In this paper, we consider the boundary-value problem and initial value problem of Bratu’s problem. It is well known that Bratu’s boundary value problem in one-dimensional planar coordinates is of the form

${u}^{\u2033}+\lambda {e}^{u}=0,\phantom{\rule{1em}{0ex}}0<x<1,$

(1)

with the boundary conditions

$u(0)=u(1)=0$. For

$\lambda >0$ is a constant, the exact solution of equation (

1) is given by [

1]

$u(x)=-2ln\left[\frac{cosh(0.5\theta (x-0.5))}{cosh(0.25\theta )}\right],$

(2)

where

*θ* satisfies

$\theta =\sqrt{2\lambda}sinh(0.25\theta ).$

(3)

The problem has zero, one or two solutions when

$\lambda >{\lambda}_{c}$,

$\lambda ={\lambda}_{c}$ and

$\lambda <{\lambda}_{c}$, respectively, where the critical value

${\lambda}_{c}$ satisfies the equation

$1=\frac{1}{4}\sqrt{2{\lambda}_{c}}cosh\left(\frac{1}{4}\theta \right).$

It was evaluated in [1–3] that the critical value ${\lambda}_{c}$ is given by ${\lambda}_{c}=3.513830719$.

In addition, an initial value problem of Bratu’s problem

${u}^{\u2033}+\lambda {e}^{u}=0,\phantom{\rule{1em}{0ex}}0<x<1,$

(4)

with the initial conditions $u(0)={u}^{\prime}(0)=0$ will be investigated.

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 [4–11].

A substantial amount of research work has been done for the study of Bratu’s problem. Boyd [2, 12] employed Chebyshev polynomial expansions and the Gegenbauer as base functions. Syam and Hamdan [8] presented the Laplace decomposition method for solving Bratu’s problem. Also, Aksoy and Pakdemirli [13] developed a perturbation solution to Bratu-type equations. Wazwaz [10] 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 [14–17].

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 [4, 16, 18]. 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 [16].

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 (1) is identified.