Solving General Fractional Lane-Emden-Fowler Differential Equations Using Haar Wavelet Collocation Method

: This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefﬁcients using Newton’s method. We have constructed the higher-order Lane-Emden-Fowler equations. We have also discussed the convergence rate and stability analysis of our technique. We have explained the applications and numerically simulated the examples graphically and in tabular format to elaborate on the accuracy and efﬁciency of this approach.


Introduction
In the past few decades, the study of singular initial value problems has attracted the attention of many physicists and mathematicians.Lane-Emden-Fowler equations are these types of equations.Lane [1] introduced the equation in 1870, and Emden and Fowler [2] generalized the equation further.Lane explained the gravitation potential of a spherically symmetric Newtonian self-gravitating star [3] using these equations.These equations have a vast amount of applications in modeling many problems in physics and dynamics.The general Lane-Emden-Fowler equation is as follows: where η > 0 is a positive real number called a shape factor, D = d dξ is the differential operator, ω is linear or nonlinear function, and β + γ gives the order of the equation.These equations are used in magnetic field models [4], classical and quantum mechanics [5], biological systems, geometry [6], and fluid mechanics problems [7,8].
Several forms of fractional initial value problems have been proposed in standard models, and there has been a significant interest in developing numerical schemes for their solutions.Fractional calculus is a generalized form of integer order calculus.There are many applications of fractional calculus.Fractional calculus has been deployed to model the oscillation of earthquakes [9], neural networks [10,11], signal processing [12], economics [13], bioengineering [14], and electromagnetism [15].
Many researchers, such as Riemann, Liouville, Caputo, Hadamard, Grunwald, and others, have published extensively about the applications of fractional calculus.Mathematical models with fractional order derivatives provide more insight because they posses the memory effect.Many techniques have been developed to solve fractional problems, such as decomposition methods [16], collocation methods [17], residual power series methods [18,19], finite differences methods [20], perturbation methods, variational iteration methods [21].Zhang and Han [22] proposed a quasi wavelet method to solve time-dependent fractional partial differential equations.Jiang et al. [23] presented a predictor-corrector difference scheme for nonlinear fractional differential equations.Yang and Zhang [24] proposed a spectral sinc-collocation method for fourth-order heat models.
Wavelet theory has made distinguished contributions to mathematical studies.It is a powerful tool for engineering.Wavelets are used in signal processing, optimal control, and time-frequency analysis [25].There are many wavelets, such as Daubechies [26], Bspline [4], Legendre [27], and Haar [28].The Haar wavelet is an orthonormal wavelet with compact support, introduced by a Hungarian mathematician, Alfred Haar, in 1910.The Haar wavelet gives accurate results for small grid points.It contains members of the Daubechies family, so it is very good for computer implementations and is easily expressed in the programming language.Chen and Hsiao [29] derived a Haar operational matrix of the integrals of Haar functions.They made a great contribution to the use of Haar wavelets in applications of dynamic systems.Lepik [30] solved the differential equations using the Haar wavelet.Islam et al. [31] solved the integro-differential equations using the Haar wavelet.Bujurke et al. [32] compute the eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets.Chang et al. [33] describe the designation of Haar wavelet matrices in the numerical solution of ODEs.This article aims to solve general fractional Emden-Fowler-type equations using the Haar wavelet collocation method.We write the highest derivative in linear combinations of Haar functions and calculate other derivatives using the integration of Haar functions.This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using the Newton method.After calculating the Haar coefficients, we can easily determine the solution.
The present study is structured as follows: Section 2 defines the Haar wavelet and recalls the basic definitions of fractional calculus.In Section 3, we discuss the construction of the general equation of the Caputo-type fractional Lane-Emden-Fowler differential equation.In Section 4, we discuss the Haar wavelet method.In Section 5, we discuss the convergence rate, stability and error analysis of the technique.In Section 6, we discuss the examples and in Section 7 we discuss the numerical simulation of all of these examples graphically and in tabular format.In the end, we conclude our results.

Preliminaries
This section will recall some necessary definitions of fractional calculus and Haar wavelets.These definitions will assist us in the next sections.Definition 1.The Riemann-Liouville fractional integral operator I σ of order σ on L 2 [0, 1] is given by dκ is gamma function and Definition 2. The Caputo fractional derivative of order σ is given by provided the integral exists, where n is the smallest integer such that n − 1 < σ ≤ n.It satisfies the following properties:

Haar Wavelet and Function Approximations
The family of Haar wavelets consists of piecewise constant functions over the real line.They contain only values −1, 0, 1.They are discontinuous, and therefore not differentiable. where We manipulate the wavelet by translating and dilating it.α here represents the level of wavelet or dilation parameter level and ς represents the translation parameter.J is the maximum level of resolution and the relationship between 2 α and r is j = In particular, the Haar wavelet is an orthogonal square wave family, generally written as , then the function is approximated using Haar functions, such as where λ τ are the Haar coefficients.
The generalized fractional integration can be calculated analytically as , where σ is a positive real number.

Construction of Lane-Emden-Fowler Equation
Consider the general form of the Lane-Emden-Fowler equation ( We can obtain higher-order equations of fourth-, fifth-and sixth-order by taking respectively.There are three possible choices of fourth-order equations, four possibilities for fifth-order equations and five possible choices for sixth-order Lane-Emden-Fowler equations.
Clearly, when β = 1 , γ = 1 and η = 2, we have the Equation ( 63) in example 7.In a similar fashion, we can extract all the examples by applying different values of β, γ, η.Now, we discuss the method.

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Step 1: We approximate the highest-order derivative using Haar functions: • Step 2: We integrate the Equation (41) again and again, and after applying the initial conditions, we get and so on.The last term is • Step 3: We collocate the points as Now, we substitute all the values of derivatives into Equation ( 40) and collocate the points, resulting in the system of differential equations as follows • Step 4: We solve the system of equations (44) using the Newton method and obtain the values of the Haar coefficients λ τ , τ = 1, 2, . . .2L.

•
Step 5: After substituting the values of Haar coefficients into Equation (43), we obtain the numerical solution of Equation ( 40).
Proof.The proof is straightforward.We can refer to [34].

Numerical Error
The maximum absolute error is given by where q exact τ and q approx.τ are exact and approximate solutions at the τth collocation point.

Rate of Convergence
Rate of convergence is defined by where E C (L) represents the maximum absolute error at L collocation points.

Stability
The condition number is significant to measure the stability of an algorithm [35].For stability, the condition number should be bounded.We consider the system of equations formed in our algorithm as HA = Υ, where H denotes the Haar weights, A denotes the unknown Haar coefficients and Υ is a known vector.The condition number bound for some examples is given in Table 1.

Definition 3 ([36]
).Let us consider the system of equations to be of type HA = Υ, if the inverse of H exists and is bounded, then the algorithm is stable; that is, where Z is a constant.
The Condition number is bounded [35]; that is,

Applications
Example 1. Taking η = 2 in Equation (8) gives us this fourth-order fractional Lane-Emden-Fowler equation: with initial conditions q(0) = 1, q (σ) (0) = 0, q (2σ) (0) = 0, q (3σ) (0) = 0. Now, we apply the method, and using initial conditions, we can write After substituting all the values of q(ξ) and its derivatives into (45) and using collocation of points, we obtain a system of nonlinear equations as follows: We can easily solve this system (47) using the Newton method.After solving this system, we have the values of the Haar coefficients λ τ , τ = 1, 2, . . .2L and substituting these values into (46) gives the approximate solution.
Example 2. When we substitute η = 2 in Equation (11), we get this fourth-order fractional Lane-Emden-Fowler equation: with initial conditions q(0) = 0, q (σ) (0) = 0, q (2σ) (0) = 0, q (3σ) (0) = 0, where Now, we apply the method, and using the initial conditions, we can write After substituting all the values of q(ξ) and its derivatives into (48) and using collocation of points, we get a system of nonlinear equations as follows: We can easily solve this system (50) using the Newton method.After solving this system, we have the values of the Haar coefficients λ τ , τ = 1, 2, . . .2L, and these values in (49) give the approximate solution.
Example 5. Consider the fractional Lane-Emden Fowler equation, [38]: with initial conditions q(0) = 3, q (σ) (0) = 0, where The exact solution of (57) is q(ξ) = 3 + ξ 2σ .Now, we apply the method, and using the initial conditions, we can write After putting all the values of q(ξ) and its derivatives into (57), we get a system of nonlinear equations as follows: We can easily solve this system (59) using the Newton method.After solving this system, we have the values of Haar coefficients λ τ , τ = 1, 2, . . .2L and putting these values in (58) gives the approximate solution.
Example 7. Consider the fractional Lane-Emden-Fowler equation, which is given in [40] with initial conditions q(0) = 0, q (σ) (0) = 0.The exact solution of (63 . Now, we apply the method Integrating (64) and applying the initial condition gives us In the same way, we can write After substituting all the values of q(ξ) and its derivatives into (63) and after collocation of points, we get a system of nonlinear equations as follows: Using the Newton method, we can easily solve this system (66).After solving this system, we have the values of the Haar coefficients λ τ , τ = 1, 2, . . .2L and putting these values in (65) gives the approximate solution.

Numerical Simulation and Conclusions
This paper uses the Haar wavelet collocation method to find the numerical solution to the general-order fractional Lane-Emden-Fowler equation.This numerical scheme is presented in general order.We have documented many examples of second-, third-, fourth-, fifth-, and sixth-order fractional differential equations.We have numerically simulated those examples graphically and in tabular format.It is clear from the simulations that this method works very nicely.Figures 1 and 2 show the comparison of exact and numerical solutions and the absolute error of Example 1 for σ = 1 respectively.Figures 3 and 4 show the HWCM solution for different values of σ and Haar coefficients of Example 1 respectively.Figures 5 and 6 show the comparison of exact and numerical solutions and the absolute error of Example 2 for σ = 1 respectively.Figures 7 and 8 show the HWCM solution for different values of σ and Haar coefficients respectively.Figures 9 and 10 show the comparison of exact and numerical solutions and the absolute error of Example 3 for σ = 1 respectively.Figures 11 and 12 show the HWCM solution for different values of σ and Haar coefficients respectively.Similarly, Figures 13 and 14 show the comparison of exact and numerical solutions and the absolute error of Example 4 for σ = 1 respectively.Figures 15 and 16 show the HWCM solution for different values of σ and Haar coefficients respectively.Figures 17 and 18 show the comparison of exact and numerical solutions and the absolute error of Example 5 for σ = 1 respectively.Figures 19 and 20 show the HWCM solution for different values of σ and Haar coefficients respectively.Figures 21 and 22 show the comparison of exact and numerical solutions and the absolute error of Example 6 for σ = 1 respectively.Figures 23 and 24 show the HWCM solution for different values of σ and Haar coefficients respectively.Figures 25 and 26 show the comparison of exact and numerical solutions and the absolute error of Example 7 for σ = 1 respectively.Figures 27 and 28 show the HWCM solution for different values of σ and Haar coefficients respectively.Tables 2-15 give us a comparison of the numerical values of Haar and the exact solution and document the numerical values for different values of σ.From all the tables and graphs, we conclude that the method is quite accurate and gives us good outcomes.

Table 1 .
Condition number bound for Examples 1 and 3.

Table 2 .
Comparison of HWCM and exact solution of Example 1 for J = 5.

Table 3 .
Error Comparison of Example 1.

Table 4 .
Comparison of HWCM and exact solution of Example 2 for J = 3.

Table 6 .
Absolute error of HWCM of Example 2 for different σ.

Table 7 .
Comparison of HWCM and exact solution of Example 3 for J = 3.

Table 9 .
Absolute error of HWCM of Example 3 for different σ.Comparison of Exact and HWCM solution of Example 4 when σ = 1.Absolute error of HWCM of Example 4 when σ = 1.

Table 10 .
Comparison of HWCM and exact solution of Example 5 for J = 3.

Table 12 .
Comparison of HWCM and exact solution of Example 6 for J = 5.

Table 14 .
Comparison of HWCM and exact solution of Example 7 for J = 3.