Numerical Solution of Emden–Fowler Heat-Type Equations Using Backward Difference Scheme and Haar Wavelet Collocation Method

Abstract

In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized using Haar basis functions. A difference scheme is also applied to approximate the time derivative. By leveraging Haar functions and the difference scheme, we form a system of equations, which is then solved for Haar coefficients using MATLAB software. The effectiveness of this technique is demonstrated through various examples. Numerical simulations are performed, and the results are presented in graphical and tabular formats. We also provide a convergence analysis and an error analysis for this method. Furthermore, approximate solutions are compared with those obtained from other methods to highlight the accuracy, efficiency, and computational convenience of this technique.

1. Introduction and Mathematical Preliminaries

Partial differential equations (PDEs) are essential mathematical instruments employed to describe a multitude of phenomena in engineering [1], physics [2,3], and other applied sciences [4]. Unlike ordinary differential equations, which focus on functions of a single variable, PDEs are inherently more complex and capable of modeling a broad spectrum of physical processes. They are vital for representing systems where changes occur with respect to multiple independent variables, such as time and space.

The study and application of PDEs are pivotal for technological advancement and the comprehension of natural phenomena (see [5]). By developing efficient methods to solve these equations, researchers can simulate and predict the behavior of intricate systems, leading to breakthroughs in fields such as aerospace engineering [6], climate modeling [7], biomedical engineering [8,9], and more.

Analytically solving PDEs is often challenging due to their complexity. Consequently, numerical methods like finite element analysis [10], finite difference methods [11], and spectral methods [12] are frequently utilized. Temimi et al. [13] used an iterative finite difference algorithm to solve a two-dimensional Bratu problem numerically. They employed the Newton–Raphson–Kantorovich approximation in function space and designed an iterative finite difference approach that provided a straightforward algorithm for computing the sequence of numerical solutions. Darvishi et al. [14] used a modification of the HPM technique to solve sine-Gordon and coupled sine-Gordon equations numerically. Darvishi [15] also used domain decomposition schemes to solve partial differential equations. These techniques approximate the solutions of PDEs through computational algorithms, enabling the resolution of real-world problems that are otherwise analytically intractable.

The time-dependent Emden–Fowler equations are a class of partial differential equations that arise in various scientific and engineering contexts, particularly in astrophysics, fluid dynamics, and thermal analysis. Named after the astronomers Robert Emden [16] and Richard Fowler [17], these equations are extensions of the classical Emden–Fowler equations, incorporating time as a dynamic variable to model evolving systems.

The classical Emden–Fowler equations typically describe static, spherically symmetric stellar structures [18,19] and other similar phenomena where spatial variables alone dictate the system’s behavior. When extended to include time dependency, these equations become powerful tools for analyzing the temporal evolution of systems under various physical conditions. This extension is crucial for understanding dynamic processes such as the thermal behavior of stars, gas dynamics in astrophysical contexts, and transient heat conduction in materials. In our work, we consider the Emden–Fowler heat-type equation [20] in the form

$${\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}+{\displaystyle \frac{\epsilon }{\varpi }}{\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}+B(\varpi ,\psi )F\left(\mathsf{\Omega }\right)={\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \psi }},$$

(1)

$$\epsilon >0,0<\varpi <1,\psi >0,$$

where $B(\varpi ,\psi )F\left(\mathsf{\Omega }\right)$ is a nonlinear function or nonlinear heat source. This equation is solved subject to boundary conditions:

$$\mathsf{\Omega }(0,\psi )=f\left(\psi \right),\mathsf{\Omega }(1,\psi )=g\left(\psi \right),$$

and initial condition $\mathsf{\Omega }(\varpi ,0)=h\left(\varpi \right).$

The time-dependent version of this equation incorporates the effect of time, allowing it to model evolving systems where properties like density, temperature, or pressure change dynamically over time. In (1), $\mathsf{\Omega }(\varpi ,\psi )$ is the primary variable of interest, which could represent physical quantities like density, temperature, or gravitational potential. $\varpi$ is a spatial coordinate often representing a radial distance in problems with spherical symmetry and $\psi$ is a time coordinate indicating that $\mathsf{\Omega }$ evolves over time or a related temporal parameter. Physically, ${\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}$ is a measure of the concavity or curvature of $\mathsf{\Omega }$ in space and often appears in diffusion processes. ${\displaystyle \frac{1}{\varpi }}$ represents a type of “damping” or scaling effect in radial diffusion processes. This could represent heat spreading from a central source, mass diffusing in a spherically symmetric object, or energy dissipating in a star. The term $B(\varpi ,\psi )F\left(\mathsf{\Omega }\right)$ acts as a source or sink, adding or removing $\mathsf{\Omega }$ based on its current state and location. This term could represent reactions like chemical decay or fusion, heat generation in a stellar context.

A significant amount of research has been conducted to examine singular boundary value problems using various methods. Culham et al. [21] used the Galerkin method to solve time-dependent nonlinear boundary value problems. Netuzhylov et al. [22] used space–time meshfree collocation method (STMCM) to solve system of partial differential equations by a consistent discretization in both space and time. Dehgan [23] explored second-order finite difference methods for addressing a non-local boundary value problem in the two-dimensional diffusion equation with Neumann boundary conditions. Wazwaz et al. [24] used a modified Adomian decomposition method to solve time-dependent Emden–Fowler equations numerically with boundary conditions. Singh et al. [25] discussed the modified homotopy perturbation method for solving nonlinear and singular time-dependent Emden–Fowler type equations with the Neumann and Dirichlet boundary conditions. Mkhatshwa et al. [26] presented the bivariate spectral collocation method with the use of overlapping grids when applying the Chebyshev spectral collocation method. Bataineh et al. [27] solved time-dependent Emden–Fowler equations using the homotopy analysis method. Shahni [28] proposed the Berstein polynomial technique to numerically solve derivative-dependent Emden–Fowler equations with boundary conditions.

We are using a Haar wavelet collocation technique with backward difference scheme to solve Emden–Fowler heat-type equations with boundary conditions. The backward difference scheme [29] is a powerful numerical tool that offers stability, ease of implementation, and robustness, making it well suited for solving a variety of differential equations. The backward difference scheme can more easily handle complex boundary conditions compared to some explicit methods. This flexibility is important in accurately modeling physical systems with intricate boundary behaviors. Haar wavelet collocation method [30] offers a combination of simplicity, efficiency, adaptability, and accuracy, making it an attractive choice for solving differential equations in various scientific and engineering applications.

Combining the backward difference scheme’s stability with the spatial adaptivity of the Haar wavelet collocation method leads to a robust overall framework. This method tends to minimize numerical dispersion and oscillations that can occur in high-gradient regions. Thus, this method is robust for a wide range of PDE problems, particularly those with sharp features or requiring stability over long time simulations. It is well suited for problems with discontinuities or local features, as the Haar wavelet collocation can adapt to these with lower computational costs due to the sparsity of the resulting system.

Haar Wavelet

The Haar wavelet functions are simple and piecewise constant, making them easy to implement and compute. Haar wavelets are widely used in computer vision and image processing applications [31], including face detection [32] and pattern recognition. The Haar wavelet function is defined as:

$$h_{\rho }\left(\varpi \right)=\left\{\begin{array}{l}1, {\kappa }_1\le \varpi <{\kappa }_2,\\ -1, {\kappa }_2\le \varpi <{\kappa }_3,\\ 0, otherwise.\end{array}\right.$$

(2)

where ${\kappa }_1=\alpha +(\gamma -\alpha ){\displaystyle \frac{r}{p}}$, ${\kappa }_2=\alpha +(\gamma -\alpha ){\displaystyle \frac{2r+1}{2p}}$, and ${\kappa }_3=\alpha +(\gamma -\alpha ){\displaystyle \frac{r+1}{p}}$. We define $p=2^s,s=0,1,2,\dots J$, and $r=0,1,2,\dots p-1,$ where J is the maximum resolution, s represents a dilation parameter, and $\rho =p+r+1.$ Specifically, the Haar wavelet constitutes a square wave family that is orthogonal, generally written as

$$h_{\rho }\left(\varpi \right)=h_2\left(p\varpi -{\displaystyle \frac{r}{p}}\right).$$

(3)

for $s\ge 3,l=p+r+1,l\ge 0,0\le r\le 2^s-1$ and

$${\displaystyle \underset{0}{\overset{1}{\int }}}{h}_{\rho }\left(\varpi \right)h_j\left(\varpi \right)d\varpi =\left\{\begin{array}{l}{2}^{-s}; \rho =s,\\ 0; s\ne j.\end{array}\right.$$

(4)

For resolution, $J=2$, the Haar function matrix is given by

$$H_{8\times 8}=\left[\begin{array}{cccccccc}{h}_1& h_2& h_3& h_4& h_5& h_6& h_7& h_8\end{array}\right].$$

We define the collocation points where we approximate any function as ${\varpi }_l={\displaystyle \frac{l-0.5}{2p}}$, for $l=1,2,\dots ,2p$.

After putting values

$$H_{8\times 8}=\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& -1& -1& -1& -1\\ 1& 1& -1& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 1& -1& -1\\ 1& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& -1\end{array}\right].$$

The integrals of wavelets are defined as

$$q_{\rho ,m}\left(\varpi \right)=\left\{\begin{array}{c}0,\varpi \in [0,{\epsilon }_1)\\ {\displaystyle \frac{1}{m!}}{(\varpi -{\kappa }_1)}^m,\varpi \in [{\kappa }_1,{\kappa }_2)\\ {\displaystyle \frac{1}{m!}}\left[{(\varpi -{\kappa }_1)}^m-2{(\varpi -{\kappa }_2)}^m\right],\varpi \in [{\kappa }_2,{\kappa }_3)\\ {\displaystyle \frac{1}{m!}}\left[{(\varpi -{\kappa }_1)}^m-2{(\varpi -{\kappa }_2)}^m+{(\varpi -{\kappa }_3)}^m\right],\varpi \in [{\kappa }_3,1)\end{array}\right.$$

(5)

where m is an integer.

$${Q^1}_{8\times 8}=\left[\begin{array}{cccccccc}{q}_{1,1}& q_{2,1}& q_{3,1}& q_{4,1}& q_{5,1}& q_{6,1}& q_{7,1}& q_{8,1}\end{array}\right].$$

$${Q^1}_{8\times 8}=\left[\begin{array}{cccccccc}0.0625& 0.1875& 0.3125& 0.4375& 0.5625& 0.6875& 0.8125& 0.9375\\ 0.0625& 0.1875& 0.3125& 0.4375& 0.4375& 0.3125& 0.1875& 0.0625\\ 0.0625& 0.1875& 0.1875& 0.0625& 0& 0& 0& 0\\ 0& 0& 0& 0& 0.0625& 0.1875& 0.1875& 0.0625\\ 0.0625& 0.0625& 0& 0& 0& 0& 0& 0\\ 0& 0& 0.0625& 0.0625& 0& 0& 0& 0\\ 0& 0& 0& 0& 0.0625& 0.0625& 0& 0\\ 0& 0& 0& 0& 0& 0& 0.0625& 0.0625\end{array}\right].$$

Similarly, we can calculate

$${Q^2}_{8\times 8}=\left[\begin{array}{cccccccc}{q}_{1,2}& q_{2,2}& q_{3,2}& q_{4,2}& q_{5,2}& q_{6,2}& q_{7,2}& q_{8,2}\end{array}\right].$$

Haar wavelets have local support, which means they are non-zero only in a small region. This property reduces the computational complexity and increases efficiency, especially for problems with localized features. Haar wavelets are piecewise constant functions that can accurately represent functions with discontinuities, sharp gradients, or localized features, unlike Fourier or polynomial-based methods, which require many terms to capture these features accurately. The hierarchical structure of Haar wavelets allows for fast computations and efficient use of resources, which is beneficial for large-scale problems. The sparsity also simplifies the process of matrix inversion or solving linear systems, as fewer non-zero elements need to be processed, making it more computationally efficient compared to methods that result in dense matrices. The adaptive mesh refinement property of Haar wavelets is advantageous over finite difference or finite element methods, which may require uniform grids or more complex adaptivity schemes to capture localized phenomena accurately.

The structure of this study is organized as follows: Section 1 presents the introduction and outlines essential mathematical preliminaries. Section 2 details the procedure employed in this article. Section 3 explores the convergence and error analysis. Section 4 applies our procedure to various examples. Section 5 addresses the numerical simulation of these examples. Finally, Section 6 concludes with a summary of our findings.

2. Numerical Procedure

In this section, we present the implementation of our technique in solving Emden–Fowler heat-type equations with boundary conditions. First, we divide the interval $[0,1]$ into $2p$ subintervals with length $\Delta ={\displaystyle \frac{1}{2p}}$. Define ${\psi }_j=j\Delta \psi$ for time $\psi$, where $0\le j\le 2p.$

Consider

$${\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}={\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{h}_{\rho }\left(\varpi \right),\varpi \in [0,1],\psi \in [{\psi }_j,{\psi }_{j+1}],$$

(6)

where ${\alpha }_{\rho }$ are Haar coefficients. Now, integrating (6) from 0 to $\varpi$ with respect to $\varpi$, we obtain

$${\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}={\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{q}_{\rho ,1}\left(\varpi \right)+{\displaystyle \frac{\partial \mathsf{\Omega }(0,\psi )}{\partial \varpi }}.$$

(7)

After integrating (7) from 0 to $\varpi$, we have

$${\displaystyle \frac{\partial \mathsf{\Omega }(0,\psi )}{\partial \varpi }}=\mathsf{\Omega }(1,\psi )-\mathsf{\Omega }(0,\psi )+{\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{Y}_{\rho ,1},$$

(8)

where $Y_{\rho ,1}=q_{\rho ,1}\left(1\right)-q_{\rho ,1}\left(0\right).$ Putting the value of (8) into (7) gives

$${\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}={\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{q}_{\rho ,1}\left(\varpi \right)+\mathsf{\Omega }(1,\psi )-\mathsf{\Omega }(0,\psi )+{\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{Y}_{\rho ,1}.$$

(9)

Again, integrating (9) from 0 to $\varpi$ with respect to $\varpi$ gives

$$\mathsf{\Omega }(\varpi ,\psi )={\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{q}_{\rho ,2}\left(\varpi \right)+\varpi (\mathsf{\Omega }(1,\psi )-\mathsf{\Omega }(0,\psi )+{\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{Y}_{\rho ,1})+\mathsf{\Omega }(0,\psi ).$$

(10)

Also, we use the backward difference scheme for the time-derivative term

$${\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \psi }}={\displaystyle \frac{\mathsf{\Omega }(\varpi ,{\psi }_{j+1})-\mathsf{\Omega }(\varpi ,{\psi }_j)}{\Delta \psi }}.$$

(11)

Putting (6), (9), (10), and (11) into (1) and using boundary conditions, we have

$$\begin{array}{l}{\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{h}_{\rho }\left(\varpi \right)+{\displaystyle \frac{\epsilon }{\varpi }}\left({\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{q}_{\rho ,1}\left(\varpi \right)+\mathsf{\Omega }(1,\psi )-\mathsf{\Omega }(0,\psi )+{\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{Y}_{\rho ,1}\right)+\\ B(\varpi ,\psi )F\left({\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{q}_{\rho ,2}\left(\varpi \right)+\varpi (\mathsf{\Omega }(1,\psi )-\mathsf{\Omega }(0,\psi )+{\displaystyle \sum _{\rho =1}^{2S}}{\alpha }_{\rho }{Y}_{\rho ,1})+\mathsf{\Omega }(0,\psi )\right)\\ -{\displaystyle \frac{\mathsf{\Omega }(\varpi ,\psi )-\mathsf{\Omega }(\varpi ,{\psi }_0)}{\Delta \psi }}=0.\end{array}$$

(12)

After collocating points in (12), we have a system of algebraic equations. This system of equations was solved for Haar coefficients using Matlab software. The values of the Haar coefficients gave us the approximate solution.

3. Convergence and Error Analysis

In this section, we concentrate on the convergence analysis approximation of $\mathsf{\Omega }(\varpi ,\psi )$. Define $0<{\psi }_0<{\psi }_1<{\psi }_2<\dots <{\psi }_{\vartheta }=\psi$, $\vartheta$ is a positive integer.

Theorem 1

([33]). Assume that the partial differential equation exists and is bounded in the interval $[a,b]\times [0,\psi ]$. For $A=2^V$, where $V=0,1,2,\dots$ and $\vartheta =0,1,2,\dots J$, then

$$Sup\Vert {\mathsf{\Omega }}_A(\varpi ,{\psi }_{\vartheta })-\mathsf{\Omega }(\varpi ,\psi ){\Vert }_{L_{\infty }(a,b)}\le O\left({\displaystyle \frac{1}{A^2}}+O\left(\Delta {\psi }^2\right)\right),V\to \infty ,J\to \infty$$

where ${\mathsf{\Omega }}_A(\varpi ,{\psi }_{\vartheta })$ denotes approximate solution, $\mathsf{\Omega }(\varpi ,\psi )$ denotes the exact solution, and $\Delta \psi ={sup}_{0\le \vartheta \le J-1}({\psi }_{\vartheta +1}-{\psi }_{\vartheta }).$

Proof. 

The proof is given in [33]. □

Error Analysis

The maximum absolute error norm is defined as follows:

$$E_{\infty }=sup\Vert {\mathsf{\Omega }}_A(\varpi ,\psi )-\mathsf{\Omega }(\varpi ,\psi )\Vert .$$

The $L_2$ and root-mean-square errors are defined as:

$$L_2=\Vert {\mathsf{\Omega }}_A(\varpi ,\psi )-\mathsf{\Omega }(\varpi ,\psi ){\Vert }_2=\sqrt{{\displaystyle \sum _{\rho =1}^{2S}}{({{\mathsf{\Omega }}^{\rho }}_A(\varpi ,\psi )-{\mathsf{\Omega }}^{\rho }(\varpi ,\psi ))}^2}$$

$$L_{rms}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{\rho =1}^{2S}}{({{\mathsf{\Omega }}^{\rho }}_A(\varpi ,\psi )-{\mathsf{\Omega }}^{\rho }(\varpi ,\psi ))}^2}{2S}}}.$$

The Figure 1 shows the comparison of maximum absolute error, $L_2$ error, and $L_{rms}$ error.

4. Applications

We know heat-type Emden–Fowler equations combine diffusion, damping, and nonlinear source or sink terms. We took examples with different source or sink terms and different damping factors.

4.1. Example 1

Consider the Emden–Fowler heat-type equation

$${\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}+{\displaystyle \frac{2}{\varpi }}{\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}-(5+4{\varpi }^2)\mathsf{\Omega }(\varpi ,\psi )={\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \psi }}-4{\varpi }^4-5{\varpi }^2+6,$$

(13)

with boundary conditions

$$\mathsf{\Omega }(0,\psi )=e^{\psi },\mathsf{\Omega }(1,\psi )=1+e^{1+\psi },$$

$$\mathsf{\Omega }(\varpi ,0)={\varpi }^2+e^{{\varpi }^2}.$$

The exact solution of (13) is

$$\mathsf{\Omega }(\varpi ,\psi )={\varpi }^2+e^{\psi +{\varpi }^2}.$$

The factor ${\displaystyle \frac{2}{\varpi }}$ in Example 1 adjusts the diffusion rate based on the radial position $\mathsf{\Omega }$. The constant term $+6$ is a uniform source term.

Then, we applied the Haar wavelet method and backward difference scheme to form a system of algebraic equations. That system was solved using MATLAB (latest v. 2024b) software to find the values of the Haar coefficients. Using the Haar coefficients, we found the numerical solution.

4.2. Example 2

Consider the Emden–Fowler heat-type equation

$${\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}+{\displaystyle \frac{6}{\varpi }}{\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}-2{\left(\mathsf{\Omega }(\varpi ,\psi )\right)}^2={\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \psi }}+5-2{\varpi }^4-2{\psi }^2-4{\varpi }^2\psi -8{\varpi }^2-8\psi ,$$

(14)

with boundary conditions

$$\mathsf{\Omega }(0,\psi )=2+\psi ,\mathsf{\Omega }(1,\psi )=3+\psi .$$

$$\mathsf{\Omega }(\varpi ,0)=2+{\varpi }^2.$$

The exact solution of (14) is

$$\mathsf{\Omega }(\varpi ,\psi )=2+{\varpi }^2+\psi .$$

The right side contains several terms, each representing source, sink, or external forcing terms that contribute independently to the dynamics of $\mathsf{\Omega }$. Then, we applied the Haar wavelet method and backward difference scheme to form a system of algebraic equations. That system was solved using MATLAB software to find the values of the Haar coefficients. Using the Haar coefficients, we found the numerical solution.

4.3. Example 3

Consider the Emden–Fowler heat-type equation

$$\begin{array}{l}{\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}+{\displaystyle \frac{\eta }{\varpi }}{\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}-\kappa \psi {\left(\varpi \psi \right)}^{-2+\kappa }({\varpi }^2-\psi (-1+\eta +\kappa ))e^{\mathsf{\Omega }(\varpi ,\psi )}={\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \psi }}\\ +{\psi }^2{\left(\varpi \psi \right)}^{-2+2\kappa }{\kappa }^2{e}^{2\mathsf{\Omega }(\varpi ,\psi )},\end{array}$$

(15)

with boundary conditions

$$\mathsf{\Omega }(0,\psi )=-ln\left(3\right),\mathsf{\Omega }(1,\psi )=-ln(3+{\psi }^2).$$

$$\mathsf{\Omega }(\varpi ,0)=-ln\left(3\right).$$

The exact solution of (15) is

$$\mathsf{\Omega }(\varpi ,\psi )=-ln(3+{\left(\varpi \psi \right)}^{\kappa }).$$

The exponential terms in the equation represent a nonlinear growth/decay, potentially modeling a reaction or feedback effect where an increase in $\mathsf{\Omega }$ accelerates its own growth. Then, we applied the Haar wavelet method and backward difference scheme to form a system of algebraic equations. That system was solved using MATLAB software to find the values of the Haar coefficients. Using the Haar coefficients, we found the numerical solution.

4.4. Example 4

Consider the Emden–Fowler heat-type equation

$${\displaystyle \frac{{\partial }^2\mathsf{\Omega }(\varpi ,\psi )}{\partial {\varpi }^2}}+{\displaystyle \frac{2}{\varpi }}{\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \varpi }}-(6+4{\varpi }^2-cos\psi )\mathsf{\Omega }(\varpi ,\psi )={\displaystyle \frac{\partial \mathsf{\Omega }(\varpi ,\psi )}{\partial \psi }},$$

(16)

with boundary conditions

$$\mathsf{\Omega }(0,\psi )=e^{sin\psi },\mathsf{\Omega }(1,\psi )=e^{1+sin\psi }.$$

$$\mathsf{\Omega }(\varpi ,0)=e^{{\varpi }^2}.$$

The exact solution of (16) is

$$\mathsf{\Omega }(\varpi ,\psi )=e^{{\varpi }^2+sin\psi }.$$

Then, we applied the Haar wavelet method and backward difference scheme to form a system of algebraic equations. That system was solved using MATLAB software to find the values of the Haar coefficients. Using the Haar coefficients, we found the numerical solution.

5. Numerical Simulation

In this section, we present our numerical findings along with 2D and 3D graphs comparing the Haar solution and the exact solution. Figure 2 shows the error and comparison of exact and Haar solutions for Example 1. Figure 3 shows 3D surface plots of exact and Haar solutions for Example 1. Figure 4 shows 3D contour plots of exact and Haar solutions for Example 1. Figure 5 shows the error and comparison of exact and Haar solutions for Example 2. Figure 6 shows 3D surface plots of exact and Haar solutions for Example 2. Figure 7 shows 3D contour plots of exact and Haar solutions for Example 2. Figure 8 shows the error and comparison of exact and Haar solutions for Example 3. Figure 9 shows 3D surface plots of exact and Haar solutions for Example 3. Figure 10 shows 3D contour plots of exact and Haar solutions for Example 3. Figure 11 shows the error and comparison of exact and Haar solutions for Example 4. Figure 12 shows 3D surface plots of exact and Haar solutions for Example 4. Figure 13 shows 3D contour plots of exact and Haar solutions for Example 4.

Table 1. Comparison of Haar solution and exact solution of Example 1 ($\Delta \psi =0.001$ and $\Delta \psi =0.0001$) when $J=3$.

$\mathit{\varpi }$	Haar Solution	Exact Solution	Error
0.03125	1.003159	1.002016	1.14 × 10−3
0.15625	1.050133	1.049320	8.12 × 10−4
0.28125	1.162487	1.161754	7.33 × 10−4
0.40625	1.345648	1.344994	6.53 × 10−4
0.53625	1.609623	1.609051	5.72 × 10−4
0.65625	1.970471	1.970000	4.71 × 10−4
0.78125	2.453259	2.452926	3.32 × 10−4
0.90625	3.09698	3.096849	1.30 × 10−4
0.96875	3.497148	3.497118	3.0 × 10−5
0.03125	1.002056	1.001959	9.7 × 10−5
0.15625	1.049231	1.049147	8.39 × 10−5
0.28125	1.161524	1.161449	7.5 × 10−5
0.40625	1.344596	1.344529	6.69 × 10−5
0.53625	1.608438	1.608380	5.8 × 10−5
0.65625	1.969096	1.969048	4.8 × 10−5
0.78125	2.451614	2.451579	3.5 × 10−5
0.90625	3.094944	3.094930	1.39 × 10−5
0.96875	3.494818	3.494816	1.99 × 10−6

gives us the numerical comparison of exact and Haar solutions for different $\Delta \psi$ for Example 1.

Table 2. Error norm of Example 1 ($\Delta \psi =0.0001$).

Level of resolution	$J=1$	$J=2$	$J=3$	$J=4$	$J=5$
$L_2$ error	8.02 × 10−5	8.86 × 10−5	7.82 × 10−5	1.76 × 10−4	7.6 × 10−4

gives the error norm for different resolutions for Example1.

Table 3. Comparison of the Haar solution and exact solution of Example 2 ($\Delta \psi =0.001$ and $\Delta \psi =0.0001$) when $J=3$.

$\mathit{\varpi }$	Haar Solution	Exact Solution	Error
0.03125	2.001075	2.001039	3.6 × 10−5
0.15625	2.02451	2.024601	9.10 × 10−5
0.28125	2.0792	2.079414	2.13 × 10−4
0.40625	2.165138	2.165476	3.38 × 10−4
0.53625	2.282326	2.282789	4.63 × 10−4
0.65625	2.430764	2.431351	5.86 × 10−4
0.78125	2.610451	2.611164	7.13 × 10−4
0.90625	2.82138	2.822226	8.46 × 10−4
0.96875	2.938655	2.939476	8.21 × 10−4
0.03125	2.001103	2.000982	1.21 × 10−4
0.15625	2.024514	2.024432	8.19 × 10−5
0.28125	2.079201	2.079132	6.89 × 10−5
0.40625	2.165139	2.165082	5.69 × 10−5
0.53625	2.282326	2.282282	4.39 × 10−5
0.65625	2.430764	2.430732	3.20 × 10−5
0.78125	2.610451	2.610432	1.89 × 10−5
0.90625	2.821389	2.821382	7.0 × 10−6
0.96875	2.938577	2.938576	1.0 × 10−6

gives us the numerical comparison of exact and Haar solutions for different $\Delta \psi$ for Example 2.

Table 4. Comparison of Haar solution and exact solution of Example 3 ($\Delta \psi =0.0001$) when $J=3$.

$\mathit{\varpi }$	Haar Solution	Exact Solution	Error
0.03125	−1.09861229	−1.09861228	2.075 × 10−12
0.15625	−1.09861229	−1.09861228	6.651 × 10−11
0.28125	−1.09861229	−-1.09861228	3.258 × 10−10
0.40625	−1.09861229	−1.09861228	8.995 × 10−10
0.53625	−1.09861229	−1.09861228	1.887 × 10−9
0.65625	−1.09861229	−1.09861228	3.370 × 10−9
0.78125	−1.0986123	−1.09861229	5.411 × 10−9
0.90625	−1.0986123	−1.09861229	8.147 × 10−9
0.96875	−1.0986123	−1.09861229	8.788 × 10−9

gives us the numerical comparison of exact and Haar solutions for different $\Delta \psi$ for Example 3. In

Table 5. Comparison of Haar solution with MADM for Example 3.

Errors	MADM [24] $\left(E_2\right)$	MADM [24] $\left(E_3\right)$	Haar (J = 1)	Haar (J = 2)	Haar (J = 3)
$\psi =0.5$	2.9 × 10−3	1.8 × 10−3	6.5 × 10−4	1.1 × 103	2.1 × 10−3
Run time (s)	-	-	0.069892	0.086797	0.138372

, we compare our results with other method for Example 3.

Table 6. Comparison of Haar solution and exact solution of Example 4 ($\Delta \psi =0.0001$) when $J=3$.

$\mathit{\varpi }$	Haar Solution	Exact Solution	Error
0.03125	1.00107989	1.00098329	9.65 × 10−5
0.15625	1.02481684	1.02473373	8.31 × 10−5
0.28125	1.0824225	1.08234806	7.44 × 10−5
0.40625	1.17955714	1.17949079	6.63 × 10−5
0.53625	1.32621167	1.32615371	5.79 × 10−5
0.65625	1.53843231	1.53838445	4.78 × 10−5
0.78125	1.84126215	1.84122813	3.40 × 10−5
0.90625	2.27365453	2.27364168	1.28 × 10−5
0.96875	2.55634128	2.55634003	1.24 × 10−6

gives us the numerical comparison of exact and Haar solutions for different $\Delta \psi$ for Example 4. In

Table 7. Comparison of Haar solution with other methods for Example 4.

Method	Haar	Haar	Haar	Haar	HAM [34]	HAM [34]	VIM [34]	VIM [34]
J/degree	$J=2$	$J=3$	$J=4$	$J=5$	deg 8	deg 16	deg 8	deg 16
Max. Abs. error	1.21 × 10−4	9.65 × 10−5	2.3 × 10−4	7.1 × 10−4	0.1933	1.22 × 10−3	0.1933	1.22 × 10−3
Run time (s)	0.891826	0.992699	1.001384	1.202591	0.188	0.327	0.218	0.343

, we compare our results with other methods and provide the CPU run time to demonstrate the efficiency and convenience of our approach.

6. Conclusions

The numerical solution of the time-dependent Emden–Fowler equation offers a powerful way to simulate and understand time-dependent processes. In diffusion processes, the numerical solution provides the density or concentration profile over time and space, showing how a substance diffuses or a population disperses under the influence of nonlinear effects. In plasma physics, solutions to the time-dependent Emden–Fowler equation are relevant to understanding plasma behavior in various configurations, such as spherical or cylindrical plasma distributions. The Haar wavelet collocation method, combined with a backward difference scheme, has proven to be an effective and efficient approach for solving time-dependent Emden–Fowler equations, including both linear and nonlinear partial differential equations. The flexibility of this method in handling complex boundary conditions makes it particularly suitable for modeling physical systems with intricate boundary behaviors. The numerical results, supported by 2D and 3D visual comparisons between the Haar solution and the exact solution, highlight the accuracy, adaptability, and computational efficiency of this technique.

Additionally, the use of Haar wavelets introduces several advantages, such as simplicity, local support, and sparse representation, which reduces computational complexity and improves efficiency. A comparative analysis, including CPU run time and error metrics, indicated that this method not only met but often surpassed the performance of alternative methods in terms of accuracy and computational convenience. In general, the Haar wavelet collocation method is a promising tool for solving differential equations in scientific and engineering applications due to its combination of precision, efficiency, and ease of implementation.