17-03-2014, 06:39 PM
Abstract—In this paper, a novel analytical method (DTMPadé)
is proposed for solving nonlinear differential equations,
especially for boundary-layer and natural convection
problems. This method is based on combination of the
differential transform method and the Padé approximant that
we use for solve the thermal boundary-layer over a flat plate
with a convective surface boundary condition. This technique
is extended to give solutions for nonlinear differential
equations whit boundary conditions at the infinity. In order to
show the effectiveness of the DTM-Padé, the results obtained
from the DTM-Padé is compared with available solutions
obtained using RungeKutta–Fehlberg fourth–fifth order
(RFK45) method to generate the numerical solution.
Numerical comparisons between the DTM-Padé and the
RFK45 reveal that the new technique introduced here is a
promising tool for solving nonlinear differential equations whit
infinity boundary conditions.
Keywords-Differential transform method (DTM); Convective
boundary condition; Padé approximant; Thermal boundary-layer
I. INTRODUCTION
Most scientific problems and physical phenomena such
as fluid mechanics, solid state physics, plasma physics,
plasma waves, thermo-elasticity and chemical physics, are
described through nonlinear equations. We know that except
a limited number of these problems have precise analytical
solution, most of them do not have analytical solution, so
these nonlinear equations should be solved using numerical
methods. There are also some analytic techniques for
nonlinear equations. The importance of obtaining the exact
or approximate solutions of nonlinear differential equations,
it is still a hot spot to seek new methods to obtain new exact
or approximate solutions. In the recent years, many authors
mainly had paid attention to study solutions of nonlinear
differential equations by using various methods. Among
these are the homotopy perturbation method (HPM) [1],
homotopy analysis method (HAM) [2] and differential
transform method [3]. Some of these methods use specific
transformations in order to reduce the equations into simpler
ones or system of equations and others give the solution in a
series form that converges to the exact solution. There are
yet other methods, which use a trial function in an iterative
scheme converging rapidly.
The concept of differential transform method was first
introduced by Zhou [3] in 1986 and it was used to solve
both linear and nonlinear initial value problems in electric
circuit analysis. The main advantage of this method is that it
can be applied directly to nonlinear differential equations
without requiring linearization, discretization and therefore,
it is not affected by errors associated to discretization also,
DTM does not require perturbation parameter. Another
important advantage is that this method reducing the size of
computational work while the Taylor series method is
computationally taken long time for the large orders. The
validity of the DTM is independent of whether or not there
exist small parameters in the considered equation.
Therefore, same as the HAM and the HPM, the DTM can
overcome the foregoing restrictions and limitations of
perturbation methods. In addition, the DTM provides an
efficient numerical solution with high accuracy, minimal
calculation and avoidance of physically unrealistic
assumptions.
Chen and Ho [4] developed this method for partial
differential equations and obtained closed form series
solutions for linear and nonlinear initial value problems and
Ayaz [5] applied it to the system of differential equations. In
recent years, the differential transform method has been
successfully employed to solve many types of nonlinear
problems such as the linear partial differential equations of
fractional order, the nonlinear oscillatory systems, multiorder
fractional differential equations, the hyperchaotic
Rossler system, the fourth-order boundary value problems,
the Volterra integral equation with separable kernels, the
difference equations, the free vibration equations of beam
on elastic soil, the integral and integro-differential equation
systems. All of these successful applications verified the
validity, effectiveness and flexibility of the DTM.
is proposed for solving nonlinear differential equations,
especially for boundary-layer and natural convection
problems. This method is based on combination of the
differential transform method and the Padé approximant that
we use for solve the thermal boundary-layer over a flat plate
with a convective surface boundary condition. This technique
is extended to give solutions for nonlinear differential
equations whit boundary conditions at the infinity. In order to
show the effectiveness of the DTM-Padé, the results obtained
from the DTM-Padé is compared with available solutions
obtained using RungeKutta–Fehlberg fourth–fifth order
(RFK45) method to generate the numerical solution.
Numerical comparisons between the DTM-Padé and the
RFK45 reveal that the new technique introduced here is a
promising tool for solving nonlinear differential equations whit
infinity boundary conditions.
Keywords-Differential transform method (DTM); Convective
boundary condition; Padé approximant; Thermal boundary-layer
I. INTRODUCTION
Most scientific problems and physical phenomena such
as fluid mechanics, solid state physics, plasma physics,
plasma waves, thermo-elasticity and chemical physics, are
described through nonlinear equations. We know that except
a limited number of these problems have precise analytical
solution, most of them do not have analytical solution, so
these nonlinear equations should be solved using numerical
methods. There are also some analytic techniques for
nonlinear equations. The importance of obtaining the exact
or approximate solutions of nonlinear differential equations,
it is still a hot spot to seek new methods to obtain new exact
or approximate solutions. In the recent years, many authors
mainly had paid attention to study solutions of nonlinear
differential equations by using various methods. Among
these are the homotopy perturbation method (HPM) [1],
homotopy analysis method (HAM) [2] and differential
transform method [3]. Some of these methods use specific
transformations in order to reduce the equations into simpler
ones or system of equations and others give the solution in a
series form that converges to the exact solution. There are
yet other methods, which use a trial function in an iterative
scheme converging rapidly.
The concept of differential transform method was first
introduced by Zhou [3] in 1986 and it was used to solve
both linear and nonlinear initial value problems in electric
circuit analysis. The main advantage of this method is that it
can be applied directly to nonlinear differential equations
without requiring linearization, discretization and therefore,
it is not affected by errors associated to discretization also,
DTM does not require perturbation parameter. Another
important advantage is that this method reducing the size of
computational work while the Taylor series method is
computationally taken long time for the large orders. The
validity of the DTM is independent of whether or not there
exist small parameters in the considered equation.
Therefore, same as the HAM and the HPM, the DTM can
overcome the foregoing restrictions and limitations of
perturbation methods. In addition, the DTM provides an
efficient numerical solution with high accuracy, minimal
calculation and avoidance of physically unrealistic
assumptions.
Chen and Ho [4] developed this method for partial
differential equations and obtained closed form series
solutions for linear and nonlinear initial value problems and
Ayaz [5] applied it to the system of differential equations. In
recent years, the differential transform method has been
successfully employed to solve many types of nonlinear
problems such as the linear partial differential equations of
fractional order, the nonlinear oscillatory systems, multiorder
fractional differential equations, the hyperchaotic
Rossler system, the fourth-order boundary value problems,
the Volterra integral equation with separable kernels, the
difference equations, the free vibration equations of beam
on elastic soil, the integral and integro-differential equation
systems. All of these successful applications verified the
validity, effectiveness and flexibility of the DTM.