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Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations

  • Kholoud Saad Albalawi , Ashish Kumar EMAIL logo , Badr Saad Alqahtani and Pranay Goswami
Published/Copyright: August 9, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

The present study introduces the Haar wavelet method, which utilizes collocation points to approximate solutions to the Emden-Fowler Pantograph delay differential equations (PDDEs) of general order. This semi-analytic method requires the transformation of the original differential equation into a system of nonlinear differential equations, which is then solved to determine the Haar coefficients. The method’s application to fourth-, fifth-, and sixth-order PDDEs is discussed, along with an examination of convergence that involves the determination of an upper bound and the formulation of the rate of convergence for the method. Numerical simulations and error tables are presented to demonstrate the effectiveness and precision of this approach. The error tables clearly illustrate that the method’s accuracy improves progressively with increasing resolution.

Keywords: Pantograph delay differential equations; Emden-Fowler-type equations; Haar wavelet method
MSC 2010: Primary 34A34; 65L60; 65T60

1 Introduction and mathematical preliminaries

The Emden-Fowler equations, named after their respective contributors, Emden [1] and Fowler [2], are a set of nonlinear differential equations that arise in astrophysics and other areas of physics, especially when examining the composition of stars [3,4] and the behavior of polytropic gases [5]. The Emden-Fowler equations can be solved numerically to determine the radial distribution of the density, pressure, temperature, and other physical quantities within the star. The Emden-Fowler equation can be expressed in the following general form:

(1) ν − ξ d ε 1 d ν ε 1 ν ξ d ε 2 d ν ε 2 w + s ( ν ) g ( w ) = 0 ,

where s ( ν ) is a nonlinear function, ε 1 + ε 2 gives the order of the equation.

Astrophysical fluid dynamics [6] is a field of study that deals with the behavior of fluids in the presence of gravity. It is a complex subject, and one way to model these behaviors [7] is through the use of Emden-Fowler equations. These equations are particularly useful when it comes to dealing with scenarios that involve polytropic equations of state. In such scenarios, the Emden-Fowler equations can be used as a tool for modeling the behavior of fluids under the influence of gravity. Emden-Fowler equations can also be applied to cosmology [8] to describe the behavior of certain types of dark matter, such as cold dark matter, which can be modeled as a polytropic fluid. The Emden-Fowler equation and its variants serve as mathematical models for various physical phenomena beyond astrophysics, including diffusion processes, population dynamics, and nonlinear wave propagation. In the context of chemical systems, similar mathematical models may arise to describe phenomena like diffusion-limited reactions in porous media.

The motivation behind studying Pantograph delay differential equations (PDDEs) [9] lies in their ability to capture and model systems where delays play a crucial role in the dynamics. These equations play a crucial role in a range of disciplines, encompassing physics [10], biology [11], and economics [12], as they can capture complex behaviors in systems with time delays. In control theory, systems with time delays are common in practical applications [13]. For example, in feedback control systems, there is often a finite time delay between the measurement of a system’s state and the application of control action. PDDEs provide a mathematical framework to model and analyze the behavior of such systems, allowing engineers to design robust controllers that can handle delays effectively. Biological neural networks exhibit complex dynamics, including time delays in signal propagation between neurons. PDDEs are employed in computational neuroscience to model the dynamics of neural networks with delayed feedback and study phenomena like synchronization, oscillations, and information processing in the brain. PDDEs are used in macroeconomic modeling, financial markets analysis, and game theory to study the dynamics of economic systems with delays and understand phenomena like business cycles, market fluctuations, and policy impacts. In the context of delay differential equations (DDEs), the Pantograph concept is used to describe systems that involve time delays in such a way that they exhibit a particular form of stability and oscillatory behavior [14].

Within this article, we delve into the Emden-Fowler-type PDDEs of general order:

(2) d ε 1 + ε 2 w ( ν ) d ν ε 1 + ε 2 + α ν d ε 1 + ε 2 − 1 w ( γ ν ) d ν ε 1 + ε 2 − 1 + β ν 2 d ε 1 + ε 2 − 2 w ( γ ν ) d ν ε 1 + ε 2 − 2 + … + s ( γ ν ) g ( w ) = 0 ,

with initial conditions w ( 0 ) = w 0 , w ′ ( 0 ) = w 1 , w ″ ( 0 ) = w 2 , … , w ( ε 1 + ε 2 − 1 ) ( 0 ) = w ε 1 + ε 2 − 1 . Here γ , α , β are some suitable constants: s ( ν ) is an unknown function. Several pertinent papers discuss the notion of generalization, such as [15,16].

The inclusion of the delayed term in the equation reflects the system’s sensitivity to its past values, introducing a notion of memory into the system’s behavior. Analyzing and solving PDDEs can be challenging due to their non-standard form and the intricate interplay between delayed and non-delayed components. Researchers use a variety of mathematical techniques, such as Laplace transforms [17], numerical methods [18,19], and phase plane analysis, and Chebyshev polynomials [20] to understand the behavior of systems described by PDDEs. Srinivasa solved nonlinear variable DDEs using Laguerre wavelets in his paper [21].

Wavelets are mathematical functions used in various real-world applications [22] to analyze and process data in both one-dimensional and multi-dimensional forms. They have gained significant popularity in signal processing, data compression [23], and image analysis due to their ability to capture both high and low-frequency data components. In medical imaging, wavelets are used for tasks like image denoising [24], feature extraction [25], and image enhancement. In geophysics, wavelets are used to analyze seismic data [26]. They can help identify subsurface structures, locate natural resources, and assess the risk of seismic events. There are various types of wavelets, such as Haar wavelet [27,28], Morlet wavelet [29], Mexican hat wavelet [30], Daubechies wavelets, and Spline wavelets [31].

The Haar wavelet is fundamental and straightforward in signal processing and image compression. Formulated by the Hungarian mathematician Alfréd Haar during the early twentieth century, the Haar wavelet is a powerful tool for detecting edges in images [32]. It can detect sharp transitions in pixel values, which indicate edges and boundaries in images.

This study uses the Haar wavelet method to solve the PDDE. This study is organized as follows: Section 2 describes the Haar wavelet. Section 3 delves into the application and operation of the method in solving differential equations. Section 4 explores the convergence theorem and the rate of convergence. In Section 5, we discuss applications. Section 6 is dedicated to numerical simulations of the provided examples, accompanied by the compilation of error tables. Section 7, in turn, encompasses our concluding remarks and findings.

2 Haar wavelet

There are many wavelets. Haar wavelets are among the simplest types of wavelets, both conceptually and computationally. Their construction involves only basic binary operations and piecewise constant functions. Different wavelets have different advantages. It depends on certain contexts. For example, Haar wavelets have compact support. Unlike Haar wavelets, ultraspherical wavelets (described by Mulimani and Srinivasa in [33]) are not necessarily compactly supported; their support depends on the parameters of the ultraspherical polynomials but ultraspherical wavelets possess more flexibility in shaping the wavelet functions compared to Haar wavelets due to the adjustable parameters of ultraspherical polynomials. In scenarios where the detection and localization of sharp transitions are crucial, Haar wavelets may outperform ultraspherical wavelets, which may not be as sensitive to abrupt changes. Haar wavelet consists of a positive value and a negative value within a specific interval, and it is zero outside that interval. The Haar wavelet is orthogonal and has a compact support. Haar functions [34] over the interval [ 0 , 1 ) are defined as:

h μ ( ν ) = 1 , ξ 1 ≤ ν < ξ 2 , − 1 , ξ 2 ≤ ν < ξ 3 , 0 , o t h e r w i s e ,

where

ξ 1 = l m , ξ 2 = 2 l + 1 2 m , ξ 3 = l + 1 m .

We define m = 2 j , j = 0 , 1 , 2 , … J , and l = 0 , 1 , 2 , … m − 1 , where J is the maximum resolution, j represents the dilation parameter, and μ = m + l + 1 . For μ = 1

h 1 ( ν ) = 1 , ν ∈ [ 0 , 1 ] , 0 , o t h e r w i s e . h 2 ( ν ) = 1 , ν ∈ 0 , 1 2 , − 1 , ν ∈ 1 2 , 1 , 0 , o t h e r w i s e .

The Haar wavelet is orthogonal so

∫ 0 1 h μ ( ν ) h b ( ν ) d ν = 2 − j ; μ = b , 0 ; μ ≠ b .

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

H 8 × 8 = h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 .

After substituting values

H 8 × 8 = 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 .

The integrals of the Haar function are defined as:

s μ , 1 ( ν ) = ∫ 0 ν h μ ( ν ) d ν ,

s μ , 2 ( ν ) = ∫ 0 ν s μ , 1 ( ν ) d ν .

In general, we write

s μ , σ ( ν ) = 0 , ν ∈ [ 0 , ξ 1 ) , 1 Γ ( σ + 1 ) ( ν − ξ 1 ) σ , ν ∈ [ ξ 1 , ξ 2 ) , 1 Γ ( σ + 1 ) [ ( ν − ξ 1 ) σ − 2 ( ν − ξ 2 ) σ ] , ν ∈ [ ξ 2 , ξ 3 ) , 1 Γ ( σ + 1 ) [ ( ν − ξ 1 ) σ − 2 ( ν − ξ 2 ) σ + ( ν − ξ 3 ) σ ] , ν ∈ [ ξ 3 , 1 ) .

For resolution J = 2 ,

( S 1 ) 8 × 8 = s 1 , 1 s 2 , 1 s 3 , 1 s 4 , 1 s 5 , 1 s 6 , 1 s 7 , 1 s 8 , 1 .

After substituting values

( S 1 ) 8 × 8 = 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 .

2.1 Function approximation

When dealing with a function w ( ν ) defined on the interval [0, 1], the approach involves approximating the function by employing Haar functions in the following manner:

w ( ν ) = ∑ μ = 1 ∞ c μ h μ ,

where c μ are the Haar coefficients.

3 Method

Let us consider the general order Emden-Fowler PDDE:

(3) d ε 1 + ε 2 w ( ν ) d ν ε 1 + ε 2 + α ν d ε 1 + ε 2 − 1 w ( γ ν ) d ν ε 1 + ε 2 − 1 + β ν 2 d ε 1 + ε 2 − 2 w ( γ ν ) d ν ε 1 + ε 2 − 2 + … + s ( γ ν ) g ( w ) = 0 ,

with initial conditions w ( 0 ) = w 0 , w ′ ( 0 ) = w 1 , w ″ ( 0 ) = w 2 , … , w ( ε 1 + ε 2 − 1 ) ( 0 ) = w ε 1 + ε 2 − 1 .

Now at first, we write

(4) d ε 1 + ε 2 w ( ν ) d ν ε 1 + ε 2 = ∑ μ = 1 2 T c μ h μ ( ν ) .

We integrate this equation (4) and apply the initial conditions. We have

(5) d ε 1 + ε 2 − 1 w ( ν ) d ν ε 1 + ε 2 − 1 = ∑ μ = 1 2 T c μ s μ , 1 ( ν ) + w ε 1 + ε 2 − 1 .

Repeatedly integrating and incorporating initial conditions, we obtain the following:

(6) w ( ν ) = ∑ μ = 1 2 T c μ s μ , ε 1 + ε 2 ( ν ) + ν ε 1 + ε 2 − 1 ( ε 1 + ε 2 − 1 ) ! w ε 1 + ε 2 − 1 + … + w 0 .

After substituting all these equations into equation (3), equation (3) will be

(7) ∑ μ = 1 2 T c μ h μ ( ν ) + ∑ μ = 1 2 T c μ s μ , 1 ( γ ν ) + w ε 1 + ε 2 − 1 + … ∑ μ = 1 2 T c μ s μ , ε 1 + ε 2 ( γ ν ) + ( γ ν ) ε 1 + ε 2 − 1 ( ε 1 + ε 2 − 1 ) ! w ε 1 + ε 2 − 1 + … + w 0 + s ( γ ν ) g ( w ) = 0 .

In the next step, we collocate the points as:

ν μ = μ − 0.5 2 T .

After collocating the points in equation (7), equation (7) will be

(8) ∑ μ = 1 2 T c μ h μ ( ν μ ) + ∑ μ = 1 2 T c μ s μ , 1 ( γ ν μ ) + w ε 1 + ε 2 − 1 + … ∑ μ = 1 2 T c μ s μ , ε 1 + ε 2 ( γ ν μ ) + ( γ ν μ ) ε 1 + ε 2 − 1 ( ε 1 + ε 2 − 1 ) ! w ε 1 + ε 2 − 1 + … + w 0 + s ( γ ν μ ) g ( w ) = 0 .

Now we have a system of equations of 2 T × 2 T . We utilize the Newton method for resolving this set of equations, resulting in the set of values of c μ . After substituting the values of c μ in equation (6) we have the numerical solution.

4 Error analysis and convergence

Lemma 4.1

[35] Suppose w ( ν ) ∈ L 2 ( ℜ ) and has a bounded first derivative, that is, ∣ w ′ ( ν ) ∣ ≤ Q , ∀ ν ∈ ( 0 , 1 ) , Q > 0 , and w ( μ ) = ∑ μ = 1 ∞ c μ h μ ( ν ) . Then, ∣ c μ ∣ ≤ Q 2 − ( 3 k − 2 ) ∕ 2 .

Theorem 4.1

[35] Consider a continuous function w ( ν ) ∈ L 2 ( ℜ ) within the interval (0, 1) that has a bounded first derivative. In this case, the error norm at the Jth level is determined by the following relationship:

∣ ∣ E J ∣ ∣ ≤ Q 7 C 2 − ( 3 ∕ 2 ) L

∣ w ′ ( ν ) ∣ ≤ Q , ∀ ν ∈ ( 0 , 1 ) , Q > 0 , and N = 2 k , where J represents the highest resolution.

The demonstration is simple and direct [35].

4.1 Error analysis

The error is determined as follows:

L ∞ = Max . ∣ w μ exact − w μ approx . ∣ ,

where w μ exact and w μ approx . represent the exact and estimated solutions at the μ th point.

4.2 Rate of convergence

The definition of the convergence rate is as follows:

R C ( J ) = log [ E C ( J ∕ 2 ) ∕ E C ( J ) ] log 2 ,

where E C ( J ) denotes the maximum absolute error across J collocation points.

The rate of convergence of a numerical method refers to how quickly the method converges to the true solution or how fast the error decreases with each iteration. Table 1 shows the rate of convergence for Example 1, Example 2, and Example 3.

Table 1

Rate of convergence for Examples

Resolution (J) 3 4 5 6
Example 1 ( R C ( J ) ) 1.897402 1.949923 1.987721 1.992677
Example 2 ( R C ( J ) ) 1.481526 1.731874 1.863017 1.9307027
Example 3 ( R C ( J ) ) 3.110294 1.306188 1.907927 2.005272

5 Applications

5.1 Application 1

Let us consider the fourth-order Emden-Fowler PDE:

(9) d 4 w ( ν ) d ν 4 + 4 ν d 3 w ( ν ∕ 2 ) d ν 3 + e w ( ν ∕ 2 ) = Z ( ν ) ,

where

Z ( ν ) = 1 1 + ν 4 4 [ 24 − 744 ν 4 + 744 ν 8 − 24 ν 12 ] + 1 16 + ν 4 3 [ 196,608 − 49,152 ν 4 + 256 ν 8 ] + 16 + ν 4 16 e ,

with initial conditions: w ( 0 ) = 1 , w ′ ( 0 ) = 0 , w ″ ( 0 ) = 0 , w ‴ ( 0 ) = 0 .

w ( ν ) = 1 + log ( 1 + ν 4 ) represents the exact solution of (9).

Now we apply the method. Consider

(10) d 4 w ( ν ) d ν 4 = ∑ μ = 1 2 T c μ h μ ( ν ) .

After performing the integration of (10) and implementing the initial condition, we have

(11) d 3 w ( ν ) d ν 3 = ∑ μ = 1 2 T c μ s μ , 1 ( ν )

and

(12) d 3 w ( ν ∕ 2 ) d ν 3 = ∑ μ = 1 2 T c μ s μ , 1 ( ν ∕ 2 ) .

Integrating equation (11) again and again and applying initial conditions, we have

(13) w ( ν ) = 1 + ∑ μ = 1 2 T c μ s μ , 4 ( ν )

and

(14) w ( ν ∕ 2 ) = 1 + ∑ μ = 1 2 T c μ s μ , 4 ( ν ∕ 2 ) .

After substituting equations (10), (12), and (14) into equation (9), we have a system of equations after the collocation of points:

(15) ∑ μ = 1 2 T c μ h μ ( ν μ ) + 4 ν μ ∑ μ = 1 2 T c μ s μ , 1 ( ν μ ∕ 2 ) + e 1 + ∑ μ = 1 2 T c μ s μ , 4 ( ν μ ∕ 2 ) − Z ( ν μ ) = 0 .

The Newton method is used to resolve this equation system and acquire the Haar coefficients denoted as c μ . Subsequently, substituting these coefficients into equation (13) yields the estimated solution.

5.2 Application 2

Let us consider fifth-order Emden-Fowler PDDE:

(16) d 5 w ( ν ) d ν 5 + 4 ν d 4 w ( ν ∕ 2 ) d ν 4 + 2 ν 2 d 3 w ( ν ∕ 2 ) d ν 3 − 5 w ( ν ∕ 2 ) 16,384 F ( ν ) = G ( ν ) ,

where

F ( ν ) = 884,736 + 506,880 ν 5 + 19,400 ν 10 + 125 ν 15 ,

G ( ν ) = 5 e ν 5 [ 24 + 3,000 ν 5 + 9,000 ν 10 + 5,000 ν 15 + 625 ν 20 ] ,

with initial conditions w ( 0 ) = 1 , w ′ ( 0 ) = 0 , w ″ ( 0 ) = 0 , w ‴ ( 0 ) = 0 , w ″ ″ ( 0 ) = 0 .

w ( ν ) = e ν 5 represents the exact solution of (16).

Now we apply the method. Consider

(17) d 5 w ( ν ) d ν 5 = ∑ μ = 1 2 T c μ h μ ( ν ) .

Integrating equation (17) again and again and applying initial conditions, we have

(18) w ( ν ) = 1 + ∑ μ = 1 2 T c μ s μ , 5 ( ν )

and

(19) w ( ν ∕ 2 ) = 1 + ∑ μ = 1 2 T c μ s μ , 5 ( ν ∕ 2 ) .

After substituting these equations into equation (16), we have a system of equations after collocation of points:

(20) ∑ μ = 1 2 T c μ h μ ( ν μ ) + 4 ν μ ∑ μ = 1 2 T c μ s μ , 1 ( ν μ ∕ 2 ) + 2 ν μ 2 ∑ μ = 1 2 T c μ s μ , 2 ( ν μ ∕ 2 ) − 5 ( 1 + ∑ μ = 1 2 T c μ s μ , 5 ( ν μ ∕ 2 ) ) 16,384 F ( ν μ ) − G ( ν μ ) = 0 .

The Newton method is used to resolve this equation system and acquire the Haar coefficients denoted as c μ . Subsequently, substituting these coefficients into equation (18) yields the estimated solution.

5.3 Application 3

Consider the sixth-order Emden-Fowler multi-term PDDE:

(21) d 6 w ( ν ) d ν 6 + 1 ν d 5 w ( ν ∕ 2 ) d ν 5 + ( w ( ν ∕ 2 ) ) 7 360 − 210 ν 2 − 105 2 ν 4 + 45 8 ν 6 − 720 ( w ( ν ) ) 7 [ − 1 + 21 ν 2 − 35 ν 4 + 7 ν 6 ] = 0 ,

with initial conditions w ( 0 ) = 1 , w ′ ( 0 ) = 0 , w ″ ( 0 ) = − 2 , w ‴ ( 0 ) = 0 , w ′ ′ ′ ′ ( 0 ) = 24 , w ′ ′ ′ ′ ′ ( 0 ) = 0 .

w ( ν ) = 1 1 + ν 2 represents the exact solution of (21).

Now we apply the method. Consider

(22) d 6 w ( ν ) d ν 6 = ∑ μ = 1 2 T c μ h μ ( ν ) .

Integrating equation (22) again and again and applying initial conditions, we have

(23) w ( ν ) = 1 − ν 2 + ν 4 + ∑ μ = 1 2 T c μ s μ , 6 ( ν )

and

(24) w ( ν ∕ 2 ) = 1 − ( ν ∕ 2 ) 2 + ( ν ∕ 2 ) 4 + ∑ μ = 1 2 T c μ s μ , 6 ( ν ∕ 2 ) .

After substituting these equations into equation (21), we have a system of equations after collocation of points:

(25) ∑ μ = 1 2 T c μ h μ ( ν μ ) + 1 ν μ ∑ μ = 1 2 T c μ s μ , 1 ( ν μ ∕ 2 ) + 1 − ( ν μ ∕ 2 ) 2 + ( ν μ ∕ 2 ) 4 + ∑ μ = 1 2 T c μ s μ , 6 ( ν μ ∕ 2 ) 7 360 − 210 ν μ 2 − 105 2 ν μ 4 + 45 8 ν μ 6 − 720 1 − ν μ 2 + ν μ 4 + ∑ μ = 1 2 T c μ s μ , 6 ( ν μ ) 7 [ − 1 + 21 ν μ 2 − 35 ν μ 4 + 7 ν μ 6 ] = 0 .

The Newton method is used to resolve this equation system and acquire the Haar coefficients denoted as c μ . Subsequently, substituting these coefficients into equation (23) yields the Haar solution.

5.4 Application 4

Consider the sixth-order Emden-Fowler multi-term PDDE:

(26) d 6 w ( ν ) d ν 6 + 12 ν d 5 w ( ν ∕ 2 ) d ν 5 + 36 ν 2 d 4 w ( ν ∕ 2 ) d ν 4 + 24 ν 3 d 3 w ( ν ∕ 2 ) d ν 3 − 8,640 = 0 ,

with initial conditions w ( 0 ) = 1 , w ′ ( 0 ) = 0 , w ″ ( 0 ) = 0 , w ‴ ( 0 ) = 0 , w ″ ″ ( 0 ) = 0 , w ″ ‴ ( 0 ) = 0 .

w ( ν ) = 1 + ν 6 represents the exact solution of (26).

Now we apply the method. Consider

(27) d 6 w ( ν ) d ν 6 = ∑ μ = 1 2 T c μ h μ ( ν ) .

Integrating equation (27) again and again and applying initial conditions, we have

(28) w ( ν ) = 1 + ∑ μ = 1 2 T c μ s μ , 6 ( ν )

and

(29) w ( ν ∕ 2 ) = 1 + ∑ μ = 1 2 T c μ s μ , 6 ( ν ∕ 2 ) .

After substituting these equations into equation (26), we have a system of equations after collocation of points:

(30) ∑ μ = 1 2 T c μ h μ ( ν μ ) + 12 ν μ ∑ μ = 1 2 T c μ s μ , 1 ( ν μ ∕ 2 ) + 36 ν μ 2 ∑ μ = 1 2 T c μ s μ , 2 ( ν μ ∕ 2 ) + 24 ν μ 3 ∑ μ = 1 2 T c μ s μ , 3 ( ν μ ∕ 2 ) − 8,640 = 0 .

The Newton method is used to resolve this equation system and acquire the Haar coefficients denoted as c μ . Subsequently, substituting these coefficients into equation (28) yields the Haar solution.

6 Numerical simulation

Figure 1 depicts the error and comparison between the exact solution and the Haar wavelet solution for Example 1 and Example 2. Figures 2 and 3 depict the error and comparison between the exact and Haar wavelet solution for example 2 and example 3 respectively. Tables 2, 3, 4, 5 illustrate the comparison between exact and Haar solution at various values of ν . Tables 6, 7, 8, 9 give the maximum absolute error for different resolutions. These tables also provide the CPU time, indicating that this method exhibits a notable speed.

Figure 1 Graphical demonstration of Application 1 and Application 2. (a) Comparison of exact solution with Haar wavelet method (HWM) solution for Application 1 when J = 3 J=3 , (b) absolute error of Application 1 when J = 3 J=3 , (c) comparison of exact solution with HWM solution for Application 2 when J = 3 J=3 , and (d) absolute error of Application 2 when J = 3 J=3 .
Figure 1

Graphical demonstration of Application 1 and Application 2. (a) Comparison of exact solution with Haar wavelet method (HWM) solution for Application 1 when J = 3 , (b) absolute error of Application 1 when J = 3 , (c) comparison of exact solution with HWM solution for Application 2 when J = 3 , and (d) absolute error of Application 2 when J = 3 .

Figure 2 Graphical demonstration of Application 3. (a) Comparison of exact solution with HWM solution for Application 3 when J = 3 J=3 and (b) absolute error of Application 3 when J = 3 J=3 .
Figure 2

Graphical demonstration of Application 3. (a) Comparison of exact solution with HWM solution for Application 3 when J = 3 and (b) absolute error of Application 3 when J = 3 .

Figure 3 Numerical simulation of Example 4. (a) Comparison of exact solution with HWM solution for Application 4 when J = 3 J=3 and (b) absolute error of Application 4 when J = 3 J=3 .
Figure 3

Numerical simulation of Example 4. (a) Comparison of exact solution with HWM solution for Application 4 when J = 3 and (b) absolute error of Application 4 when J = 3 .

Table 2

Comparison of Haar wavelet solution with exact solution for Application 1 at J = 3

ν HWM Exact Error
0.1 1.00009999 1.00009999 1.553 × 1 0 ‒ 9
0.2 1.00159844 1.00159872 2.715 × 1 0 ‒ 7
0.3 1.00806459 1.00806737 2.775 × 1 0 ‒ 6
0.4 1.02526313 1.02527780 1.467 × 1 0 ‒ 5
0.5 1.06057291 1.06062462 5.171 × 1 0 ‒ 5
0.6 1.12172689 1.12186358 1.366 × 1 0 ‒ 4
0.7 1.21490555 1.21519202 2.864 × 1 0 ‒ 4
0.8 1.34281450 1.34330597 4.914 × 1 0 ‒ 4
0.9 1.50376271 1.50446544 7.027 × 1 0 ‒ 4
Table 3

Comparison of Haar wavelet solution with exact solution for Application 2 at J = 3

ν HWM Exact Error
0.1 1.00001000 1.00001000 2.979 × 1 0 ‒ 11
0.2 1.00032007 1.00032005 1.940 × 1 0 ‒ 8
0.3 1.00243341 1.00243295 4.559 × 1 0 ‒ 7
0.4 1.01029712 1.01029260 4.512 × 1 0 ‒ 6
0.5 1.03177062 1.03174340 2.721 × 1 0 ‒ 5
0.6 1.08098489 1.08086322 1.216 × 1 0 ‒ 4
0.7 1.18347043 1.18301941 4.510 × 1 0 ‒ 4
0.8 1.38925012 1.38774482 1.505 × 1 0 ‒ 3
0.9 1.80972919 1.80487258 4.856 × 1 0 ‒ 3
Table 4

Comparison of Haar wavelet solution with exact solution for Application 3 at J = 3

ν HWM Exact Error
0.1 0.99009901 0.99009900 9.544 × 1 0 ‒ 9
0.2 0.96153870 0.96153846 2.401 × 1 0 ‒ 7
0.3 0.91743329 0.91743119 2.101 × 1 0 ‒ 6
0.4 0.86207902 0.86206896 1.005 × 1 0 ‒ 5
0.5 0.80003134 0.80000000 3.134 × 1 0 ‒ 5
0.6 0.73536528 0.73529411 7.116 × 1 0 ‒ 5
0.7 0.67126510 0.67114093 1.241 × 1 0 ‒ 4
0.8 0.60992023 0.60975609 1.641 × 1 0 ‒ 4
0.9 0.55262038 0.55248618 1.341 × 1 0 ‒ 4
Table 5

Comparison of Haar wavelet solution with exact solution for Application 4 at J = 3

ν HWM Exact Error
0.1 1.00000100 1.00000100 0
0.2 1.00006400 1.00006400 0
0.3 1.00072900 1.00072900 0
0.4 1.00409600 1.00409600 0
0.5 1.01562500 1.01562500 0
0.6 1.04665600 1.04665600 0
0.7 1.11764900 1.11764900 0
0.8 1.26214400 1.26214400 0
0.9 1.53144100 1.53144100 0
Table 6

Maximum absolute error of Application 1

Resolution System of equations HWM error CPU time in seconds
J = 3 16 × 16 8.100 × 1 0 ‒ 4 1.124490
J = 4 32 × 32 2.090 × 1 0 ‒ 4 1.338281
J = 5 64 × 64 5.286 × 1 0 ‒ 5 1.728761
J = 6 128 × 128 1.328 × 1 0 ‒ 5 3.793212
J = 7 256 × 256 3.346 × 1 0 ‒ 6 15.148980
J = 8 512 × 512 8.562 × 1 0 ‒ 7 99.325444
Table 7

Maximum absolute error of Application 2

Resolution System of equations HWM error CPU time in seconds
J = 3 16 × 16 1.103 × 1 0 − 2 1.162713
J = 4 32 × 32 3.321 × 1 0 − 3 1.380484
J = 5 64 × 64 9.131 × 1 0 − 4 2.009864
J = 6 128 × 128 2.395 × 1 0 − 4 5.376645
J = 7 256 × 256 6.134 × 1 0 − 5 25.112066
J = 8 512 × 512 1.552 × 1 0 − 5 167.294925
Table 8

Maximum absolute error of Application 3

Resolution System of equations HWM error CPU time in seconds
J = 3 16 × 16 1.643 × 1 0 ‒ 4 1.294916
J = 4 32 × 32 6.644 × 1 0 ‒ 5 1.650631
J = 5 64 × 64 1.770 × 1 0 ‒ 5 2.602486
J = 6 128 × 128 4.410 × 1 0 ‒ 6 7.337785
J = 7 256 × 256 1.099 × 1 0 ‒ 6 35.188070
J = 8 512 × 512 2.748 × 1 0 ‒ 7 236.130223
Table 9

Maximum absolute error of Application 4

Resolution System of equations HWM error CPU time in seconds
J = 3 16 × 16 4.4409 × 1 0 ‒ 16 1.284342
J = 4 32 × 32 1.3323 × 1 0 ‒ 15 1.769604
J = 5 64 × 64 4.4409 × 1 0 ‒ 16 3.876561
J = 6 128 × 128 3.1086 × 1 0 ‒ 15 14.472351
J = 7 256 × 256 3.9968 × 1 0 ‒ 15 165.167635

7 Conclusion

We investigate the use of the Haar wavelet method as a valuable numerical technique to solve differential equations and perform various mathematical and scientific calculations in this study. We have also delved into the fascinating and challenging realm of PDDE, exploring their mathematical properties, analytical solutions, and numerical approximations. The complexity of these equations, combined with their wide-ranging applications, makes them a captivating subject for mathematicians, scientists, and engineers. The Haar wavelet collocation method leverages the simplicity and piecewise constant nature of Haar wavelets to approximate solutions to differential equations efficiently and accurately. It excels at handling problems with discontinuities and sharp changes in the solution, making it a suitable choice for various applications in science and engineering. In this study, we have demonstrated the method’s effectiveness.

In conclusion, the Haar wavelet collocation method is a valuable numerical analysis tool. Its simplicity, ability to handle discontinuities, and potential for accurate approximations make it a compelling choice for a broad spectrum of issues. The Haar wavelet collocation method, while not without its limitations, offers a promising path toward solving complex mathematical and scientific challenges. Its efficiency and accuracy in approximating solutions make it a valuable addition to the numerical methods available to researchers and practitioners.

  1. Funding information: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23028).

  2. Author contributions: P.G. spearheaded the investigation and coordinated the necessary literature. A.K. composed the manuscript, interpreted the findings, performed all numerical calculations, and generated the graphs. K.S.A. and B.S.A. compiled the data for tables, crafted the study site map, and formatted the final document. All authors have reviewed and consented to the published version of the manuscript.

  3. Conflict of interest: The article presented is declared to be free from any conflicts of interest by the researchers.

  4. Ethical approval: The conducted research is not related to either human or animals use.

  5. Data availability statement: Not applicable.

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Received: 2023-11-28
Revised: 2024-03-16
Accepted: 2024-05-11
Published Online: 2024-08-09

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2024-0023
Keywords for this article
Pantograph delay differential equations; Emden-Fowler-type equations; Haar wavelet method
Creative Commons
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  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
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  99. On the generalized Mellin integral operators
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  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
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