An Efficient Approach for Solving Nonlinear Troesch’s and Bratu’s Problems by Wavelet Analysis Method

Abstract

We introduce Chebyshev wavelet analysis method to solve the nonlinear Troesch and Bratu problems. Chebyshev wavelets expansions together with operational matrix of derivative are employed to reduce the computation of nonlinear problems to a system of algebraic equations. Several examples are given to validate the efficiency and accuracy of the proposed technique. We compare the results with those ones reported in the literature in order to demonstrate that the method converges rapidly and approximates the exact solution very accurately by using only a small number of Chebyshev wavelet basis functions. Convergence analysis is also included.

1. Introduction

Nonlinear equations occurring in a wide variety of problems in engineering and science have received a great deal of attention in the recent decades. Consider the Troesch nonlinear two-point boundary value problem [1–3] in the following form: 𝑢^′′(𝑡)((𝑡[],𝑢(0)(1)+𝜆sinh𝜆𝑢))=0,𝑡∈0,1=0,𝑢=1, (1) where 𝜆 is a positive constant. Troesch’s problem comes from the investigation of the confinement of a plasma column under radiation pressure. Many authors have paid considerable attention to solving Troesch’s problem. Weibel was the first to explain and solve the problem [4]. Roberts and Shipman [3] obtained the closed form solution of the problem in terms of the Jacobi elliptic function. Troesch obtained numerical solution of this problem by the shooting method [5]. Deeba et al. [6] proposed a numerical method based on the decomposition technique. A numerical scheme based on the modified homotopy perturbation method is deduced by Feng et al. [7]. Variational iteration method was proposed by Momani et al. [8]. Khuri solved this problem numerically based on Laplace transform and a modified decomposition technique [9] and also proposed B-spline collocation approach for solving Troesch’s problem [10]. The Sinc-Galerkin method was introduced by Zarebnia and Sajjadian [11]. S.-H. Chang and I.-L. Chang [12] proposed a new technique based on differential transform for solving Troesch’s problem. A new algorithm based on the variational method and variable transformation was proposed by Chang [13]. More detailed information and references to other discussions on Troesch’s problem may be found in [14–18].

The closed form solution of Troesch’s problem is given in [3], where 𝑢^′(0) is the derivative at 0 given by 𝑢^′(0)=2(1−𝑚)^1/2. (2) The constant 𝑚 satisfies the transcendental equation sinh(𝜆/2)(1−𝑚)^1/2=sc(𝜆𝑚), (3) where sc(𝜆∣𝑚) is the Jacobi function defined by sc(𝜆∣𝑚)=sin𝜙/cos𝜙, where 𝜙, 𝜆, and 𝑚 satisfy the following integral: 𝜆=^𝜙₀11−𝑚sin²𝜃𝑑𝜃. (4)

It has been indicated in [3, 5] that 𝑢(𝑡) has a singularity located approximately at 𝑡_𝑠=12𝜆ln16=11−𝑚𝜆8ln𝑢^′(0), (5) which implies that the singularity lies within the integration range if 𝑢^′(0)>8𝑒^𝜆. This characteristic of Troesch’s problem makes it impossible to be solved by some methods when 𝜆>1 [6, 7, 9, 19].

Another well-known problem is the Bratu nonlinear two-point boundary value problem given in the following form: 𝑢^′′(𝑡)+𝜆𝑒^𝑢(𝑡)(),𝑢(0)(1)=0,𝑡∈0,1=0,𝑢=0, (6) where 𝜆 is a physical parameter and the prime denotes the differentiation with respect to 𝑡. The Bratu problem is indeed a special case of the following well-known classical Gelfand problem by assuming 𝑁=1: 𝑢^′′(𝑡)+𝑁−1𝑟𝑢^′(𝑡)+𝜆𝑒^𝑢(𝑡)(]𝑢(0)(1)=0,𝑟∈0,1,𝑁=1,2,3,…,=0,𝑢=0, (7)(8) where 𝑁=1,2, and 3 correspond to the infinite slab, infinite circular cylinder, and sphere, respectively. The numerical solutions for all 𝑁=1,2,3,… were obtained by Joseph and Lundgren [20] for the domain of a unit ball.

The Bratu problem has an analytical solution given in the following form: 𝑢(𝑡)(=−2lncosh((𝑡−1/2))(𝜃/2))(),cosh𝜃/4 (9) where 𝜃 is the solution of 𝜃=√2𝜆cosh(𝜃/4).

The Bratu problem has zero, one, or two solutions when 𝜆>𝜆_𝑐, 𝜆=𝜆_𝑐, and 𝜆<𝜆_𝑐, respectively, where the critical value 𝜆_𝑐 satisfies the equation 1=(1/4)2𝜆_𝑐sinh(𝜃_𝑐/4) and it was obtained in [21, 22] that the critical value 𝜆_𝑐 is given by 𝜆_𝑐=3.513830719.

The Bratu problem is used in a different variety of applications such as the fuel ignition of the thermal combustion theory, the model thermal reaction process, the Chandrasekhar model of the expansion of the universe, chemical reaction theory, radiative heat transfer, and nanotechnology [23–29].

Solving the Bratu Problem by analytical and numerical methods has gained considerable attention from many authors. Aregbesola applied weighted residual method [22] and Wazwaz has employed the Adomian decomposition method to get exact solutions [23]. Homotopy analysis method was developed by Liao and Tan [29] and Laplace transform decomposition method was used in [30]. Nonpolynomial spline method has been applied by Jalilian to obtain smooth approximate solution of the one-dimensional Bratu problem [31]. Caglar et al. [32] developed the B-spline method. Variational method and differential transform method were used in [33, 34]. Also Abbasbandy et al. [35] tried to solve the problem using the Lie-group shooting method.

In the recent years, wavelets have received considerable attention by researchers in different fields of science and engineering. One advantage of wavelet analysis is the ability to perform local analysis [36]. Wavelet analysis is able to reveal signal aspects that other analysis methods miss, such as trends, breakdown points, and discontinuities. In comparison with other orthogonal functions, multiresolution analysis aspect of wavelets permits the accurate representation of a variety of functions and operators. In other words, we can change 𝑀 and 𝑘 simultaneously to get more accurate solution. In addition, the coefficient matrix of algebric equations obtained after dicretization is sparse. So it is computationally efficient to use wavelet methods for solving equations. In addition, the solution is convergent.

We organize our paper as follows. In Section 2, we introduce the Chebyshev wavelets, and the operational matrix of derivative for Chebyshev wavelets is defined. In Section 3, convergence analysis is included. In Section 4, we introduce the method of solving Troesch’s and Bratu’s problems by wavelet analysis method. Several numerical examples are included in Section 5 to confirm that our method is efficient and accurate. Some conclusions are drawn in Section 6.

2. Chebyshev Wavelets and Their Properties

2.1. Wavelets and Chebyshev Wavelets

Wavelets have been very successfully used in many scientific and engineering fields. They constitute a family of functions constructed from dilation and transformation of a single function called the mother wavelet 𝜓(𝑡); we have the following family of continuous wavelets: 𝜓_𝑎,𝑏(𝑡)=|𝑎|^−1/2𝜓𝑡−𝑏𝑎,𝑎,𝑏∈𝑅,𝑎≠0. (10) Chebyshev wavelets, 𝜓_𝑛,𝑚=𝜓(𝑘,𝑛,𝑚,𝑡), have four arguments, 𝑛=0,1,…,2^𝑘−1, where 𝑘 can assume any positive integer, 𝑚 is the degree of Chebyshev polynomials of the first kind, and 𝑡 denotes the time.

Consider 𝜓_𝑛,𝑚(𝑡)=2^(𝑘+1)/2𝑇_𝑚2^𝑘+1,𝑛𝑡−2𝑛−12^𝑘≤𝑡<𝑛+12^𝑘,0,otherwise, (11) where 𝑇_𝑚(𝑡)=1√𝜋,𝑚=0,2𝜋𝑇_𝑚(𝑡),𝑚≥1, (12) and 𝑚=0,1,…,𝑀, and 𝑛=0,1,…,2^𝑘−1. In (10) the coefficients are used for orthonormality. Here 𝑇_𝑚(𝑡) are Chebyshev polynomials of the first kind of degree 𝑚 which are orthogonal with respect to the weight function, 𝑤(𝑡)=1/√1−𝑡², and satisfy the following recursive formula: 𝑇₀(𝑡)=1,𝑇₁(𝑡)𝑇=𝑡,_𝑚+1(𝑡)=2𝑡𝑇_𝑚(𝑡)−𝑇_𝑚−1(𝑡),𝑚=1,2,…. (13) Note that, in dealing with Chebyshev wavelets, the weight function 𝑤(𝑡) has to be dilated and translated as follows: 𝑤_𝑛,𝑘(𝑡)=𝑤2^𝑘+1𝑡−2𝑛−1 (14) in order to get orthogonal wavelets.

2.2. Function Approximation

A function 𝑢(𝑡)∈𝐿𝑤²[0,1] (where 𝑤(𝑡)=𝑤(2𝑡−1)) may be expanded as 𝑢(𝑡)=^∞∞_𝑛=0𝑚=0𝑐_𝑛,𝑚𝜓_𝑛,𝑚(𝑡), (15) where 𝑐_𝑛,𝑚=(𝑢(𝑡),𝜓_𝑛,𝑚(𝑡))_{𝑤𝑛,𝑘}, in which (⋅,⋅) denotes the inner product in 𝐿𝑤²[0,1]. If we consider truncated series in (15), we obtain 𝑢(𝑡)=^{2𝑘−1𝑀}_𝑛=0𝑚=0𝑐_𝑛,𝑚𝜓_𝑛,𝑚(𝑡)=𝐶^𝑇Ψ(𝑡), (16) where 𝐶 and Ψ(𝑡) are 2^𝑘(𝑀+1)×1 matrices given by 𝐶=𝑐_0,0,𝑐_0,1,…,𝑐_0,𝑀,𝑐_1,0,𝑐_1,1,…,𝑐_1,𝑀𝑐,…,_2𝑘−1,0,𝑐₂^𝑘_−1,1,…,𝑐₂^𝑘_−1,𝑀^𝑇,Ψ(𝑡)=𝜓_0,0,𝜓_0,1,…,𝜓_0,𝑀,𝜓_1,0,𝜓_1,1,…,𝜓_1,𝑀𝜓,…,_2𝑘−1,0,𝜓_2𝑘−1,1,…,𝜓_{2𝑘−1,𝑀}^𝑇. (17)

2.3. Operational Matrix of Derivative (OMD)

The derivative of the vector Ψ(𝑡), which is defined in (17), can be expressed by 𝑑𝜓(𝑡)(𝑡),𝑑𝑡=𝐷𝜓 (18) where 𝐷 is 2^𝑘(𝑀+1)×2^𝑘(𝑀+1) operational matrix of derivative defined as follows: 𝐷=𝐸0⋯00𝐸⋯0⋮⋮⋱⋮000𝐸, (19) in which 𝐸 is (𝑀+1)×(𝑀+1) matrix and its (𝑖,𝑗)th element is defined as follows: 𝐸_𝑖,𝑗=2^𝑘+2𝑚𝛾_𝑖−1𝛾_𝑗−1,𝑖=2,…,(𝑀+1),𝑗=1,…,𝑗−1;𝑖+𝑗isodd,0,otherwise, (20) where 𝛾_𝑛=2,𝑛=0,1,𝑛≥1. (21) The method of calculation of 𝐷 is illustrated in [37].

Corollary 1. The operational matrix for nth derivative can be obtained using (18) as 𝑑^𝑛Ψ(𝑡)𝑑𝑥^𝑛=𝐷^𝑛Ψ(𝑡), (22) where 𝐷^𝑛 is the 𝑛th power of matrix 𝐷.

3. Convergence Analysis

Lemma 2. If the Chebyshev wavelet expansion of a continuous function 𝑢(𝑡) converges uniformly, then the Chebyshev wavelet expansion converges to the function 𝑢(𝑡).

Proof. Suppose the Chebyshev wavelet expansion of the continuous function 𝑢(𝑡) converges to function 𝑣(𝑡)𝑣(𝑡)=^{2𝑘−1𝑀}_𝑛=1𝑚=0𝑐̂_𝑛𝑚𝜓_𝑛𝑚(𝑡), (23) where 𝑐̂_𝑛,𝑚=(𝑢(𝑡),𝜓_𝑛,𝑚(𝑡))_{𝑤𝑛,𝑘}. Multiply both sides of (23) by 𝜓_𝑟,𝑠(𝑡)𝑤_𝑟,𝑘(𝑡), where 𝑟 and 𝑠 are fixed. Due to the uniform convergence, we can then integrate termwise on interval [0,1]¹₀𝑣(𝑡)𝜓_𝑟,𝑠(𝑡)𝑤_𝑟,𝑘=𝑑𝑡_{2𝑘−1𝑀𝑛=1𝑚=010}𝑐̂_𝑛𝑚𝜓_𝑛𝑚(𝑡)𝜓_𝑟,𝑠(𝑡)𝑤_𝑟,𝑘𝑑𝑡. (24) By the orthonormality of wavelet basis functions, we have ¹₀𝑣(𝑡)𝜓_𝑟,𝑠(𝑡)𝑤_𝑟,𝑘𝑑𝑡=𝑐̂_𝑟,𝑠. (25) Thus (𝑣(𝑡),𝜓_𝑛,𝑚(𝑡))_{𝑤𝑛,𝑘}=𝑐̂_𝑛,𝑚 for 𝑛=1,2,…,2^𝑘−1 and 𝑚=0,1,…,𝑀; consequently, 𝑢(𝑡) and 𝑣(𝑡) have the same Fourier expansions with Chebyshev wavelet basis; therefore, 𝑢(𝑡)=𝑣(𝑡), for 𝑡∈[0,1] [38].

Theorem 3. A function 𝑢(𝑡)∈𝐿²_{𝑤𝑛,𝑘}([0,1]), with bounded second derivative, say |𝑢^′′(𝑡)|≤𝐵, can be expanded as an infinite sum of Chebyshev wavelets, and the series converges uniformly to 𝑢(𝑡); that is, 𝑢(𝑡)=^∞∞_𝑛=1𝑚=0𝑐̂_𝑛𝑚𝜓_𝑛𝑚(𝑡). (26)

Proof. We have 𝑐̂_𝑛,𝑚=𝑢(𝑡),𝜓_𝑛,𝑚(𝑡)_{𝑤𝑛,𝑘}=¹₀𝑢(𝑡)𝜓_𝑛𝑚(𝑡)𝑤_𝑛,𝑘(𝑡)=𝑑𝑡_{𝑛/2𝑘−1𝑛−1/2𝑘−1}2^𝑘/2𝑝_𝑚𝑢(𝑡)𝑇_𝑚2^𝑘2𝑡−2𝑛+1×𝑤^𝑘𝑡−2𝑛+1𝑑𝑡. (27) For 𝑚>1, by substituting 2^𝑘𝑡−2𝑛+1=cos𝛼, it yields 𝑐̂_𝑛𝑚=12^𝑘/2𝜋₀𝑢cos𝛼+2𝑛−12^𝑘2𝜋cos𝑚𝛼𝑑𝛼. (28) Using integration by parts, we get 𝑐̂_𝑛𝑚=√22^𝑘/2√𝜋𝑢cos𝛼+2𝑛−12^𝑘sin𝑚𝛼𝑚^𝜋₀+√22^3𝑘/2𝑚√𝜋^𝜋₀𝑢^′cos𝛼+2𝑛−12^𝑘sin𝑚𝛼sin𝛼𝑑𝛼. (29) The first part is zero; therefore, 𝑐̂_𝑛𝑚=√22^3𝑘/2𝑚√𝜋^𝜋₀𝑢^′cos𝛼+2𝑛−12^𝑘sin𝑚𝛼sin𝛼𝑑𝛼. (30) Using integration by parts again, it yields 𝑐̂_𝑛𝑚=12^3𝑘/2𝑚√2𝜋×𝑢^′cos𝛼+2𝑛−12^𝑘sin(𝑚−1)𝛼−()𝛼𝑚−1sin𝑚+1𝑚+1^𝜋₀+12^5𝑘/2𝑚√2𝜋^𝜋0𝑢^′′cos𝛼+2𝑛−12^𝑘𝑟_𝑚(𝛼)𝑑𝛼, (31) where 𝑟_𝑚(𝛼)()𝛼=sin𝛼sin𝑚−1−()𝛼𝑚−1sin𝑚+1𝑚+1. (32) Thus, we get 𝑐̂_𝑛𝑚=12^5𝑘/2𝑚√2𝜋^𝜋0𝑢^′′cos𝛼+2𝑛−12^𝑘𝑟_𝑚(𝛼)≤1𝑑𝛼2^5𝑘/2𝑚√2𝜋^𝜋0𝑢^′′cos𝛼+2𝑛−12^𝑘𝑟_𝑚(𝛼≤𝐵)𝑑𝛼2^5𝑘/2𝑚√2𝜋^𝜋0𝑟_𝑚(𝛼)𝑑𝛼. (33) However ^𝜋₀𝑟_𝑚(𝛼=)𝑑𝛼_𝜋0()𝛼sin𝛼sin𝑚−1−()𝛼𝑚−1sin𝑚+1𝑚+1𝑑𝛼≤^𝜋₀sin𝛼sin(𝑚−1)𝛼+()𝛼𝑚−1sin𝛼sin𝑚+1≤𝑚+1𝑑𝛼2𝑚𝜋𝑚².−1 (34) Since 𝑛≤2^𝑘−1, we obtain 𝑐̂_𝑛𝑚≤√2𝜋𝐵(2𝑛)^5/2𝑚².−1 (35) Now, if 𝑚=1, by using (30), we have 𝑐̂_𝑛1<√2𝜋(2𝑛)^3/2max_0≤𝑡≤1𝑢^′(𝑡). (36) It is mentioned in [39] that {𝜓_𝑛0}^∞_𝑛=1 form an orthogonal system constructed by Haar scaling function with respect to the weight function 𝑤(𝑡), and so ∑^∞_𝑛=1𝑐̂_𝑛0𝜓_𝑛0(𝑡) is convergent. Hence, we will have 𝑐̂^∞∞_{𝑛=1𝑚=0𝑛𝑚}𝜓_𝑛𝑚(𝑡≤_∞𝑛=1)𝑐̂_𝑛0𝜓_𝑛0(𝑡)+^∞∞_𝑛=1𝑚=1𝑐̂_𝑛𝑚𝜓_𝑛𝑚(𝑡≤_∞𝑛=1)𝑐̂_𝑛0𝜓_𝑛0(𝑡)+^∞∞_𝑛=1𝑚=1𝑐̂_𝑛𝑚<∞. (37) Therefore, in view of Lemma 2, the series ∑^∞_𝑛=1∑^∞_𝑚=1𝑐̂_𝑛𝑚𝜓_𝑛𝑚(𝑡) converges to 𝑢(𝑡) uniformly.

4. The Chebyshev Wavelet Analysis Method

4.1. Troesch’s Problem

Troesch’s problems (1) can be considered as follows: 𝑢^′𝑤=𝑤,_′(),𝑢(0)(1)=𝜆sinh𝜆𝑢=0,𝑢=1. (38) Then the Jacobian matrix of system (38) is given by 𝐽()=𝜆𝑢,𝑤01₂cosh(𝜆𝑢)0. (39) Therefore, the eigenvalues of the Jacobian matrix at the end points of the interval [0,1] are 𝜆^′(0)=±𝜆,𝜆^′(1)=±𝜆cosh(𝜆). (40) If we choose large 𝜆, we will have large eigenvalues; for example, if 𝜆=10, then the eigenvalue becomes 𝜆^′=±1049. On the other hand, the Jacobian matrix 𝐽 is normal if and only if 𝜆²cosh(𝜆𝑢)=1. This equality is satisfied only for relatively small values of 𝜆. This clarifies why conventional methods, such as finite differences, are not convenient for large values of 𝜆 [10]. To solve Troesch’s problems where there exist, a strong nonlinear term sinh(𝜆𝑢) and boundary layer only at the right endpoint (𝜆>1) [10], we propose Chebyshev wavelet analysis method especially to handle larger eigenvalues.

Consider the Troesch nonlinear boundary value problem 𝑢^′′(𝑡)+𝜆sinh(𝜆𝑢(𝑡))=0,𝑡∈(0,1), (41) with the boundary conditions 𝑢(0)=0,𝑢(1)=1. (42) In order to solve the problem, we first approximate all functions 𝑢(𝑡) and 𝑢^′′(𝑡) using basis functions Ψ(𝑡) as 𝑢(𝑡)=𝐶^𝑇Ψ(𝑡),𝑢^′′(𝑡)=𝐶^𝑇𝐷²Ψ(𝑡). (43) Substituting (43) in (41), we obtain 𝐶^𝑇𝐷²Ψ(𝑡)−𝜆sinh𝜆𝐶^𝑇Ψ(𝑡)=0. (44) Now, we have 2^𝑘(𝑀+1)−2 nonlinear equations by collocating (41) at 2^𝑘(𝑀+1)−2 suitable collocation, points. From boundary conditions we also get two equations. So we have a nonlinear system of 2^𝑘(𝑀+1) equations with the same number of unknowns which can be solved by Newton’s iterative method to obtain the vector 𝐶 and consequently the approximated solution 𝑢(𝑡).

4.2. Bratu’s Problem

Consider the Bratu nonlinear boundary value problem 𝑢^′′(𝑡)+𝜆𝑒^𝑢(𝑡)=0,𝑡∈(0,1), (45) with the boundary conditions 𝑢(0)=0,𝑢(1)=0, (46) or initial conditions 𝑢(0)=0,𝑢^′(0)=0. (47) In order to solve the problem, we first approximate all functions 𝑢(𝑡), 𝑢^′(𝑡), and 𝑢^′′(𝑡) using basis functions Ψ(𝑡) as 𝑢(𝑡)=𝐶^𝑇Ψ(𝑡),𝑢^′(𝑡)=𝐶^𝑇𝐷Ψ(𝑡),𝑢^′′(𝑡)=𝐶^𝑇𝐷²Ψ(𝑡). (48) Substituting (48) in (45), we obtain 𝐶^𝑇𝐷²Ψ(𝑡)+𝜆𝑒^{𝐶𝑇Ψ(𝑡)}=0. (49) Now, we have 2^𝑘(𝑀+1)−2 nonlinear equations by collocating (49) at 2^𝑘(𝑀+1)−2 suitable collocation points. From boundary conditions, or initial conditions, we also get two equations. So we have a nonlinear system of 2^𝑘(𝑀+1) equations with the same number of unknowns which can be solved by Newton’s iterative method to obtain the vector 𝐶 and consequently the approximated solution 𝑢(𝑡).

5. Numerical Examples

In this section, we solve Troesch’s and Bratu’s problems for different values of the parameter 𝜆 using the computer algebra system Maple and make a comparison between our results and those ones reported in the literature to confirm the efficiency and accuracy of our method.

Example 1. Troesch’s problem for 𝜆=0.5,1, and 10.
In Tables 1 and 2, the absolute errors in solutions obtained by the introduced method for 𝜆=0.5 and 𝜆=1, respectively, are compared with those ones reported by other existing methods. We observe that the wavelet analysis method with only a few number of basis functions is comparable to Laplace, HAM, and spline methods but is slightly better than perturbation method and much better than variational method in terms of accuracy.

Table 1 

Obtained absolute errors of Troesch's problem for 𝜆=0.5.

Table 2 

Obtained absolute errors of Troesch's problem for 𝜆=1.0.

Due to the nonlinear term sinh(𝜆𝑢(𝑡)), which is not analytic, some methods like Laplace, variational iteration method, and homotopy are not able to solve the Troesch problem when 𝜆≥5. In Table 3, the numerical solution for 𝜆=5 obtained by the current method is compared with the numerical approximation of the exact solutions given by a Fortran code called TWPBVP and B-spline method [10]. It can be seen that our obtained results are much more accurate than those obtained by B-spline method.

Table 3 

Numerical solution of Troesch's problem for 𝜆=5.

In Table 4, the numerical solution obtained by the current method using 𝑀=20 and 𝑘=5 (672-term approximant), for 𝜆=10, is compared with the results obtained by B-spline method over a nonuniform mesh using 𝑛=790 mesh points [10], with those in [12] computed using 1750-term approximant, and with those obtained by [13] using a method based on the variational iteration method and variable transformation. It can be seen that the results obtained by the present method with much lesser number of terms to approximate the solution are compatible and in well agreement with those ones obtained by Chang [13].

Table 4 

Numerical solution of Troesch's problem for 𝜆=10.

Example 2. Consider the Bratu problem for 𝜆=1.
We solve the problem by applying the technique described in Section 4 with 𝑀=10 and 𝑘=3. The absolute errors in solutions are tabulated in Table 5. As can be seen in Table 5, only a small number of Chebyshev wavelet basis functions are needed to get the approximate solution which is in full agreement with the exact solution up to 18 digits while, using other methods, we can find a numerical approximation to the exact solution which is the same at most in 10 digits. We display the exact and obtained solutions in Figure 1. Absolute errors in solutions are plotted in Figure 2.

Table 5 

Obtained absolute errors for 𝜆=1.

Figure 1 

Plot of exact and approximated solutions for 𝜆=1.

Figure 2 

Plot of absolute errors for 𝜆=1.

Example 3. Consider the Bratu problem for 𝜆=2.
We solve the problem with 𝑀=10 and 𝑘=3. The absolute errors in solutions are tabulated in Table 6. We display the exact and obtained solutions in Figure 3. The plot of absolute errors in solutions in Figure 4 confirms the priority of our method over other methods in terms of efficiency and accuracy.

Table 6 

Obtained absolute errors for 𝜆=2.

Figure 3 

Plot of exact and approximated solutions for 𝜆=2.

Figure 4 

Plot of absolute errors for 𝜆=2.

Example 4. Consider the Bratu problem for 𝜆=3.51.
In this example, we set 𝑀=10 and 𝑘=3. The absolute errors in solutions are tabulated in Table 7. As can be seen in Table 7, when 𝜆 is close to the critical value 𝜆_𝑐, some of the mentioned methods are not able to handle the problem very well. However, using wavelet analysis method, we get the approximate solution which is much more accurate than nonpolynomial spline method and Lie-group shooting method. We display the exact and obtained solutions in Figure 5. Absolute errors in solutions are illustrated in Figure 6.

Table 7 

Obtained absolute errors for 𝜆=3.51.

Figure 5 

Plot of exact and approximated solutions for 𝜆=3.51.

Figure 6 

Plot of absolute errors for 𝜆=3.51.

The maximum absolute errors in solutions for different values of 𝑀, 𝑘, and 𝜆 are tabulated in Table 8. According to Table 8, we can conclude that more accurate results can be obtained by increasing the values of 𝑀 and 𝑘 properly.

Table 8 

Obtained maximum absolute errors.

6. Conclusion

The well-known nonlinear Troesch and Bratu problems arise in a different variety of applications, and many researchers have drawn attention to solve them. The difficulty in this type of problems, due to existing strong nonlinear terms, is overcome here. The main characteristic of the proposed method is reducing the given problems to those of solving a system of algebraic equations, thus greatly simplifying the problems. Sparseness of the coefficients matrix of algebraic equations makes it computationally efficient to solve these problems using the current method. It is also seen that increasing the number of subintervals or the number of collocation points in subintervals results in improving the accuracy. Numerical results confirm that our method is much better than other reported ones in the literature in the sense of accuracy and efficiency. According to Tables 1–4, our approach is applicable to solve Troesch’s problem especially when 𝜆 is large while some other methods fail to do so. As shown in Tables 5–8, we can obtain the results for Bratu’s problem only by using a small number of Chebyshev wavelet basis functions. When 𝜆 is close to the critical value 𝜆_𝑐, the wavelet analysis method was also accurate to the ninth order, whereas other methods especially the B-spline method yielded poorer results.

Acknowledgment

The authors gratefully acknowledge that this research was partially supported by the Universiti Putra Malaysia under the ERGS Grant Scheme having project number 5527068.