching Grids Method for Nonlinear Elliptic Boundary Value Problem −Δu = f(u)

**Authors:**
Abida Harbi

**Abstract:**

**Keywords:**
Finite elements,
nonlinear PDEs,
error estimates,
Schwarz method

##### 53 An Application of the Sinc-Collocation Method to a Three-Dimensional Oceanography Model

**Authors:**
Y. Mohseniahouei,
K. Abdella,
M. Pollanen

**Abstract:**

**Keywords:**
Differential Equations,
Boundary Value Problems,
Wind-Driven Currents,
Sinc Numerical Methods

##### 52 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