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Solution of third-order nonlinear integro-differential equations with parallel computing for intelligent IoT and wireless networks using the Haar wavelet method

  • Rohul Amin EMAIL logo , Muhammad Awais , Kamal Shah , Shah Nazir and Thabet Abdeljawad
Published/Copyright: November 18, 2024
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

We investigate a class of third-order nonlinear integro-differential equations (IDEs) with parallel computing of intelligent Internet of Things and wireless networks for numerical solutions. A numerical scheme based on the Haar wavelet has been established to compute the approximate solution for the problem under our consideration. By utilizing the mentioned tool, we discretize the involved derivatives and integrals. In this way, a sophisticated scheme is derived. Formulations for maximum root mean square and absolute errors have been given. Also, the convergent method has been discussed. In engineering, such as structural dynamics and control systems, third-order IDEs can improve modelling precision and solution effectiveness. Various examples have been testified by the aforementioned method. Additionally, by using different Gauss and collocation points (CPs), the aforementioned error terms were recorded. The convergence rate using distinct numbers of CPs has also been calculated, which is nearly equal to 2.

Keywords: nonlinear integro-differential equations; IoT; Gauss elimination method; Gauss and collocation points; Haar wavelet

1 Introduction

Integro-differential equations (IDEs) are problems involving both the integral and derivative of a function. Researchers have found a wide range of applications of the aforementioned area in engineering [1], astronomy [2], economics [3], mechanics [4], and fluid dynamics [5]. There are three categories of IDEs: Fredholm, Volterra, and Volterra-Fredholm. Researchers in applied mathematics are interested in nonlinear IDEs. Researchers have developed many numerical schemes to solve IDEs. Hashmi et al. [6] deduced the exact solution of Fredholm IDEs using the homotopy asymptotic technique. Chandel et al. [7] developed the numerical solution of higher order Volterra IDEs via the Legendary wavelet method. The authors reduced the problem to the nonlinear algebraic equation by utilizing the Legendary wavelets method. Hemeda [8] presented an iterative technique for nth-order IDEs. Majid [9] used a reliable algorithm for solving boundary value problems of higher order IDEs. Wang [10] presented an algorithm for the solution of higher order nonlinear Volterra–Fredholm IDEs with mechanization. Shang and Han [11] used a variational iteration technique for the solution of nth-order IDEs. In the aforesaid method, the problem has been changed into a system of ordinary IDEs and then investigated by using the variational iterative method. Hou and Yang [12] computed the solution of Fredholm IDEs by utilizing a hybrid function technique. The properties of Chebyshev polynomials and block pulse functions were used to reduce the IDEs to algebraic equations in the said method. Zarebnia and Nikpour [13] computed the solution of Volterra IDEs via the sin functions method. The authors have used the properties of the Sin-collocation technique to reduce the problem to the algebraic equation. Abubakar and Taiwo [14] applied an integral approximation collocation technique for the solution of higher order Fredholm–Volterra IDEs.

Davaeifar et al. [15] used the Bernstein polynomial scheme for the numerical solution of a higher order Fredholm ID-difference equation. The authors have described that the method depends on approximation by the truncated Bernstein series that converts the equations into a system of algebraic equations. Rashidinia and Tahmeasebi [16] computed approximate solution for IDEs by using a modified Taylor expansion technique. Macdonald and Ibrahim [17] presented treatment for higher order linear Fredholm IDEs of degenerated kernels. Atabakan et al. [18] used the spectral homotopy analysis scheme to investigate the solution of nonlinear Fredholm IDEs. Qualitative analysis for a nonlinear variable order problem was investigated by Khan et al. [19]. Khan et al. [20] established essential criteria for the existence of the solution to a class of hybrid problems. Ahmad et al. [21] studied the dynamical model with sliding mode control by using the numerical method based on the wavelet technique. Matinfar and Riahifar [22] calculated the analytic solution of nonlinear Volterra IDEs. In the aforementioned scheme, the nonlinear part of IDEs was approximated by the Adomian polynomials and reduced the equations into simple equations. Kady and Mahmoud [23] investigated the solution of Volterra and Fredholm IDEs through an optimal control approach. Nejafzadeh et al. [24] computed solution of the higher order IDEs by the variational iteration method. With the help of the aforesaid method, they converted higher order IDEs to a system of integral equations, which were then solved by the variational iteration method. During the solutions of differential equations, usually we reduce the system to some integral equations first. For instance, various dynamical systems have been treated by using the mentioned tools.

Keeping in mind the usefulness of the Haar wavelet collocation (HWC) scheme, we investigate the numerical solution of third-order nonlinear Volterra–Fredholm IDEs with parallel computing of intelligent Internet of Things (IoT) and wireless networks. Our considered problem can be described as follows:

(1) w ‴ ( x ) + a ( x ) w ″ ( x ) + b ( x ) w ′ ( x ) + c ( x ) w ( x ) = μ 1 ∫ a x k 1 ( x , s , w ( s ) , w ′ ( s ) , w ″ ( s ) ) d s + μ 2 ∫ a b k 2 ( x , s , w ( s ) , w ′ ( s ) , w ″ ( s ) ) d s + f ( x ) ,

with initial conditions (ICs)

(2) w ( 0 ) = α , w ′ ( 0 ) = β , w ″ ( 0 ) = γ ,

where k 1 and k 2 are kernels of integration and μ 1 and μ 2 are constants. In addition, a , b , c ∈ C [ 0 , T ] are coefficient functions, and f ∈ C [ 0 , T ] is a source function. The unknown function w ∈ C [ 0 , T ] is to be found out.

This article is organized as follows: Section 2 contains Haar functions details. The HWC method for numerical solution of third-order nonlinear IDEs is presented in Section 3. In Section 4, some problems from the existing literature are presented for validation of the HWC scheme. Conclusion is given in Section 6.

2 Haar wavelet

The scaling function [25] on [ a 1 , a 2 ) is

(3) h 1 ( x ) = 1 if x ∈ [ a 1 , a 2 ) , 0 elsewhere .

The mother wavelet on [ a 1 , a 2 ) is

(4) h 2 ( x ) = 1 if x ∈ a 1 , a 1 + a 2 2 , − 1 if x ∈ a 1 + a 2 2 , a 2 , 0 elsewhere .

The other terms except the scaling function in this series are represented as

(5) h i ( x ) = 1 if ξ 1 ≤ x < ξ 2 , − 1 if ξ 2 ≤ x < ξ 3 , 0 otherwise ,

where ξ 1 = a 1 + ϱ d ( a 2 − a 1 ) , ξ 2 = a 1 + ϱ + 0.5 d ( a 2 − a 1 ) , ξ 3 = a 1 + ϱ + 1 d ( a 2 − a 1 ) , d = 2 r , r = 0 , … , r ′ , and ϱ = 0 , … , d − 1 . The formula i = d + ϱ + 1 is used to obtain the value of number i . Any member of L 2 [ 0 , 1 ) is expressed as

(6) w ( x ) = ∑ k = 1 ∞ μ k h k ( x ) .

In Eq. (6), the approximation series is terminated at N terms

w ( x ) ≈ ∑ k = 1 N μ k h k ( x ) .

Using notation

(7) p i , 1 ( x ) = ∫ 0 x h i ( s ) d s

and

(8) p i , 1 ( x ) = x − ξ 1 if ξ 1 ≤ x < ξ 2 , ξ 3 − x if ξ 2 ≤ x < ξ 3 , 0 elsewhere .

Thus,

p i , 2 ( x ) = ∫ 0 x p i , 1 ( s ) d s ,

we have

(9) p i , 2 ( x ) = 1 2 ( x − ξ 1 ) 2 for ξ 1 ≤ x < ξ 2 , 1 4 m 2 − 1 2 ( ξ 3 − x ) 2 for ξ 2 ≤ x < ξ 3 , 1 4 m 2 for x ∈ [ ξ 3 , 1 ) , 0 elsewhere .

Also,

p i , 3 ( x ) = ∫ 0 x p i , 2 ( s ) d s ,

we obtain

(10) p i , 3 ( x ) = 1 6 ( x − ξ 1 ) 3 if ξ 1 ≤ x < ξ 2 , 1 4 m 2 ( x − ξ 2 ) − 1 6 ( ξ 3 − x ) 3 if ξ 2 ≤ x < ξ 3 , 1 4 m 2 ( x − ξ 2 ) if ξ 3 ≤ x < 1 , 0 elsewhere .

For the HWC method, the interval [ α , β ] is discretized by utilizing

(11) x m = α + ( β − α ) m − 1 ⁄ 2 2 M .

Eq. (11) is called collocation points (CPs). Gauss points (GPs) are represented as

G i = h 3 − 3 6 + i − 1 2 , G i + 1 = h 3 + 3 6 + i − 1 2 .

Some of the work using the HWC scheme can be found in previous studies [26–37].

3 Numerical method

In this section, we develop HWC scheme for approximate solution of Eq. (1). We use notation Θ = ∑ i = 1 N . Let w ‴ ( x ) ∈ L 2 [ 0 , 1 ) , so

(12) w ‴ ( x ) = Θ μ i h i ( x ) .

Integrating three times and using Eq. (2), we obtain

(13) w ″ ( x ) = γ + Θ μ i p i , 1 ( x ) ,

(14) w ′ ( x ) = β + γ x + Θ μ i p i , 2 ( x ) ,

(15) w ( x ) = α + β x + γ x 2 2 + Θ μ i p i , 3 ( x ) .

Eq. (15) is the approximate solution of Eq. (1).

Putting Haar approximations in Eq. (1), we obtain

Θ μ i h i ( x ) + a ( x ) ( γ + Θ μ i p i , 1 ( x ) ) + b ( x ) ( β + γ x + Θ μ i p i , 2 ( x ) ) + c ( x ) α + β x + γ x 2 2 + Θ μ i p i , 3 ( x ) = μ 1 ∫ a x k 1 ( x , t , γ + Θ μ i p i , 1 ( t ) , β + γ t + Θ μ i p i , 2 ( t ) , α + β t + γ t 2 2 + Θ μ i p i , 3 ( t ) d t + μ 2 ∫ a b k 2 ( x , t , γ + Θ μ i p i , 1 ( t ) , β + γ t + Θ μ i p i , 2 ( t ) , α + β t + γ t 2 2 + Θ μ i p i , 3 ( t ) d t + f ( x ) .

The integrals are approximated as

(16) ∫ a 1 a 2 f ( s ) d s ≈ a 2 − a 1 N ∑ m = 1 N f ( s m ) = ∑ m = 1 N f a 1 + ( a 2 − a 1 ) m − 0.5 N .

So

Θ μ i h i ( x ) + a ( x ) ( γ + Θ μ i p i , 1 ( x ) ) + b ( x ) ( β + γ x + Θ μ i p i , 2 ( x ) ) + c ( x ) α + β x + γ x 2 2 + Θ μ i p i , 3 ( x ) = μ 1 a − x N ∑ m = 1 N k 1 ( x , t m , γ + Θ μ i p i , 1 ( t m ) , β + γ t m + Θ μ i p i , 2 ( t m ) , α + β t m + γ t m 2 2 + Θ μ i p i , 3 ( t m ) + μ 2 b − a N ∑ m = 1 N k 2 ( x , t m , γ + Θ μ i p i , 1 ( t m ) , β + γ t m + Θ μ i p i , 2 ( t m ) , α + β t m + γ t m 2 2 + Θ μ i p i , 3 ( t m ) + f ( x ) .

After simplification and putting the CPs, we obtain

Θ μ i h i ( x j ) + ( γ + Θ μ i p i , 1 ( x j ) ) a ( x j ) + ( β + γ x j + Θ μ i p i , 2 ( x j ) ) b ( x j ) + α + β x j + γ x j 2 2 + Θ μ i p i , 3 ( x j ) c ( x j ) − μ 1 a − x j N ∑ m = 1 N k 1 ( x j , t m , γ + Θ μ i p i , 1 ( t m ) , β + γ t m + Θ μ i p i , 2 ( t m ) , α + β t m + γ t m 2 2 + Θ μ i p i , 3 ( t m ) − μ 2 b − a N ∑ m = 1 N k 2 ( x j , t m , γ + Θ μ i p i , 1 ( t m ) , β + γ t m + Θ μ i p i , 2 ( t m ) , α + β t m + γ t m 2 2 + Θ μ i p i , 3 ( t m ) − f ( x j ) = 0 .

Let

F x j = Θ μ i h i ( x j ) + ( γ + Θ μ i p i , 1 ( x j ) ) a ( x j ) + ( β + γ x j + Θ μ i p i , 2 ( x j ) ) b ( x j ) + α + β x j + γ x j 2 2 + Θ μ i p i , 3 ( x j ) c ( x j ) − μ 1 a − x j N ∑ m = 1 N k 1 ( x j , t m , γ + Θ μ i p i , 1 ( t m ) , β + γ t m + Θ μ i p i , 2 ( t m ) , α + β t m + γ t m 2 2 + Θ μ i p i , 3 ( t m ) − μ 2 b − a N ∑ m = 1 N k 2 ( x j , t m , γ + Θ μ i p i , 1 ( t m ) , β + γ t m + Θ μ i p i , 2 ( t m ) , α + β t m + γ t m 2 2 + Θ μ i p i , 3 ( t m ) − f ( x j ) .

This is nonlinear system with unknown μ i ’s and Broyden’s scheme for solution of this system is used. Putting these coefficients in Eq. (15), we obtain numerical solution of Eq. (1). For Broyden’s scheme, the Jacobian is required, and it is computed using partial derivatives; specifically, the Jacobian of the system is calculated as:

∂ F x j ∂ a k = h k ( x j ) + p k , 1 ( x j ) a ( x j ) + p k , 2 ( x j ) b ( x j ) + p k , 3 ( x j ) c ( x j ) − ∑ m = 1 N μ 1 a − x j N + μ 2 b − a N × ( p k , 1 ( t m ) + p k , 2 ( t m ) + p k , 3 ( t m ) ) .

4 Numerical examples

If w a p is used for numerical solution and w e x is used for analytical solution, then maximum absolute errors E 1 at CPs and E 2 at GPs are calculated as E 1 = max ∣ w a p c − w e x c ∣ and E 2 = max ∣ w a p g − w e x g ∣ . Root mean square error E 3 at CPs and E 4 at GP are E 3 = 1 N ∑ i = 1 N ∣ w a p c − w e x c ∣ 2 and E 4 = 1 N ∑ i = 1 N ∣ w a p g − w e x g ∣ 2 . Convergence rate R 1 at CPs and R 2 at GPs is defined as [38]: R 1 = log [ w a p c ( N ⁄ 2 ) ⁄ w a p c ( N ) ] log 2 and R 2 = log [ w a p g ( N ⁄ 2 ) ⁄ w a p g ( N ) ] log 2 .

Problem 1

Consider Fredholm IDE [39]:

(17) w ‴ ( x ) = − e x + ∫ − 1 1 e x − 2 t w 2 ( t ) d t ,

with ICs w ″ ( 0 ) = w ′ ( 0 ) = w ( 0 ) = 1 . Exact solution is w ( x ) = e x . We compute the convergence rate and mean square errors in Table 1 and present graphical illustration in Figure 1.

Table 1

Convergence rate and errors for Problem 1

J N = 2 J + 1 E 1 R 1 E 2 R 2 E 3 E 4
0 2 2.65641 × 1 0 ‒ 03 — 6.04560 × 1 0 ‒ 04 — 1.47984 × 1 0 ‒ 03 3.33883 × 1 0 ‒ 04
1 4 7.80322 × 1 0 ‒ 04 1.7673 1.69911 × 1 0 ‒ 04 1.8311 3.80282 × 1 0 ‒ 04 8.31493 × 1 0 ‒ 05
2 8 2.10711 × 1 0 ‒ 04 1.8888 4.50182 × 1 0 ‒ 05 1.9162 9.57222 × 1 0 ‒ 05 2.07661 × 1 0 ‒ 05
3 16 5.47051 × 1 0 ‒ 05 1.9455 1.15834 × 1 0 ‒ 05 1.9585 2.39711 × 1 0 ‒ 05 5.19040 × 1 0 ‒ 06
4 32 1.39343 × 1 0 ‒ 05 1.9731 2.93781 × 1 0 ‒ 06 1.9792 5.99531 × 1 0 ‒ 06 1.29750 × 1 0 ‒ 06
5 64 3.51604 × 1 0 ‒ 06 1.9866 7.39741 × 1 0 ‒ 07 1.9896 1.49900 × 1 0 ‒ 06 3.24371 × 1 0 ‒ 07
6 128 4.22161 × 1 0 ‒ 07 2.0581 4.57160 × 1 0 ‒ 08 2.0162 1.22261 × 1 0 ‒ 07 2.98872 × 1 0 ‒ 08
Figure 1 Exact and approximate solution comparison of Problem 1.
Figure 1

Exact and approximate solution comparison of Problem 1.

Problem 2

Consider

(18) w ‴ ( x ) = 1 16 ∫ 0 1 e x − 4 t w 2 ( t ) d t + 8 e 2 x − 1 16 e x ,

w ″ ( 0 ) = 4 , w ′ ( 0 ) = 2 , w ( 0 ) = 1 .

Exact solution of given problem is w ( x ) = e 2 x . We compute the convergence rate and mean square errors in Table 2. In addition, we present graphical illustration in Figure 2.

Table 2

Convergence rate and errors for Problem 2

J N = 2 J + 1 E 1 R 1 E 2 R 2 E 3 E 4
0 2 4.72660 × 1 0 ‒ 02 — 1.01470 × 1 0 ‒ 02 — 2.60900 × 1 0 ‒ 2 5.58900 × 1 0 ‒ 03
1 4 1.40901 × 1 0 ‒ 02 1.7461 2.82820 × 1 0 ‒ 03 1.8431 6.73444 × 1 0 ‒ 3 1.37621 × 1 0 ‒ 03
2 8 3.83512 × 1 0 ‒ 03 1.8774 7.48822 × 1 0 ‒ 04 1.9172 1.69724 × 1 0 ‒ 3 3.42872 × 1 0 ‒ 04
3 16 9.99672 × 1 0 ‒ 04 1.9397 1.92741 × 1 0 ‒ 04 1.9579 4.25161 × 1 0 ‒ 4 8.56451 × 1 0 ‒ 05
4 32 2.55142 × 1 0 ‒ 04 1.9701 4.88991 × 1 0 ‒ 05 1.9788 1.06342 × 1 0 ‒ 4 2.14071 × 1 0 ‒ 05
5 64 6.44474 × 1 0 ‒ 05 1.9851 1.23152 × 1 0 ‒ 05 1.9894 2.65892 × 1 0 ‒ 5 5.35142 × 1 0 ‒ 06
6 128 1.61383 × 1 0 ‒ 05 1.9977 3.08391 × 1 0 ‒ 06 1.9976 6.62672 × 1 0 ‒ 6 1.33562 × 1 0 ‒ 06
Figure 2 Exact and approximate solution comparison of Problem 2.
Figure 2

Exact and approximate solution comparison of Problem 2.

Problem 3

Next, we have the following nonlinear Volterra IDE:

(19) w ‴ ( x ) = 3 2 e x − 1 2 e 3 x + ∫ 0 x e x − t w 3 ( t ) d t ,

w ″ ( 0 ) = w ′ ( 0 ) = w ( 0 ) = 1 . Exact solution is w ( x ) = e x . We compute the convergence rate and mean square errors in Table 3. Also, we present graphical illustration in Figure 3.

Table 3

Convergence rate and errors for Problem 3

J N = 2 J + 1 E 1 R 1 E 2 R 2 E 3 E 4
0 2 2.25254 × 1 0 ‒ 03 — 5.67091 × 1 0 ‒ 04 — 1.27402 × 1 0 ‒ 03 3.14510 × 1 0 ‒ 04
1 4 6.48080 × 1 0 ‒ 04 1.7973 1.58932 × 1 0 ‒ 04 1.8352 3.26311 × 1 0 ‒ 04 7.84045 × 1 0 ‒ 05
2 8 1.72683 × 1 0 ‒ 04 1.9081 4.20332 × 1 0 ‒ 05 1.9188 8.20685 × 1 0 ‒ 05 1.95873 × 1 0 ‒ 05
3 16 4.44951 × 1 0 ‒ 05 1.9564 1.08053 × 1 0 ‒ 05 1.9598 2.05490 × 1 0 ‒ 05 4.89603 × 1 0 ‒ 06
4 32 1.12930 × 1 0 ‒ 05 1.9782 2.73901 × 1 0 ‒ 06 1.9800 5.14045 × 1 0 ‒ 06 1.22392 × 1 0 ‒ 06
5 64 2.85092 × 1 0 ‒ 06 1.9859 6.89500 × 1 0 ‒ 07 1.9900 1.28655 × 1 0 ‒ 06 3.05981 × 1 0 ‒ 07
6 128 7.22660 × 1 0 ‒ 07 1.9800 1.72981 × 1 0 ‒ 07 1.9949 3.22982 × 1 0 ‒ 07 7.64990 × 1 0 ‒ 08
Figure 3 Exact and approximate solution comparison of Problem 3.
Figure 3

Exact and approximate solution comparison of Problem 3.

Problem 4

Consider Volterra IDE:

(20) w ‴ ( x ) = 1 4 ( − 2 x + sin 2 x − 4 cos x ) + ∫ 0 x w 2 ( s ) d s ,

w ( 0 ) = w ″ ( 0 ) = 0 , and w ′ ( 0 ) = 1 . The above problem has exact solution w ( x ) = sin x . We compute the convergence rate and mean square errors in Table 4. Graphical illustration is shown in Figure 4.

Table 4

Convergence rate and errors for Problem 4

J N = 2 J + 1 E 1 R 1 E 2 R 2 E 3 E 4
0 2 2.72905 × 1 0 ‒ 04 — 4.63421 × 1 0 ‒ 05 — 1.47455 × 1 0 ‒ 04 2.48625 × 1 0 ‒ 05
1 4 8.02774 × 1 0 ‒ 05 1.7653 1.17992 × 1 0 ‒ 05 1.9737 3.70704 × 1 0 ‒ 05 5.31774 × 1 0 ‒ 06
2 8 2.18071 × 1 0 ‒ 05 1.8802 3.09080 × 1 0 ‒ 06 1.9326 9.28092 × 1 0 ‒ 06 1.27610 × 1 0 ‒ 06
3 16 5.68322 × 1 0 ‒ 06 1.9400 7.97540 × 1 0 ‒ 07 1.9544 2.32091 × 1 0 ‒ 06 3.15742 × 1 0 ‒ 07
4 32 1.44983 × 1 0 ‒ 06 1.9708 2.02941 × 1 0 ‒ 07 1.9745 5.80081 × 1 0 ‒ 07 7.87302 × 1 0 ‒ 08
5 64 3.65260 × 1 0 ‒ 07 1.9889 5.12081 × 1 0 ‒ 08 1.9866 1.44830 × 1 0 ‒ 07 1.96691 × 1 0 ‒ 08
6 128 9.07884 × 1 0 ‒ 08 2.0084 1.28621 × 1 0 ‒ 08 1.9933 3.60165 × 1 0 ‒ 08 4.91644 × 1 0 ‒ 09
Figure 4 Exact and approximate solution comparison of Problem 4.
Figure 4

Exact and approximate solution comparison of Problem 4.

Problem 5

Consider Volterra–Fredholm IDE

(21) w ‴ ( x ) + w 2 ( x ) = 1 12 ( 398 e 3 x + 18 x + e 3 + 12 e 6 x + 114 ) − ∫ 0 x w ( t ) d t − ∫ 0 1 w ( t ) d t ,

w ″ ( 0 ) = 9 , w ( 0 ) = 4 , and w ′ ( 0 ) = 3 . Exact solution is w ( x ) = e 3 x + 3 . In addition, the convergence rate and mean square errors are given in Table 5. Further, graphical presentation is given in Figure 5.

Table 5

Convergence rate and errors for Problem 5

J N = 2 J + 1 E 1 R 1 E 2 R 2 E 3 E 4
0 2 2.79070 × 1 0 ‒ 01 — 5.70800 × 1 0 ‒ 02 — 1.53191 × 1 0 ‒ 01 3.12333 × 1 0 0 ‒ 2
1 4 8.42825 × 1 0 ‒ 02 1.7273 1.58544 × 1 0 ‒ 02 1.8481 3.99682 × 1 0 ‒ 02 7.62382 × 1 0 ‒ 03
2 8 2.29983 × 1 0 ‒ 02 1.8737 4.20272 × 1 0 ‒ 03 1.9155 1.01024 × 1 0 ‒ 02 1.89592 × 1 0 ‒ 03
3 16 5.99442 × 1 0 ‒ 03 1.9398 1.08301 × 1 0 ‒ 03 1.9563 2.53285 × 1 0 ‒ 03 4.73371 × 1 0 ‒ 04
4 32 1.52961 × 1 0 ‒ 03 1.9705 2.74951 × 1 0 ‒ 04 1.9778 6.33754 × 1 0 ‒ 04 1.18300 × 1 0 ‒ 04
5 64 3.86750 × 1 0 ‒ 04 1.9837 6.92713 × 1 0 ‒ 05 1.9888 1.58563 × 1 0 ‒ 04 2.95742 × 1 0 ‒ 05
6 128 9.76821 × 1 0 ‒ 05 1.9852 1.73851 × 1 0 ‒ 05 1.9944 3.97461 × 1 0 ‒ 05 7.39361 × 1 0 ‒ 06
Figure 5 Exact and approximate solution comparison of Problem 5.
Figure 5

Exact and approximate solution comparison of Problem 5.

5 Discussion

The third-order derivative in IDEs has been approximated by the Haar series. The obtained results were then used to approximate the second- and first-order derivatives by integration. Using the HWC method and putting CPs in given nonlinear IDEs, we obtain a system of nonlinear equations. Broyden’s algorithm has been utilized for solution of nonlinear system. The solution at CPs and GPs is obtained by using these coefficients. Errors E 1 , E 2 , E 3 , and E 4 for distinct number of CPs and GPs are given in Table 1 for Problem 1, Table 2 for Problem 2, Table 3 for Problem 3, Table 4 for Problem 4, and Table 5 for Problem 5. Errors can be decreased by taking more discrete CPs and GPs. R 1 has also calculated, we see that R 1 is nearly equal to 2, which confirms the results of Majak et al. [38]. An important property of the HWC method, which has been observed from all tables, is that if we take more CPs and GPs, the accuracy increases. The initial guess for Broyden’s scheme was set to zero and iterations ended when the convergent criterion was satisfied. The comparison of exact and numerical solutions at N = 32 CPs for Problem 1 is shown in Figure 1, for Problem 2 in Figure 2, for Problem 3 in Figure 3, for Problem 4 in Figure 4, and for Problem 5 in Figure 5.

6 Conclusion

In this work, solution of nonlinear IDEs with parallel computing of intelligent IoT and wireless networks using HWC techniques has been discussed. E 1 , E 2 , E 3 , and E 4 errors were calculated for different numbers of CPs and GPs for each example to demonstrate the efficiency of our adopted numerical scheme. CP R 1 and R 2 at different CPs and GPs were calculated, which is nearly equal to 2. Numerical and exact solution comparison for 32 CPs has been given for each example in various figures. The numerical results were obtained using MATLAB program. The HWC method can be applied to higher order linear and nonlinear IDEs with different types of initial and boundary conditions. In addition, the aforementioned numerical scheme can be extended to various classes of IDEs with fractional orders involving different kinds of kernels in the future.

Acknowledgement

Kamal Shah and Thabet Abdeljawad would like to thank Prince Sultan University for APC and support through TAS research lab.

  1. Funding information: No funding is available to support this study.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. R.A., M.A., K.S., S.N., and T.A. wrote the main manuscript, R.A. and M.A. prepared the tables figures and reviewed the manuscript.

  3. Conflict of interest: No conflict of interest exists.

  4. Data availability statement: The data used in this research is included within the article.

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Received: 2024-01-17
Revised: 2024-08-24
Accepted: 2024-09-13
Published Online: 2024-11-18

© 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/nleng-2024-0039
Keywords for this article
nonlinear integro-differential equations; IoT; Gauss elimination method; Gauss and collocation points; Haar wavelet
Creative Commons
BY 4.0

Articles in the same Issue

  1. Editorial
  2. Focus on NLENG 2023 Volume 12 Issue 1
  3. Research Articles
  4. Seismic vulnerability signal analysis of low tower cable-stayed bridges method based on convolutional attention network
  5. Robust passivity-based nonlinear controller design for bilateral teleoperation system under variable time delay and variable load disturbance
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  17. Traveling wave solutions, numerical solutions, and stability analysis of the (2+1) conformal time-fractional generalized q-deformed sinh-Gordon equation
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