Chebyshev wavelets method for solving Bratu’s problem
Boundary Value Problems
Chebyshev wavelets method for solving Bratu's problem
Changqing Yang 0
Jianhua Hou 0
0 Department of Science, Huaihai Institute of Technology , Lianyungang, Jiangsu 222005 , China
A numerical method for one-dimensional Bratu's problem is presented in this work. The method is based on Chebyshev wavelets approximates. The operational matrix of derivative of Chebyshev wavelets is introduced. The matrix together with the collocation method are then utilized to transform the differential equation into a system of algebraic equations. Numerical examples are presented to verify the efficiency and accuracy of the proposed algorithm. The results reveal that the method is accurate and easy to implement.
1 Introduction
=
with the boundary conditions u() = u() = . For λ > is a constant, the exact solution of
equation () is given by []
u(x) = – ln
cosh(.θ (x – .))
cosh(.θ )
It was evaluated in [–] that the critical value λc is given by λc = ..
In addition, an initial value problem of Bratu’s problem
Bratu’s problem is also used in a large variety of applications such as the fuel ignition
model of the thermal combustion theory, the model of thermal reaction process, the
Chandrasekhar model of the expansion of the universe, questions in geometry and
relativity about the Chandrasekhar model, chemical reaction theory, radiative heat transfer
and nanotechnology [–].
A substantial amount of research work has been done for the study of Bratu’s problem.
Boyd [, ] employed Chebyshev polynomial expansions and the Gegenbauer as base
functions. Syam and Hamdan [] presented the Laplace decomposition method for solving
Bratu’s problem. Also, Aksoy and Pakdemirli [] developed a perturbation solution to
Bratu-type equations. Wazwaz [] presented the Adomian decomposition method for
solving Bratu’s problem. In addition, the applications of spline method, wavelet method
and Sinc-Galerkin method for solution of Bratu’s problem have been used by [–].
In recent years, the wavelet applications in dealing with dynamic system problems,
especially in solving differential equations with two-point boundary value constraints have
been discussed in many papers [, , ]. By transforming differential equations into
algebraic equations, the solution may be found by determining the corresponding coefficients
that satisfy the algebraic equations. Some efforts have been made to solve Bratu’s problem
by using the wavelet collocation method [].
In the present article, we apply the Chebyshev wavelets method to find the approximate
solution of Bratu’s problem. The method is based on expanding the solution by Chebyshev
wavelets with unknown coefficients. The properties of Chebyshev wavelets together with
the collocation method are utilized to evaluate the unknown coefficients and then an
approximate solution to () is identified.
2 Chebyshev wavelets and their properties
2.1 Wavelets and Chebyshev wavelets
In recent years, wavelets have been very successful in many science and engineering fields.
They constitute a family of functions constructed from dilation and translation of a single
function called the mother wavelet ψ (x). When the dilation parameter a and the
translation parameter b vary continuously, we have the following family of continuous wavelets
[]:
Chebyshev wavelets ψn,m = ψ (k, n, m, x) have four arguments, n = , , . . . , k–, k can
assume any positive integer, m is the degree of Chebyshev polynomials of first kind and x
denotes the time.
nk–– ≤ x < kn– ;
and m = , , , . . . , M – , n = , , . . . , k–. Here Tm(x) are the well-known Chebyshev
polynomials of order m, which are orthogonal with respect to the weight function ω(x) =
√
/ – x and satisfy the following recursive formula:
T(x) = ,
T(x) = x,
Tm+(x) = xTm(x) – Tm–(x).
We should note that the set of Chebyshev wavelets is orthogonal with respect to the weight
function ωn(x) = ω(kx – n + ).
The derivative of Chebyshev polynomials is a linear combination of lower-order
Chebyshev polynomials, in fact [],
km=– Tk(x),
km=– Tk(x) + mT(x), m odd.
2.2 Function approximation
A function u(x) defined over [, ) may be expanded as
where cnm = (u(x), ψnm(x)), in which (·, ·) denotes the inner product with the weight
function ωn(x). If u(x) in () is truncated, then () can be written as
∞ ∞
n= m=
n= m=
u(x) ≈
C = [c, c, . . . , ck– ]T ,
ci = [ci, ci, . . . , ci,M–],
3 Chebyshev wavelets operational matrix of derivative
km=– Tk(kx – n + ),
ψi (x) = ψi(x)M, i = , , . . . , k–,
⎜⎜
⎝
⎜⎜
⎝
√ √ √ · · · (M – )√⎞
· · · ⎟⎟
· · · (M – ) ⎟⎟
... ... ... ... ... ·.·. ·. (M... – ) ⎟⎟⎟⎟⎟⎠
√ √ √ · · · ⎞
· · · ⎟⎟
... ... ... ... ... ··.··. ··. ((MM... –– ))⎟⎟⎟⎟⎠⎟⎟⎟
for even M,
for odd M.
In fact we have shown that
D = diag MT , MT , . . . , MT .
(...truncated)