Two Schemes Based on the Collocation Method Using Müntz–Legendre Wavelets for Solving the Fractional Bratu Equation

Abstract

Our goal in this work is to solve the fractional Bratu equation, where the fractional derivative is of the Caputo type. As we know, the nonlinearity and derivative of the fractional type are two challenging subjects in solving various equations. In this paper, two approaches based on the collocation method using Müntz–Legendre wavelets are introduced and implemented to solve the desired equation. Three different types of collocation points are utilized, including Legendre and Chebyshev nodes, as well as uniform meshes. According to the experimental observations, we can confirm that the presented schemes efficiently solve the equation and yield superior results compared to other existing methods. Also, the schemes are convergent.

1. Introduction

Bratu equations in science and engineering have captured the interest of many due to their valuable applications. We can find insights into the origins and importance of Bratu equations in [1,2]. This equation is used to model numerous chemical and physical processes, including the Chandrasekhar model of the expansion of the universe, the thermal reaction process in combustibles, the electro-spinning process for the production of ultrafine polymer fibers, nanotechnology, chemical reactor theory, and the solid in the fuel ignition model [3,4,5,6,7].

For most nonlinear differential equations, analytical methods to solve them either do not exist or are very difficult to apply. Numerical methods have consistently proven to be effective [8,9,10,11,12]. To mention but a few, the Bratu equation has been solved using the Adomian decomposition method [13], the finite difference method [14], the variational iteration method [15], the successive differentiation method [16], the Block Nyström method [17], Taylor’s decomposition on two points [18], the differential transformation method [19], the Laplace transform decomposition method [20], the shooting method [21], the Laplace Adomain decomposition method [22], the Jacobi–Gauss collocation method [23], the perturbation iteration method [24], the Spline method [25], the Sinc–Galerkin method [26], the Legendre wavelets method [27], an algorithm based on Chebyshev polynomial expansion [28], the hybrid block method [29,30], and the Taylor wavelets method [31].

The fractional Bratu equation can be expressed as a fractional nonlinear differential equation

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\left(t\right)\right)+\rho e^{w\left(t\right)}=0,t\in [0,1],1<\upsilon \le 2,$$

(1)

with initial conditions

$$w\left(0\right)=a,w^{\prime }\left(0\right)=b,$$

(2)

where $\rho$, a, and b are constant. The fractional derivative denoted by ${ }^c{\mathcal{D}}_0^{\upsilon }$ is of the Caputo type, and this concept will be introduced later. Note that the standard Bratu equation can be obtained by choosing $\upsilon =2$.

In general, the existence and uniqueness of the solution of equation

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\left(t\right)\right)=f(t,w\left(t\right)),\upsilon \in {\mathbb{R}}^{+},$$

(3)

have been established in space

$$C_{\gamma }^{\upsilon ,\beta }\left([0,1]\right):=\{w\left(t\right)\in C^{\beta }\left([0,1]\right):{ }^c{D}_0^{\upsilon }\left(w\right)\in C_{\gamma }\left([0,1]\right)\},\beta \in \mathbb{N},\upsilon >0,0\le \gamma <1,$$

where $C_{\gamma }\left([0,1]\right)$ is the space of functions w with $t^{\gamma }w\left(t\right)\in C\left([0,1]\right)$.

Theorem 1 

(cf [32]). Assume that $0\le \gamma <1$, $\gamma \le \upsilon$, $G\subset \mathbb{R}$ is an open set, and $f:[0,1]\times G\to \mathbb{R}$, with $f(t,w)\in C_{\gamma }\left([0,1]\right)$, where $w\in G$, satisfies the Lipschitz condition

$$\left|f\right(t,w\left)\right(t)-f(w,v\left)\right(t\left)\right|\le \mathsf{\Lambda }|w-v|,w,v\in G,$$

(4)

where $\mathsf{\Lambda }>0$ is independent of t. Let $w\left(0\right)=a,w^{\prime }\left(0\right)=b$. If $\kappa -1<\upsilon <\kappa =\left[\upsilon \right]+1$, then (3) has a unique solution $w\in C_{\gamma }^{\upsilon ,\kappa -1}$.

In recent years, fractional calculus has become increasingly important in modeling various physical phenomena in engineering and other fields. Extensive research has been conducted on various equations involving fractional derivatives, leading to the development of numerous numerical methods aimed at their solving. To mention but a few, the wavelet spectral element has been used to solve fractional Cauchy-type problems [33], and the wavelet collocation method has been applied to solve the fractional Cauchy problem [34]. In [35], the authors studied the fractional differential equation with multi-order fractional derivatives. The fractional differential equation with multi-order fractional derivatives is solved using the fractional Lucas optimization method in [36]. For further studies of fractional differential equations and proposed methods for solving them, we refer the readers to [37,38,39,40,41].

The fractional Bratu equation proves invaluable in understanding combustion theory, electrospinning processes, and heat transfer as well as the expansion of the universe. Implementing numerical schemes for this equation is challenging due to its nonlinearity and fractional derivative. There are some semi-analytical and numerical schemes for solving this equation, including the Chebyshev collocation method [42], the differential transform method [43], and the homotopy perturbation transform method [44].

Wavelets have become an effective tool in solving differential equations and representing various operators in recent years [45]. Two types of wavelets are used for solving equations through numerical schemes: multiwavelets and scalar wavelets. Unlike scalar wavelets that use one generator to generate wavelet spaces, multiwavelets use multiple generators [46]. Consequently, they offer certain advantages over scalar wavelets in different applications. It is enough to mention that they possess high vanishing moment, orthogonality, a closed form, symmetry, and more, all at the same time. The Alpert multiwavelet is a widely recognized wavelet with diverse uses in image processing and numerical computations [45,47,48,49]. In numerical works, the utilization of Müntz–Legendre (ML) wavelets [50,51], a type of multiwavelet, has been on the rise. This includes solving problems such as multi-order fractional differential equations [34], fractional optimal control problems [51], and pantograph equations with fractional derivatives [52].

In this paper, Section 2 is dedicated to the study of ML wavelets and their properties. We then implement the wavelet collocation method for solving the fractional Bratu equation in Section 3. In Section 4, we conduct several numerical experiments to demonstrate the accuracy and usefulness of the method. Section 5 of our paper provides a concise summary of our findings.

2. Müntz–Legendre Wavelets

Given $\mathcal{M}=\{0={\upsilon }_0<{\upsilon }_1<\dots \}$ as an increasing sequence, let $T_n\left(\mathcal{M}\right):=span\{t^{{\upsilon }_0}, t^{{\upsilon }_1},\dots ,t^{{\upsilon }_n}\}$ [53]. The space $T\left(\mathcal{M}\right)$ is introduced so that it is spanned by the function ${\left\{t^{{\upsilon }_n}\right\}}_{n=0}^{\infty }$, i.e.,

$$T\left(\mathcal{M}\right):=\bigcup _{n=0}^{\infty }{T}_n\left(\mathcal{M}\right)=span\{t^{{\upsilon }_n},n=0,1,\dots \},t\in (0,1).$$

(5)

To confirm $\overline{T\left(\mathcal{M}\right)}=C[0,1]$ (the space of continuous functions on $[0,1]$), there are sufficient and necessary criteria introduced by S. N. Bernstein, as

$$\sum _{{\upsilon }_n>0}{\displaystyle \frac{1+log{\upsilon }_n}{{\upsilon }_n}}=\infty ,$$

(6)

and

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\upsilon }_n}{nlogn}}=0,$$

(7)

respectively [54]. So it can be concluded that $T\left(\mathcal{M}\right)$ is dense in $C[0,1]$. It is also worth noting that Bernstein proposed the condition

$$\sum _{n=1}^{\infty }{\displaystyle \frac{1}{{\upsilon }_n}}=\infty ,$$

(8)

for the existence and uniqueness of the increasing sequence of $\mathcal{M}=\{0={\upsilon }_0<{\upsilon }_1<\dots \}$. Müntz proved this conjecture two years later [55]. Also, one can find the sufficient and necessary conditions to confirm $\overline{T\left(\mathcal{M}\right)}=L^2(0,1)$ in [54].

It is crucial to understand that utilizing the functions ${\left\{t^{{\upsilon }_n}\right\}}_{n=0}^{\infty }$ as bases is not advisable. That is why, in the following, the ML functions are defined in such a way that they are orthogonal and easy to evaluate.

Assuming $\mathsf{ϝ}$ as a simple contour that encircles all zeros in the integrand’s denominator, the ML polynomials have the following closed form [54,56].

$$L_n(t;\mathcal{M}):={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{ϝ}}\prod _{k=1}^{n-1}{\displaystyle \frac{x+{\upsilon }_k+1}{x-{\upsilon }_k}}{\displaystyle \frac{t^x}{x-{\upsilon }_n}}dx,$$

(9)

Taking into account the Residue Theorem, we can immediately conclude that the ML polynomials can be written as

$$L_n(t;\mathcal{M})=\sum _{k=0}^n{l}_{k,n}{t}^{{\upsilon }_k},{\upsilon }_k>-1/2,t\in [0,1].$$

(10)

Here, it is assumed that the sequence ${\left\{{\upsilon }_k\right\}}_{k=1}^{\infty }$ is increasing and ${\upsilon }_k$ are specified as ${\upsilon }_k:=\{k\eta :\eta \in \mathbb{R},k=0,\dots n\}$. The coefficients in this definition can be calculated by [56]

$$l_{k,n}:={\displaystyle \frac{{\prod }_{i=0}^{n-1}({\upsilon }_k+{\upsilon }_i+1)}{{\prod }_{i=0,i\ne k}^n({\upsilon }_k-{\upsilon }_i)}}.$$

(11)

Motivated by Theorem 2.4 in [56], the orthogonality can be obtained for Müntz–Legendre polynomials as

$${\int }_0^1{L}_n\left(t\right)L_{n^{\prime }}\left(t\right)dt={\delta }_{n,n^{\prime }}/({\upsilon }_n+{\upsilon }_{n^{\prime }}+1),$$

(12)

where ${\delta }_{n,n^{\prime }}$ is the Kronecker delta function, and where, for the sake of simplicity in writing, it has been considered that $L_n\left(t\right):=L_n(t;\mathcal{M})$.

Considering $m\in \mathbb{N}$ and $s\in {\mathbb{N}}_0$, which are called the multiplicity and refinement levels, respectively, and according to the multi-resolution analysis (MRA) definition [57], there is a nested sequence of subspaces ${\left\{V_s\right\}}_{s\in {\mathbb{N}}_0}\subset L^2\left([0,1]\right)$ determined by

$$V_s=span\{{\varphi }_{s,\tau }^n:\tau \in \mathcal{A},n\in \mathcal{R}\},$$

(13)

in which $\mathcal{R}:=\{0,1,\dots ,m-1\}$, $\mathcal{A}:=\{0,1,\dots ,2^s-1\}$, and ${\varphi }_{s,\tau }^n\left(t\right)$ specifies the Müntz–Legendre wavelets [34]

$${\varphi }_{s,\tau }^n\left(t\right)=\left\{\begin{array}{cc}\hfill 2^{s/2}\sqrt{2{\upsilon }_n+1}{L}_n(2^{s}t-\tau ),\hfill & \hfill {\displaystyle \frac{\tau }{2^s}}\le t\le {\displaystyle \frac{\tau +1}{2^s}},\hfill \\ \hfill 0,\hfill & \hfill otherwise.\hfill \end{array}\right.$$

(14)

To approximate the function $w\in L^2[0,1]$, we consider the projection operator ${\mathcal{P}}_s$ that maps $w\left(t\right)$ onto $V_s$, that is,

$$w\left(t\right)\approx {\mathcal{P}}_s\left(w\right)\left(t\right)=\sum _{\tau =0}^{2^s-1}\sum _{n=0}^{m-1}{w}_{\tau ,n}{\varphi }_{s,\tau }^n\left(t\right)=W^T\mathsf{\Phi }\left(t\right)\in V_s,$$

(15)

where ${\left[\mathsf{\Phi }\left(t\right)\right]}_{\tau m+n+1,1}:={\varphi }_{s,\tau }^n\left(t\right)$, and the coefficients $w_{\tau ,n}$ can be attained by

$$w_{\tau ,n}=⟨w,{\varphi }_{s,\tau }^n⟩={\int }_0^{1}w\left(t\right){\varphi }_{s,\tau }^n\left(t\right)dt.$$

(16)

There is an estimation of the bound of approximation (15) that can be stated by the following lemma [51].

Lemma 1. 

Given $s\in {\mathbb{N}}_0$ and $m\in \mathbb{N}$, let $w\in H^{\nu }[0,1]$ for any $\nu <m$. Then, it can be verified that

$$\parallel w-{\mathcal{P}}_s{\left(w\right)\parallel }_2\le c{\left(2^{s-1}\right)}^{-\nu }{(m-1)}^{-\nu }{\parallel w^{\left(\nu \right)}\parallel }_2,$$

(17)

and when ${\nu }^{\prime }\ge 1$, we get

$$\parallel w-{\mathcal{P}}_s{\left(w\right)\parallel }_{H^{{\nu }^{\prime }}\left([0,1]\right)}\le c{(m-1)}^{2{\nu }^{\prime }-{\displaystyle \frac{1}{2}}-\nu }{\left(2^{s-1}\right)}^{{\nu }^{\prime }-\nu }{\parallel w^{\left(\nu \right)}\parallel }_2.$$

(18)

where c is a constant. Here $H^{\nu }\left([0,1]\right)$ expresses the Sobolev space that is equipped with norm

$${\parallel w\parallel }_{H^{\nu }\left([0,1]\right)}={\left(\sum _{i=0}^{\nu }{\parallel w^{\left(i\right)}\parallel }_2^2\right)}^{1/2}.$$

(19)

2.1. Operational Matrix of Fractional Integration

In this subsection, we will try to find a matrix for representing the Riemann–Liouville (RL) fractional integration (FI). On the other hand, finding the elements of matrix $I_{\upsilon }$, called the operational matrix of fractional integral, is the main subject of this subsection. Strictly speaking, to specify the elements of matrix $I_{\upsilon }$, all elements of matrix $\mathsf{\Phi }\left(t\right)$ must be approximated by ML wavelets, that is,

$${\mathcal{P}}_s\left({\mathcal{I}}_0^{\upsilon }\right)\left(\mathsf{\Phi }\left(t\right)\right)\approx I_{\upsilon }\left(t\right)\mathsf{\Phi }\left(t\right),$$

(20)

in which ${\mathcal{I}}_0^{\upsilon }$ indicates the FI operator

$${\mathcal{I}}_0^{\upsilon }\left(w\right)\left(t\right):={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\upsilon \right)}}{\int }_0^t{(t-z)}^{\upsilon -1}w\left(z\right)dz,0\le t\le 1,\upsilon \in {\mathbb{R}}^{+},$$

(21)

where $\mathsf{\Gamma }\left(\upsilon \right)$ characterizes the Gamma function.

To obtain the elements of matrix $I_{\upsilon }$, it is helpful that the piecewise fractional-order Taylor functions $p_{s,\tau }^n\left(t\right)$ are utilized instead of ML wavelets ${\varphi }_{s,\tau }^n\left(t\right)$. We can demonstrate a closed relationship between the ML wavelet and piecewise fractional-order Taylor functions as follows:

$$\mathsf{\Phi }\left(t\right)={\mathsf{{\rm Y}}}^{-1}P\left(t\right),$$

(22)

in which the elements of the square matrix $\mathsf{{\rm Y}}$ are calculated by

$${\left[\mathsf{{\rm Y}}\right]}_{i,j}=⟨{\mathsf{\Phi }}_i\left(t\right),P_j\left(t\right)⟩={\int }_0^1{P}_j\left(t\right){\mathsf{\Phi }}_i\left(t\right)dt,i,j=1,\dots ,N,N=2^{s}m.$$

(23)

Here, the $(\tau m+n+1)$-th element of the vector function $P\left(t\right)$ is introduced by

$$p_{s,\tau }^l=\left\{\begin{array}{cc}\hfill t^{{\upsilon }_n},\hfill & \hfill {\displaystyle \frac{\tau }{2^s}}\le t\le {\displaystyle \frac{\tau +1}{2^s}},\hfill \\ \hfill 0,\hfill & \hfill otherwise,\hfill \end{array}\right.\tau \in \mathcal{A},n\in \mathcal{R},s\in {\mathbb{N}}_0.$$

(24)

Assuming that U is an m-dimensional vector whose entries are ${\left\{t^{{\upsilon }_n}\right\}}_{n=1}^m$, it is not a challenging task to verify that

$$P\left(t\right)={\left[U,\dots ,U\right]}^T.$$

(25)

Motivated by [32], it has been demonstrated that the fractional integral of a power function results in a power function of the same structure, albeit with modified coefficients introduced by a certain factor, that is,

$${\mathcal{I}}_0^{\upsilon }\left(t^{\alpha }\right)={\displaystyle \frac{\mathsf{\Gamma }(\alpha +1)}{\mathsf{\Gamma }(\alpha +\upsilon +1)}}{t}^{\alpha +\upsilon }.$$

(26)

As a result, it follows from (24) and (26) that

$${\mathcal{I}}_0^{\upsilon }\left(P_i\right)\left(t\right)={\displaystyle \frac{\mathsf{\Gamma }({\upsilon }_i+1)}{\mathsf{\Gamma }({\upsilon }_i+\upsilon +1)}}{t}^{{\upsilon }_i+\upsilon },i=1,2,\dots ,N.$$

(27)

Equation (27) leads to introducing the matrix $I_{P,\upsilon }\left(t\right)$ that satisfies

$${\mathcal{I}}_0^{\upsilon }\left(P\right)\left(t\right)=I_{P,\upsilon }\left(t\right)P\left(t\right).$$

(28)

Assuming the diagonal matrix K

$${\left[K\right]}_{i,i}=\mathsf{\Gamma }({\upsilon }_i+1)/\mathsf{\Gamma }({\upsilon }_i+\upsilon +1),i=1,2,\dots ,N,$$

(29)

the matrix $I_{P,\upsilon }\left(t\right)$ can be specified by

$$I_{P,\upsilon }\left(t\right)=\left(\begin{array}{ccc}{B}_{\upsilon }\left(t\right)& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & B_{\upsilon }\left(t\right)\end{array}\right)$$

(30)

in which $B_{\upsilon }\left(t\right):=t^{\upsilon }K$, $({\mathcal{I}}_0^{\upsilon }\left(U\right)\left(t\right)=B_{\upsilon }\left(t\right)U\left(t\right))$.

Now, we can specify the FI operational matrix $I_{\upsilon }\left(t\right)$ as follows.

$$\begin{array}{cc}\hfill {\mathcal{P}}_s\left({\mathcal{I}}_0^{\upsilon }\right)\left(\mathsf{\Phi }\left(t\right)\right)& \hfill ={\mathcal{P}}_s\left({\mathcal{I}}_0^{\upsilon }\right)\left({\mathsf{{\rm Y}}}^{-1}P\left(t\right)\right)\hfill \\ \hfill & \hfill ={\mathsf{{\rm Y}}}^{-1}{I}_{P,\upsilon }\left(t\right)P\left(t\right)\hfill \\ \hfill & \hfill ={\mathsf{{\rm Y}}}^{-1}{I}_{P,\upsilon }\left(t\right)\mathsf{{\rm Y}}\mathsf{\Phi }\left(t\right).\hfill \end{array}$$

(31)

Therefore, we have

$$I_{\upsilon }\left(t\right):={\mathsf{{\rm Y}}}^{-1}{I}_{P,\upsilon }\left(t\right)\mathsf{{\rm Y}}.$$

(32)

2.2. Matrix Representation of Caputo Fractional Derivative (CFD) Operator

Recall that the CFD ${ }^c{\mathcal{D}}_0^{\upsilon }$ of $w\left(t\right)$ is characterized by [32]

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\right)\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(\kappa -\upsilon )}}{\int }_0^t{\displaystyle \frac{w^{\left(\kappa \right)}\left(x\right)dx}{{(t-x)}^{\upsilon -\kappa +1}}}=:{\mathcal{I}}_0^{\kappa -\upsilon }{\mathcal{D}}^{\kappa }\left(w\right)\left(t\right),\upsilon \in {\mathbb{R}}^{+},$$

(33)

in which $\kappa =-[-\upsilon ]$. This integral exists for almost every $t\in [0,1]$. Given this definition, it is helpful to verify that

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(x^{\alpha -1}\right)\left(t\right)={\displaystyle \frac{\mathsf{\Gamma }\left(\alpha \right)}{\mathsf{\Gamma }(\alpha -\upsilon )}}{t}^{\alpha -\upsilon },(\alpha >\kappa ).$$

(34)

As we mentioned, the object of this subsection is to introduce the square matrix $D_{\upsilon }$ that represents CFD, that is,

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(\mathsf{\Phi }\left(t\right)\right)\approx D_{\upsilon }\mathsf{\Phi }\left(t\right).$$

(35)

To find the elements of $D_{\upsilon }$, unlike the direct method proposed for the fractional integral operator, one can use the relation between the CFD operator and the fractional integral operator (33). To achieve the desired result, namely, introducing the matrix $D_{\upsilon }$, we have the following process:

$$\begin{array}{cc}\hfill { }^c{\mathcal{D}}_0^{\upsilon }\left(\mathsf{\Phi }\left(t\right)\right)={\mathcal{I}}_0^{\kappa -\upsilon }{\mathcal{D}}^{\kappa }\left(\mathsf{\Phi }\left(t\right)\right)& \approx {\mathcal{I}}_0^{\kappa -\upsilon }\left(D^{\kappa }\mathsf{\Phi }\left(t\right)\right)\hfill \\ \hfill & \hfill =D^{\kappa }{\mathcal{I}}_0^{\kappa -\upsilon }\left(\mathsf{\Phi }\left(t\right)\right)\approx D^{\kappa }{I}_{\kappa -\upsilon }\left(\mathsf{\Phi }\left(t\right)\right),\hfill \end{array}$$

where the expression D refers to the operational matrix of the derivative introduced in reference [58]. Thus, the operational matrix $D_{\upsilon }$ can be specified by

$$D_{\upsilon }:=D^{\kappa }{I}_{\kappa -\upsilon }.$$

(36)

Consequently, employing spectral methods, there is no necessity to determine the CFD of the bases; instead, the matrix $D_{\upsilon }$ can be utilized.

3. Outline of Method

Recall the fractional Bratu equation

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\left(t\right)\right)+\rho e^{w\left(t\right)}=0,t\in [0,1],1<\upsilon \le 2,$$

(37)

with initial conditions

$$w\left(0\right)=a,w^{\prime }\left(0\right)=b,$$

(38)

where $\rho$, a, and b are constant. The outline of the collocation method involves the selection of a finite-dimensional space comprising potential solutions and a designated set of points within the domain, referred to as collocation points. The objective is to identify the solution that satisfies the specified equation at these collocation points.

In this paper, our objective is to introduce and implement two methodologies for addressing the fractional Bratu equation. The specifics of these approaches will be expounded upon as follows.

• First approach
  We seek the approximation $w_N\in V_s$ of the solution of Equation (37), which can be obtained as

  $$w\left(t\right)\approx w_N\left(t\right):={\mathcal{P}}_s\left(w\right)\left(t\right)=\sum _{\tau =0}^{2^s-1}\sum _{n=0}^{m-1}{w}_{\tau ,n}{\varphi }_{s,\tau }^n\left(t\right)=W^T\mathsf{\Phi }\left(t\right)\in V_s,$$

  (39)

  where $W\in {\mathbb{R}}^N$, the elements of which must be determined. To implement the collocation method, the residual $r_N$ is introduced as

  $$\begin{array}{cc}\hfill r_N\left(t\right)& \hfill ={ }^c{\mathcal{D}}_0^{\upsilon }\left(w_N\left(t\right)\right)+\rho e^{w_N\left(t\right)},\hfill \\ \hfill & \hfill \approx W^T{D}_{\upsilon }\mathsf{\Phi }\left(t\right)+\rho e^{W^T\mathsf{\Phi }\left(t\right)}.\hfill \end{array}$$

  (40)
• Second approach
  In this approach, we approximate the fractional derivative of the unknown solution $w\left(t\right)$ as

  $${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\left(t\right)\right)\approx w_{\upsilon ,N}\left(t\right):=\sum _{\tau =0}^{2^s-1}\sum _{n=0}^{m-1}{q}_{\tau ,n}{\varphi }_{s,\tau }^n\left(t\right)=Q^T\mathsf{\Phi }\left(t\right)\in V_s,$$

  (41)

  in which $Q\in {\mathbb{R}}^N$, whose elements are unknown.
  According to Lemma 2.22 in [32], we have

  $$\left({\mathcal{I}}_0^{\upsilon }{ }^c{\mathcal{D}}_0^{\upsilon }\right)\left(w\right)\left(t\right)=w\left(t\right)-\sum _{i=0}^{\kappa -1}{\displaystyle \frac{w^{\left(i\right)}\left(0\right)}{i!}}{t}^i,w\in C^{\kappa }[0,1].$$

  (42)
  So we can determine $w\left(t\right)$, that is,

  $$w\left(t\right)\approx w_N\left(t\right):=Q^T{I}_{\upsilon }\mathsf{\Phi }\left(t\right)+w_0\left(t\right),$$

  (43)

  where $w_0:={\sum }_{i=0}^{\kappa -1}{\displaystyle \frac{w^{\left(i\right)}\left(0\right)}{i!}}{t}^i$. Substituting (41) and (43) into (37) gives rise to specifying the residual

  $$\begin{array}{cc}\hfill r_N\left(t\right)& \hfill =w_{\upsilon ,N}\left(t\right)+\rho e^{w_N\left(t\right)},\hfill \\ \hfill & \hfill \approx Q^T\mathsf{\Phi }\left(t\right)+\rho e^{Q^T{I}_{\upsilon }\mathsf{\Phi }\left(t\right)+w_0\left(t\right)}.\hfill \end{array}$$

  (44)

The goal and expectation are to obtain the expansion coefficients $\{w_{\tau ,n},\tau =0,\dots ,2^s-1,n=0,\dots ,m-1\}$ by forcing $r_N\left(t\right)$ to be approximately zero. To this end, picking the distinct collocation points $t_1,\dots ,t_N\in [0,1]$, the collocation scheme requires

$$r_N\left(t_j\right)=0,j=1,\dots ,N.$$

(45)

This leads to specifying the unknown coefficients as the solution of the nonlinear system

$$\mathcal{F}\left(W\right):=\left\{\begin{array}{ll}{ }^c{\mathcal{D}}_0^{\upsilon }\left(w_N\left(t_j\right)\right)+\rho e^{w_N\left(t_j\right)}=0,& \mathrm{first} \mathrm{approach},\\ w_{\upsilon ,N}\left(t\right)+\rho e^{w_N\left(t\right)}=0,& \mathrm{second} \mathrm{approach},\\ j=1,\dots ,N.\end{array}\right.$$

(46)

We use Newton’s method to solve this system. It is worth mentioning that Newton’s method is implemented with starting point $W_0=O$ (null vector) and the termination criterion is selected to be absolute residual, i.e.,

$$\parallel \mathcal{F}\left(W_i\right)\parallel \le {10}^{-16},i\ge 1.$$

An immediate question is how the collocation points are chosen. To answer this question, three options have been considered: the roots of Chebyshev or Lagrange polynomials and evenly spaced nodes.

4. Numerical Results

By providing some numerical simulations, we can showcase the effectiveness of the present method. These examples will demonstrate how the method can provide practical solutions to various problems. To provide an overview of method efficiency, tables, and figures, we report the absolute error

$$e_N=|w\left(t_i\right)-w_N\left(t_i\right)|,$$

and $L^2$ error

$$L^{2}error={\left({\int }_0^1{|w\left(t\right)-w_N\left(t\right)|}^2\right)}^{1/2}.$$

All examples are carried out with the combined use of Maple (version 2022) and Matlab software (version R2022a) with an Intel(R) Core(TM) i7-7700k CPU 4.20 GHz (RAM 32 GB). To have higher precision, we increase precision beyond 50 digits.

Example 1. 

Let us consider the following fractional Bratu equation [42]

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\left(t\right)\right)-2e^{w\left(t\right)}=0,t\in [0,1],1<\upsilon \le 2,$$

with initial conditions

$$w\left(0\right)=0,w^{\prime }\left(0\right)=0.$$

For this equation, the exact solution is $w\left(t\right)=-2ln(cos(t\left)\right)$, for $\alpha =2$ [31,42]. This equation has been chosen deliberately so that we can more easily assess the accuracy of our approximations.

Table 1 and Table 2 are reported to demonstrate the $L^2$ error and CPU time for the first and second approaches, respectively. These quantities have been obtained by applying the presented schemes for different values of N when the upsilon is 2. The numerical approximation with the associated $L^2$ error is plotted in Figure 1, where ${log}_{10}$ of the $L^2$ error is illustrated instead of the $L^2$ error. We observe that both approaches are convergent, and by increasing N, the $L^2$ error approaches zero. The accuracy of the methods is also compared with existing methods reported in [42,59] (Table 3). The results demonstrate that the presented schemes yield better results than others. To demonstrate the efficiency of the presented schemes for non-integer values of υ, Figure 2 is plotted. Note that [32]

$$\begin{array}{c}\hfill \underset{\upsilon \to \kappa }{lim}{ }^c{D}_0^{\upsilon }w\left(t\right)=w^{\left(\kappa \right)}\left(t\right),\hfill \\ \\ \hfill \underset{\upsilon \to \kappa -1}{lim}{ }^c{D}_0^{\upsilon }w\left(t\right)=w^{(\kappa -1)}\left(t\right)-w^{(\kappa -1)}\left(0\right).\hfill \end{array}$$

Our results confirm this, and we can verify that when $\upsilon \to \kappa$, the estimated solutions with increasing υ tend to the results for κ. To show the ability and accuracy of the presented approaches, we report the residual errors

$${\parallel r\left(t\right)\parallel }_2={\left({\int }_0^1{\left|r\left(t\right)\right|}^{2}dt\right)}^{1/2},$$

(47)

in Table 4.

Table 1. The $L^2$ error obtained from the first approach for different N, taking $\upsilon =2$ (Example 1).

	N	12	14	16	18	20
Chebyshev nodes	$L^2$-error	$2.37\times {10}^{-6}$	$3.11\times {10}^{-7}$	$1.29\times {10}^{-8}$	$1.56\times {10}^{-9}$	$5.94\times {10}^{-11}$
	$\mathrm{CPU} \mathrm{time}$	$0.172$	$0.235$	$0.319$	$0.488$	$0.569$
Legendre nodes	$L^2$-error	$1.42\times {10}^{-6}$	$1.03\times {10}^{-7}$	$7.21\times {10}^{-9}$	$5.00\times {10}^{-10}$	$3.42\times {10}^{-11}$
	$\mathrm{CPU} \mathrm{time}$	$0.128$	$0.250$	$0.328$	$0.375$	$0.563$
Uniform meshes	$L^2$-error	$2.78\times {10}^{-5}$	$8.74\times {10}^{-6}$	$1.30\times {10}^{-6}$	$1.89\times {10}^{-7}$	$2.72\times {10}^{-8}$
	$\mathrm{CPU} \mathrm{time}$	$0.159$	$0.172$	$0.375$	$0.313$	$0.485$

Table 2. The $L^2$ error obtained from the second approach for different N, taking $\upsilon =2$ (Example 1).

	N	8	10	12	14	16
Chebyshev nodes	$L^2$-error	$2.34\times {10}^{-6}$	$1.10\times {10}^{-7}$	$5.51\times {10}^{-9}$	$2.86\times {10}^{-10}$	$1.52\times {10}^{-11}$
	$\mathrm{CPU} \mathrm{time}$	$0.094$	$0.110$	$0.281$	$0.313$	$0.391$
Legendre nodes	$L^2$-error	$6.95\times {10}^{-7}$	$3.40\times {10}^{-8}$	$1.73\times {10}^{-9}$	$9.11\times {10}^{-11}$	$4.90\times {10}^{-12}$
	$\mathrm{CPU} \mathrm{time}$	$0.094$	$0.171$	$0.250$	$0.344$	$0.344$
Uniform meshes	$L^2$-error	$1.74\times {10}^{-4}$	$2.38\times {10}^{-5}$	$3.20\times {10}^{-6}$	$4.28\times {10}^{-7}$	$5.69\times {10}^{-8}$
	$\mathrm{CPU} \mathrm{time}$	$0.157$	$0.204$	$0.282$	$0.297$	$0.453$

Figure 1. The $L^2$ error obtained from the first approach (left) and the second approach (right) for different choices of N (Example 1).

Table 3. Numerical results obtained from different methods and their comparison for Example 1.

	Proposed Method (N = 8)
t	First Approach	Second Approach	[42] (N = 8)	[59] ($\mathit{N}=\mathbf{8}$)
$0.1$	$2.30\times {10}^{-7}$	$1.21\times {10}^{-7}$	$1.43\times {10}^{-5}$	$1.41\times {10}^{-5}$
$0.2$	$1.60\times {10}^{-7}$	$3.18\times {10}^{-7}$	$3.25\times {10}^{-5}$	$2.94\times {10}^{-5}$
$0.3$	$3.25\times {10}^{-6}$	$8.38\times {10}^{-7}$	$5.14\times {10}^{-5}$	$1.79\times {10}^{-5}$
$0.4$	$1.21\times {10}^{-5}$	$1.26\times {10}^{-8}$	$7.20\times {10}^{-5}$	$1.20\times {10}^{-4}$
$0.5$	$7.09\times {10}^{-5}$	$1.61\times {10}^{-6}$	$9.34\times {10}^{-5}$	$6.59\times {10}^{-4}$
$0.6$	$1.67\times {10}^{-4}$	$2.92\times {10}^{-7}$	$1.18\times {10}^{-4}$	$2.21\times {10}^{-3}$
$0.7$	$2.66\times {10}^{-4}$	$1.20\times {10}^{-6}$	$1.46\times {10}^{-4}$	$6.01\times {10}^{-3}$
$0.8$	$3.56\times {10}^{-4}$	$8.43\times {10}^{-7}$	$1.78\times {10}^{-4}$	$1.43\times {10}^{-2}$
$0.9$	$4.83\times {10}^{-4}$	$1.23\times {10}^{-7}$	$2.19\times {10}^{-4}$	$3.13\times {10}^{-2}$
$1.0$	$5.53\times {10}^{-4}$	$7.89\times {10}^{-9}$	$2.71\times {10}^{-4}$	$6.43\times {10}^{-2}$

Figure 2. The approximate solution for different choices of $\upsilon$ (Example 1).

Table 4. The residual errors for the presented approaches to solve the fractional Bratu equation (Example 1).

			$\mathit{\upsilon }=1.8$	$\mathit{\upsilon }=1.9$	$\mathit{\upsilon }=1.95$
First approach	Chebyshev nodes	$N=18$	$8.88\times {10}^{-4}$	$6.30\times {10}^{-4}$	$5.28\times {10}^{-4}$
		$N=20$	$4.02\times {10}^{-4}$	$3.51\times {10}^{-4}$	$2.17\times {10}^{-4}$
	Legendre nodes	$N=18$	$7.91\times {10}^{-4}$	$4.18\times {10}^{-4}$	$3.65\times {10}^{-4}$
		$N=20$	$3.02\times {10}^{-4}$	$1.87\times {10}^{-4}$	$1.04\times {10}^{-4}$
	Uniform meshes	$N=18$	$7.10\times {10}^{-3}$	$5.98\times {10}^{-3}$	$5.24\times {10}^{-3}$
		$N=20$	$4.45\times {10}^{-3}$	$2.61\times {10}^{-3}$	$1.62\times {10}^{-3}$
Second approach	Chebyshev nodes	$N=18$	$1.13\times {10}^{-6}$	$3.54\times {10}^{-7}$	$1.39\times {10}^{-7}$
		$N=20$	$6.87\times {10}^{-7}$	$2.12\times {10}^{-7}$	$8.24\times {10}^{-8}$
	Legendre nodes	$N=18$	$1.28\times {10}^{-6}$	$3.99\times {10}^{-7}$	$1.56\times {10}^{-7}$
		$N=20$	$7.87\times {10}^{-7}$	$2.42\times {10}^{-7}$	$9.36\times {10}^{-8}$
	Uniform meshes	$N=18$	$2.66\times {10}^{-5}$	$7.43\times {10}^{-6}$	$3.02\times {10}^{-6}$
		$N=20$	$1.57\times {10}^{-5}$	$5.01\times {10}^{-6}$	$2.03\times {10}^{-6}$

Example 2. 

For the second example, we apply the presented method to solve the Bratu equation

$${ }^c{\mathcal{D}}_0^{\upsilon }\left(w\left(t\right)\right)-e^{2w\left(t\right)}=0,t\in [0,1],1<\upsilon \le 2,$$

with initial conditions

$$w\left(0\right)=0,w^{\prime }\left(0\right)=0.$$

For this equation, the exact solution is reported in [31,42], when $\alpha =2$, as follows

$$w\left(t\right)=ln(sec(t\left)\right).$$

We report the L error in Table 5 and Table 6, and Figure 3 to verify that the presented schemes are convergent. Observations confirm results similar to Example 1. A comparison with other methods is also reported for this example in Table 7. The superiority of the presented schemes over other existing methods is also evident in this example. For this example, we can also verify that when $\upsilon \to \kappa$, the estimated solutions with increasing υ tend to the results for κ. To this end, we plotted Figure 4. Table 8 is tabulated to show the residual errors (47) for this example.

Table 5. The $L^2$ error obtained from the first approach for different N, taking $\upsilon =2$ (Example 2).

	N	10	12	14	16	18
Chebyshev nodes	$L^2$-error	$1.83\times {10}^{-5}$	$1.19\times {10}^{-6}$	$1.55\times {10}^{-7}$	$6.47\times {10}^{-9}$	$7.81\times {10}^{-10}$
	$\mathrm{CPU} \mathrm{time}$	$0.078$	$0.250$	$0.313$	$0.428$	$0.500$
Legendre nodes	$L^2$-error	$9.68\times {10}^{-6}$	$7.11\times {10}^{-7}$	$5.13\times {10}^{-8}$	$3.60\times {10}^{-9}$	$2.50\times {10}^{-10}$
	$\mathrm{CPU} \mathrm{time}$	$0.157$	$0.297$	$0.356$	$0.399$	$0.401$
Uniform meshes	$L^2$-error	$1.86\times {10}^{-4}$	$2.89\times {10}^{-5}$	$4.37\times {10}^{-6}$	$6.48\times {10}^{-7}$	$9.45\times {10}^{-8}$
	$\mathrm{CPU} \mathrm{time}$	$0.156$	$0.266$	$0.281$	$0.422$	$0.438$

Table 6. The $L^2$ error obtained from the second approach for different N, taking $\upsilon =2$ (Example 2).

	N	12	13	14	15	16
Chebyshev nodes	$L^2$-error	$2.76\times {10}^{-9}$	$6.24\times {10}^{-10}$	$1.43\times {10}^{-10}$	$3.28\times {10}^{-11}$	$7.60\times {10}^{-12}$
	$\mathrm{CPU} \mathrm{time}$	$0.219$	$0.250$	$0.313$	$0.360$	$0.453$
Legendre nodes	$L^2$-error	$8.67\times {10}^{-10}$	$1.98\times {10}^{-10}$	$4.56\times {10}^{-11}$	$1.05\times {10}^{-11}$	$2.45\times {10}^{-12}$
	$\mathrm{CPU} \mathrm{time}$	$0.102$	$0.171$	$0.328$	$0.403$	$0.741$
Uniform meshes	$L^2$-error	$1.60\times {10}^{-6}$	$5.85\times {10}^{-7}$	$2.14\times {10}^{-7}$	$7.80\times {10}^{-8}$	$2.85\times {10}^{-8}$
	$\mathrm{CPU} \mathrm{time}$	$0.218$	$0.254$	$0.391$	$0.428$	$0.875$

Figure 3. The $L^2$ error obtained from the first approach (left) and the second approach (right) for different choices of N (Example 2).

Table 7. Numerical results obtained from different methods and their comparison for Example 2.

	Proposed Method (N = 8)
t	First Approach	Second Approach	[42] (N = 8)	[60] (N = 8)
$0.1$	$1.65\times {10}^{-7}$	$6.06\times {10}^{-8}$	$7.14\times {10}^{-6}$	$1.40\times {10}^{-5}$
$0.3$	$1.63\times {10}^{-6}$	$4.19\times {10}^{-7}$	$2.57\times {10}^{-5}$	$2.63\times {10}^{-5}$
$0.5$	$3.55\times {10}^{-5}$	$5.80\times {10}^{-7}$	$4.67\times {10}^{-5}$	$1.08\times {10}^{-4}$
$0.7$	$1.33\times {10}^{-4}$	$6.01\times {10}^{-7}$	$7.29\times {10}^{-5}$	$2.51\times {10}^{-4}$
$0.9$	$2.42\times {10}^{-4}$	$6.15\times {10}^{-8}$	$1.10\times {10}^{-4}$	$3.98\times {10}^{-4}$

Figure 4. The approximate solution for different choices of $\upsilon$ (Example 2).

Table 8. The residual errors for the presented approaches to solve the fractional Bratu equation (Example 2).

			$\mathit{\upsilon }=1.8$	$\mathit{\upsilon }=1.9$	$\mathit{\upsilon }=1.95$
First approach	Chebyshev nodes	$N=18$	$9.11\times {10}^{-4}$	$7.23\times {10}^{-4}$	$5.98\times {10}^{-4}$
		$N=20$	$6.10\times {10}^{-4}$	$5.63\times {10}^{-4}$	$1.82\times {10}^{-4}$
	Legendre nodes	$N=18$	$8.56\times {10}^{-4}$	$5.41\times {10}^{-4}$	$4.42\times {10}^{-4}$
		$N=20$	$6.74\times {10}^{-4}$	$4.11\times {10}^{-4}$	$2.94\times {10}^{-4}$
	Uniform meshes	$N=18$	$3.12\times {10}^{-2}$	$1.37\times {10}^{-2}$	$8.19\times {10}^{-3}$
		$N=20$	$9.68\times {10}^{-3}$	$6.43\times {10}^{-3}$	$5.06\times {10}^{-3}$
Second approach	Chebyshev nodes	$N=18$	$5.68\times {10}^{-7}$	$1.77\times {10}^{-7}$	$6.96\times {10}^{-8}$
		$N=20$	$3.43\times {10}^{-7}$	$1.06\times {10}^{-7}$	$4.12\times {10}^{-8}$
	Legendre nodes	$N=18$	$6.41\times {10}^{-7}$	$3.00\times {10}^{-7}$	$7.82\times {10}^{-8}$
		$N=20$	$3.94\times {10}^{-7}$	$1.21\times {10}^{-7}$	$4.68\times {10}^{-8}$
	Uniform meshes	$N=18$	$1.33\times {10}^{-5}$	$3.71\times {10}^{-6}$	$1.51\times {10}^{-6}$
		$N=20$	$7.84\times {10}^{-6}$	$2.51\times {10}^{-6}$	$1.01\times {10}^{-6}$

5. Conclusions

This paper is devoted to solving the well-known Bratu equation with Caputo fractional derivative. Two different approaches based on the collocation method using Müntz–Legendre wavelets with varying choices of collocation points are presented in this study. The results verify that the proposed schemes for solving this problem are efficient and give results more accurate than others.

According to the experimental observations, the following can be concluded:

• The proposed schemes are effective in solving these types of equations.
• Our presented approaches yield better results compared to other existing methods.
• The second approach is better than the first one, due to the obtained results in Tables and Figures.
• The best choices of collocation points for this equation are the Legendre nodes.
• Both approaches are convergent. This is independent of choosing the Chebyshev, Legendre nodes, and uniform meshes as the collocation points.