Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

Abstract

Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

Introduction

In functional differential equations, the rate of change of unknown function not only depends upon the values of unknown functions at present time but also on previous time values. Delay differential equations also require information of the unknown function at different time levels including the previous levels also. Numerous applications of delay differential equations occur in mathematical modeling ¹ including chemical kinetics, the navigational control of ships, infectious diseases, and population dynamics.

Fractional calculus^2–8 has gained a considerable importance over the past few decades, due to the fact that it represents real-time physical models better and more accurately as compared with integer-order derivatives and integrals. Fractional delay differential equation is the generalization of delay differential equation. Over the past few decades, several researchers had devoted their study in finding numerical solutions of fractional delay differential equations due to non-availability of exact solutions in most of the cases. Therefore, different numerical and analytical methods^9–12 have been developed and applied for finding approximate solutions. Method of steps ¹³ is commonly used to convert a delay differential equation into an ordinary differential equation. The procedure of implementing method of steps is easy to understand and simple to use. In Falbo, ¹⁴ the method of steps is utilized to solve linear and nonlinear discrete delay differential equations with different types of delay.

Wavelet techniques are one of the relatively new techniques used for finding solutions of differential equations. Due to better accuracy of wavelets over other methods, a number of researchers are being attracted toward these techniques. The most common related schemes are harmonic wavelets of successive approximation, ¹⁵ Legendre wavelets,^16,17 cosine and sine (CAS) wavelets, ¹⁸ Haar wavelets, ¹⁹ and Chebyshev wavelets.^20,21 Chen and Hsiao ²² used Haar wavelets method for solving lumped and distributed parameter systems. Islam et al. ²³ used Haar wavelet collocation method for the numerical solutions of boundary layer fluid flow problems. Rawashdeh ¹⁷ implemented Legendre wavelets method to obtain solutions of fractional integro-differential equations. Razzaghi and Yousefi ²⁴ used Legendre wavelets method for constrained optimal control problems. E Babolian and FF Zadeh ²⁰ obtained numerical solutions of differential equations using Chebyshev wavelet operational matrix of integration.

Since these methods are newly developed, these methods had few shortcomings while dealing with nonlinear differential equations of fractional order. In this work, we propose a new method called Gegenbauer Wavelet Steps Method for solving nonlinear fractional delay differential equations and also show that it is strongly reliable method for such nonlinear problems than the other existing methods.

Preliminaries

Some basic definitions of fractional differentiation and integration are as follows:

1. Riemann–Liouville fractional integral operator of order $\alpha$ :

   Riemann–Liouville fractional-order integral of order $\alpha \in {\mathbb{R}}^{+}$ is defined as

   $$I_a^{\alpha }y\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{x}{\int }}{(x-t)}^{\alpha }y\left(t\right)\mathrm{dt}$$

   for $a<x\le b$ , and $I_a^{\alpha }$ becomes zero for $x=a$ .

2. Riemann–Liouville fractional derivative operator of order $\alpha$ :

   Riemann–Liouville fractional-order derivative of order $\alpha \in {\mathbb{R}}^{+}$ is defined as

   $$D_{R,a}^{\alpha }y\left(x\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}{\left(\frac{d}{\mathrm{dx}}\right)}^n\underset{a}{\overset{x}{\int }}{(x-t)}^{n-\alpha -1}y\left(t\right)\mathrm{dt}$$

   for $a<x\le b$ , where $n-1<\alpha <n$ , $n\in \mathbb{N}$ , and $n=[\alpha ]$ .

3. Caputo fractional derivative operator of order $\alpha$ :

   Caputo fractional-order derivative of order $\alpha \in {\mathbb{R}}^{+}$ is defined as

   $$D_{C,a}^{\alpha }y\left(x\right)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}\underset{a}{\overset{x}{\int }}{(x-t)}^{n-\alpha -1}{\left(\frac{d}{\mathrm{dt}}\right)}^{n}y\left(t\right)\mathrm{dt}$$

   for $a<x\le b$ , where $n-1<\alpha <n$ , $n\in \mathbb{N}$ , and $n=[\alpha ]$ . Also $D_{R,a}^{\alpha }y(x)$ and $D_{C,a}^{\alpha }y(x)$ become zero for $x=a$ .

4. Xiao-Jun Yang’s fractional derivative operator of order $\alpha$ :

   Xiao-Jun Yang’s fractional derivative of order $\alpha$ is defined as

   $$D_x^{\alpha }y\left(x_0\right)=y^{\left(\alpha \right)}\left(x_0\right)={\left(\frac{d^{\alpha }y}{d{x}^{\alpha }}\right)|}_{x=x_0}=\underset{x\to x_0}{lim}\frac{{\Delta }^{\alpha }(y\left(x\right)-y\left(x_0\right))}{{(x-x_0)}^{\alpha }}$$

   where ${\Delta }^{\alpha }(y(x)-y(x_0))\cong \mathrm{\Gamma }(1+\alpha )\Delta (y(x)-y(x_0))$ .

5. Ji-Huan He’s fractal derivative:

   According to Ji-Huan He, the fractal derivative can be defined as

   $$\frac{D^{\alpha }y\left(x\right)}{D{x}^{\alpha }}=\mathrm{\Gamma }(1+\alpha )\underset{\Delta x=x_1-x_2\to L}{\lim }\frac{y\left(x_1\right)-y\left(x_2\right)}{{(x_1-x_2)}^{\alpha }}$$

   where $\Delta x$ does not tend to zero.

Gegenbauer polynomials and Gegenbauer wavelets

Gegenbauer polynomials, or ultra spherical harmonic polynomials, $G_m^{\lambda }(x)$ , of order $m$ are defined for $\lambda >-(1/2)$ , m ∈  $[-1,1]$ , and are given by the following recurrence formulae

$$G_0^{\lambda }\left(x\right)=1,G_1^{\lambda }\left(x\right)=2\lambda x$$

$$G_{m+1}^{\lambda }\left(x\right)=\frac{1}{m+1}\left(2\left(m+\lambda \right)x{G}_m^{\lambda }\left(x\right)-\left(m+2\lambda -1\right)G_{m-1}^{\lambda }\left(x\right)\right),m=1,2,3,\mathrm{\dots }$$

Gegenbauer polynomials are orthogonal on $[-1,1]$ with respect to the weight function $w(x)=(1-x^2{)}^{(\lambda -(1/2))}$ as

$$\underset{-1}{\overset{1}{\int }}{(1-x^2)}^{\lambda -\frac{1}{2}}{G}_m^{\lambda }\left(x\right)G_n^{\lambda }\left(x\right)\mathrm{dx}=N_m^{\lambda }{\delta }_{\mathrm{mn}},\lambda >-\frac{1}{2}$$

where

$$N_m^{\lambda }=\frac{\pi 2^{1-2\lambda }\mathrm{\Gamma }(m+2\lambda )}{m!(m+\lambda ){\left(\mathrm{\Gamma }\left(\lambda \right)\right)}^2}$$

is the normalizing factor, and $\delta$ is the Kronecker delta function.

Legendre polynomials and Chebyshev polynomials are special cases of Gegenbauer polynomials. For $\lambda =0$ , we get Chebyshev polynomials of first kind; for $\lambda =1$ , we get Chebyshev polynomials of second kind; and for $\lambda =(1/2)$ , we get Legendre polynomials.

Mother wavelet (basic wavelet) is defined by the basis using scaling and translation parameters as

$${\psi }_{r,s}\left(x\right)=\frac{1}{\sqrt{\left|x\right|}}\psi \left(\frac{x-s}{r}\right),r,s\in \mathbb{R}\mathrm{and}r\ne 0$$

where r and s are the scaling and translation parameters, respectively.

If we restrict the parameters r and s to discrete values as $r_0^{-k}$ , $s=n{s}_0{r}_0^{-k}$ , where $r_0>1$ , $s_0>0$ , and $k,n\in \mathbb{N}$ , we have the following family of discrete wavelets

$${\psi }_{k,n}\left(x\right)={\left|r\right|}^{\frac{k}{2}}\psi (r_0^{-k}x-n{s}_0)$$

where ${\psi }_{k,n}(x)$ forms a wavelet basis for $L^2(R)$ . In particular, when $r_0=2$ and $s_0=1$ , then ${\psi }_{k,n}(x)$ forms an orthogonal basis.

Gegenbauer wavelets are defined on interval [0,1) by

$${\psi }_{n,m}^{\lambda }\left(x\right)=\{\begin{array}{cc}\frac{1}{\sqrt{N_m^{\lambda }}}{2}^{\frac{k}{2}}{G}_m^{\lambda }(2^{k}x-2n+1),\hfill & \frac{2n-2}{2^k}\le x\le \frac{2n}{2^k}\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{otherwise}\hfill \end{array}$$

(1)

where $k=1,2,3,\dots ,$ is the level of resolution, $n=1,2,3,\dots ,2^{k-1}$ is the translation parameter, and $m=0,1,2,3,\dots ,M-1$ is the order of Gegenbauer polynomials, $M>0$ .

In order to obtain the orthogonality of wavelets, Gegenbauer polynomials have to be dilated and translated using a weight function

$$w_{n,k}\left(x\right)={(1-{(2^{k}x-2n+1)}^2)}^{\lambda -\frac{1}{2}}$$

At fixed level of resolution say $k=a$ , we get

$$\{\begin{array}{cc}{w}_{1,a}\left(x\right),\hfill & 0\le x\le \frac{1}{2^{a-1}}\hfill \\ \multicolumn{1}{c}{w_{2,a}\left(x\right),}& \frac{1}{2^{a-1}}\le x\le \frac{2}{2^{a-1}}\hfill \\ \multicolumn{1}{c}{w_{3,a}\left(x\right),}& \frac{2}{2^{a-1}}\le x\le \frac{3}{2^{a-1}}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill \\ \multicolumn{1}{c}{w_{2^{a-1},a}\left(x\right),}& \frac{2^{a-1}-1}{2^{a-1}}\le x\le 1\hfill \end{array}$$

The solution of the nonlinear fractional delay differential equation can be expanded as a Gegenbauer wavelets series as follows

$$y\left(x\right)=\sum _{n=1}^{\infty }\sum _{m=0}^{\infty }{d}_{n,m}^{\lambda }{\psi }_{n,m}^{\lambda }\left(x\right)$$

where ${\psi }_{n,m}^{\lambda }(x)$ is given by equation (1). We approximate $y(x)$ by the truncated series

$$y_{k,M}\left(x\right)=\sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}{d}_{n,m}^{\lambda }{\psi }_{n,m}^{\lambda }\left(x\right)={\mathit{d}}^{{\lambda }^T}{\mathbf{\Psi }}^{\lambda }\left(\mathit{x}\right)$$

(2)

where ${\mathit{d}}^{{\lambda }^T}$ and ${\mathbf{\Psi }}^{\lambda }(\mathit{x})$ are matrices of order $2^{k-1}M\times 1$ , given as

$$\begin{array}{c}{\mathit{d}}^{{\lambda }^T}={[d_{10}^{\lambda },d_{11}^{\lambda },\dots ,d_{1M-1}^{\lambda },d_{20}^{\lambda },d_{21}^{\lambda },\dots ,d_{2M-1}^{\lambda },\dots ,d_{2^{k-1}0}^{\lambda },d_{2^{k-1}1}^{\lambda },\dots ,d_{2^{k-1}M-1}^{\lambda }]}^T\hfill \\ {\mathbf{\Psi }}^{\lambda }\left(\mathit{x}\right)={\left[\begin{array}{c}{\psi }_{1,0}^{\lambda }\left(x\right),{\psi }_{1,1}^{\lambda }\left(x\right),\dots ,{\psi }_{1,M-1}^{\lambda }\left(x\right),{\psi }_{2,0}^{\lambda }\left(x\right),{\psi }_{2,1}^{\lambda }\left(x\right),\dots ,{\psi }_{2,M-1}^{\lambda }\left(x\right),\dots ,\hfill \\ \multicolumn{1}{c}{{\psi }_{2^{k-1},0}^{\lambda }\left(x\right),{\psi }_{2^{k-1},1}^{\lambda }\left(x\right),\dots ,{\psi }_{2^{k-1},M-1}^{\lambda }\left(x\right)}\end{array}\right]}^T\hfill \end{array}$$

The collocation points for Gegenbauer wavelets are taken as $x_i=\cos (2(i+1)\pi /2^{k}M)$ , where $i=1,2,\dots ,2^{k-1}M$ .

Procedure of implementation

In the present word, we consider the delay differential equation of the form

$$y^{\alpha }\left(x\right)=f\left(y\right)+g\left(x\right)y(\frac{x}{a}-c),0<x<b\mathrm{and}1<\alpha \le 2$$

(3)

where $g(x)$ is a source term function, and $f(y)$ is a given continuous linear or nonlinear delay function.

According to the proposed method, first use the method of step to convert the delay differential equation (3) to inhomogeneous ordinary differential equation using initial function, $\varphi (x)$ , equation (3) implies

$$y^{\alpha }\left(x\right)=f\left(y\right)+g\left(x\right)\varphi (\frac{x}{a}-c),1<\alpha \le 2$$

(4)

Using series solution of Gegenbauer wavelets given in equation (2), we have

$$\begin{array}{l}\frac{d^{\alpha }}{d{x}^{\alpha }}{\displaystyle \sum _{n=1}^{2^{k-1}}{\displaystyle \sum _{m=0}^{M-1}{d}_{n,m}^{\lambda }{\psi }_{n,m}^{\lambda }\left(x\right)}}\\ =f\left({\displaystyle \sum _{n=1}^{2^{k-1}}{\displaystyle \sum _{m=0}^{M-1}{d}_{n,m}^{\lambda }{\psi }_{n,m}^{\lambda }\left(x\right)}}\right)+g\left(x\right)p\left(\frac{x}{a}-c\right)\end{array}$$

(5)

Solving system of equations given in equation (5), we get values of unknowns $d_{m,n}^{\lambda }$ , and replacing the values of these $d_{m,n}^{\lambda }$ , we get the required series solution of these nonlinear fractional delay differential equations.

Numerical problems

In this section, different nonlinear delay differential equations of fractional order including singular-type problems are considered and the developed method is implemented on them.

Problem 1

Consider fractional nonlinear pantograph equation ²⁵ is

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)+y\left(x\right)=5{\left(y\left(\frac{x}{2}\right)\right)}^2,1<\alpha \le 2$$

(6)

with initial conditions (ICs)

$$y\left(0\right)=1\mathrm{and}\frac{d}{\mathrm{dx}}y\left(0\right)=-2$$

The exact solution for $\alpha =2$ is

$$y\left(x\right)={\mathrm{e}}^{-2x}$$

Applying method of steps on equation (6), we have

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)+y\left(x\right)=5{\left(\varphi \left(\frac{x}{2}\right)\right)}^2,1<\alpha \le 2$$

(7)

Now considering $\varphi (x)={\mathrm{e}}^{-2x}$ , equation (7) can be written as

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)+y\left(x\right)=5{\left(e^{-x}\right)}^2,1<\alpha \le 2$$

(8)

Replacing value of $y(x)$ in the series solution of Gegenbauer wavelets, equation (8) becomes

$$\frac{d^{\alpha }}{d{x}^{\alpha }}\left(\sum _{n=1}^{2^{k-1}}\sum _{\omega =0}^{M-1}{c}_{n,\omega }{\psi }_{n,\omega }^{\lambda }\left(x\right)\right)+\left(\sum _{n=1}^{2^{k-1}}\sum _{\omega =0}^{M-1}{c}_{n,\omega }{\psi }_{n,\omega }^{\lambda }\left(x\right)\right)=5{\left(e^{-x}\right)}^2$$

(9)

Solving equation (9) using MAPLE, after simplification for $\lambda =(3/2)$ , M = 10, and $\alpha =2$ , we get the following series solution by Gegenbauer wavelet method

$$\begin{array}{c}\multicolumn{1}{c}{y\left(x\right)=0.99999999+x+9.591503\times {10}^{-30}{x}^2}\\ -1.000000x^3-9.491656\times {10}^{-29}{x}^4\hfill \\ +3.624266\times {10}^{-28}{x}^5-4.009744\times {10}^{-28}{x}^6\hfill \\ +1.50088\times {10}^{-28}{x}^7\hfill \end{array}$$

(10)

In Table 1, solution by Gegenbauer Wavelet Steps Method for different values of polynomials M is given when α = 2. In Table 1, $\mathrm{Error}{y}_{M=5}$ and $\mathrm{Error}{y}_{M=8}$ represent the error when M = 5 and 8, respectively.

Table 1.

Comparison of errors in solutions for different values of M.

$x$	Exact solution	$\mathrm{Error}{y}_{M=5}$	$\mathrm{Error}{y}_{M=8}$
0.0	1.00000000	0.00000E+00	1.20000E−29
0.1	1.09900000	0.00000E+00	1.00000E−29
0.2	1.19200000	0.00000E+00	1.00000E−29
0.3	1.27300000	0.00000E+00	1.00000E−29
0.4	1.33600000	0.00000E+00	1.00000E−29
0.5	1.37500000	0.00000E+00	1.00000E−29
0.6	1.38400000	1.20000E−14	1.00000E−29
0.7	1.35700000	1.30000E−14	1.00000E−29
0.8	1.28800000	1.60000E−14	2.00000E−29
0.9	1.17100000	1.90000E−14	2.00000E−29
1.0	1.00000000	1.00000E−15	1.50000E−29

Solution at different fractional values of $\alpha$ for equation (6) is given in Figure 1. According to figure, the solutions at different values of $\alpha$ converge toward the exact solution when $\alpha$ approaches integer value.

Figure 1.

Comparison of solutions for different fractional values of α for equation (6).

Problem 2

Consider the following fractional inhomogeneous nonlinear delay differential equation ²⁶

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)=\frac{8}{3}\frac{d}{dx}\left(y\left(\frac{x}{2}\right)\right)y\left(x\right)+8x^{2}y\left(\frac{x}{2}\right)-\frac{4}{3}-\frac{22}{3}x-7x^2-\frac{5}{3}{x}^3,1<\alpha \le 2$$

(11)

with boundary conditions (BCs)

$$y\left(0\right)=1\mathrm{and}y\left(1\right)=1$$

The actual solution for $\alpha =2$ is

$$y\left(x\right)=1+x-x^3$$

Applying method of steps on equation (11), we have

$$\begin{array}{l}\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)=\frac{8}{3}\frac{d}{dx}\left(\varphi \left(\frac{x}{2}\right)\right)y\left(x\right)\\ +8x^2\varphi \left(\frac{x}{2}\right)-\frac{4}{3}-\frac{22}{3}x-7x^2-\frac{5}{3}{x}^3\end{array}$$

(12)

Now considering $\varphi (x)=1+x-x^3$ , equation (12) can be written as

$$\begin{array}{l}\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)=\frac{8}{3}\frac{d}{dx}\left(1+\frac{x}{2}-\frac{x^3}{8}\right)y\left(x\right)\\ +8x^2\left(1+\frac{x}{2}-\frac{x^3}{8}\right)-\frac{4}{3}-\frac{22}{3}x-7x^2-\frac{5}{3}{x}^3\end{array}$$

(13)

Replacing value of $y(x)$ in the series solution of Gegenbauer wavelets, equation (13) becomes

$$\begin{array}{c}\frac{d^{\alpha }}{d{x}^{\alpha }}\left(\sum _{n=1}^{2^{k-1}}\sum _{\omega =0}^{M-1}{c}_{n,\omega }{\psi }_{n,\omega }^{\lambda }\left(x\right)\right)\hfill \\ =\frac{8}{3}\frac{d}{\mathrm{dx}}(1+\frac{x}{2}-\frac{x^3}{8})\left(\sum _{n=1}^{2^{k-1}}\sum _{\omega =0}^{M-1}{c}_{n,\omega }{\psi }_{n,\omega }^{\lambda }\left(x\right)\right)\hfill \\ +8x^2(1+\frac{x}{2}-\frac{x^3}{8})-\frac{4}{3}-\frac{22}{3}x-7x^2-\frac{5}{3}{x}^3\hfill \end{array}$$

(14)

Solving equation (14) using MAPLE, after simplification for $\lambda =(3/2)$ , M = 8, and $\alpha =2$ , we get the following series solution by Gegenbauer wavelet method

$$\begin{array}{c}\multicolumn{1}{c}{y\left(x\right)=1+x+9.591503\times {10}^{-30}{x}^2-1x^3}\\ -9.491656\times {10}^{-29}{x}^4+3.624266\times {10}^{-28}{x}^5\hfill \\ -4.00974\times {10}^{-28}{x}^6+1.5008\times {10}^{-28}{x}^7\hfill \\ +5.45812\times {10}^{-15}{x}^8-1.67685\times {10}^{-15}{x}^9\hfill \end{array}$$

(15)

In Table 2, solution by Gegenbauer Wavelets Steps Method for different values of polynomials $M$ is given when $\alpha =2$ . In Table 2, $\mathrm{Error}{y}_{M=5}$ and $\mathrm{Error}{y}_{M=8}$ represent the error when M = 5 and 8, respectively.

Table 2.

Comparison of errors in solutions for different values of M.

$x$	Exact solution	$\mathrm{Error}{y}_{M=5}$	$\mathrm{Error}{y}_{M=8}$
0.0	1.00000000	0.00000E+00	1.20000E−29
0.1	1.09900000	0.00000E+00	1.00000E−29
0.2	1.19200000	0.00000E+00	1.00000E−29
0.3	1.27300000	0.00000E+00	1.00000E−29
0.4	1.33600000	0.00000E+00	1.00000E−29
0.5	1.37500000	0.00000E+00	1.00000E−29
0.6	1.38400000	1.20000E−14	1.00000E−29
0.7	1.35700000	1.30000E−14	1.00000E−29
0.8	1.28800000	1.60000E−14	2.00000E−29
0.9	1.17100000	1.90000E−14	2.00000E−29
1.0	1.00000000	1.00000E−15	1.50000E−29

Solution at different fractional values of $\alpha$ for equation (11), when M = 5, is given in Figure 2. According to figure, the solutions at different values of $\alpha$ converge toward the exact solution when $\alpha$ approaches integer value.

Figure 2.

Comparison of solutions for different fractional values of α for equation (11).

Problem 3

Consider the following singular fractional pantograph equation

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)+\frac{8}{x^2}{\left(y\left(\frac{x}{2}\right)\right)}^2=y\left(x\right),1<\alpha \le 2$$

(16)

with conditions

$$y\left(0\right)=0\mathrm{and}\frac{d}{\mathrm{dx}}y\left(0\right)=1$$

Exact solution of equation (16) is $y(x)=x{e}^{-x}$ (Figure 3).

Figure 3.

Comparison of solutions for different fractional values of α for equation (16).

Applying method of steps on equation (16), we have

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)+\frac{8}{x^2}{\left(\varphi \left(\frac{x}{2}\right)\right)}^2=y\left(x\right),1<\alpha \le 2$$

(17)

Now considering $\varphi (x)=x{e}^{-x}$ , equation (17) can be written as

$$\frac{d^{\alpha }}{d{x}^{\alpha }}y\left(x\right)+\frac{8}{x^2}{\left(\frac{x}{2}{e}^{-\frac{x}{2}}\right)}^2=y\left(x\right),1<\alpha \le 2$$

(18)

Replacing value of $y(x)$ in the series solution of Gegenbauer wavelets, equation (18) becomes

$$\frac{d^{\alpha }}{d{x}^{\alpha }}\left({\displaystyle \sum _{n=1}^{2^{k-1}}{\displaystyle \sum _{\omega =0}^{M-1}{c}_{n,\omega }{\psi }_{n,\omega }^{\lambda }\left(x\right)}}\right)+\frac{8}{x^2}{\left(\frac{x}{2}{e}^{-\frac{x}{2}}\right)}^2=\left({\displaystyle \sum _{n=1}^{2^{k-1}}{\displaystyle \sum _{\omega =0}^{M-1}{c}_{n,\omega }{\psi }_{n,\omega }^{\lambda }\left(x\right)}}\right),1<\alpha \le 2$$

(19)

Solving equation (19) using MAPLE, after simplification for $\lambda =(3/2)$ , M = 15, and $\alpha =2$ , we get the following series solution by Gegenbauer wavelet method

$$\begin{array}{c}\multicolumn{1}{c}{y\left(x\right)=1.018288{10}^{-25}+0.99999999x}\\ -0.99999999x^2+0.499999999x^3\hfill \\ -0.16666666x^4+0.041666x^5\hfill \\ -0.008333x^6+0.001388x^7-0.000198x^8\hfill \\ +0.0000248x^9-0.00000275x^{10}\hfill \\ +2.753155\times {10}^{-7}{x}^{11}-2.480813\times {10}^{-8}{x}^{12}\hfill \\ +1.934897\times {10}^{-9}{x}^{13}-1.02209277\times {10}^{-10}{x}^{14}\hfill \end{array}$$

(20)

In Table 3, solution by Gegenbauer Wavelets Steps Method for different values of polynomials $M$ is given when $\alpha =2$ . In Table 3, $\mathrm{Error}{y}_{M=5}$ , $\mathrm{Error}{y}_{M=8}$ , and $\mathrm{Error}{y}_{M=15}$ represent the error when M = 5, 8, and 15, respectively.

Table 3.

Comparison of errors in solutions for different values of M.

$x$	Exact solution	$\mathrm{Error}{y}_{M=5}$	$\mathrm{Error}{y}_{M=8}$	$\mathrm{Error}{y}_{M=15}$
0.0	0.00000000000	1.56000E−15	1.54347E−16	1.01829E−25
0.1	0.09048374180	1.35055E−06	6.72620E−10	6.32190E−22
0.2	0.16374615061	1.90470E−05	3.23234E−10	6.24200E−22
0.3	0.22224546620	3.72001E−05	1.29264E−09	3.03900E−21
0.4	0.26812801841	2.42490E−05	1.11706E−09	2.46180E−21
0.5	0.30326532985	2.99359E−05	3.14194E−09	3.71720E−21
0.6	0.32928698165	8.11874E−05	3.72014E−09	2.87720E−21
0.7	0.34760971265	1.94195E−06	7.44202E−09	6.19150E−21
0.8	0.35946317129	4.44512E−04	1.51659E−08	3.13432E−20
0.9	0.36591269376	1.62681E−03	4.35801E−08	1.16828E−19
1.0	0.36787944117	4.06566E−03	8.36406E−07	1.08849E−17

Conclusion

In this work, proposed method, namely, Gegenbauer Wavelet Steps Method is developed for nonlinear fractional delay differential equations and is successfully implemented on three different nonlinear fractional delay differential equations including singular-type problems also. The developed method is simple and is easy to implement. The results obtained by the developed method are very encouraging. One important feature of the developed method is that by increasing the degree of Gegenbauer polynomials, the accuracy of the solution can be enhanced, which is verified in each considered problem shown in Tables 1–3. Since the results obtained are quite encouraging, the proposed method can be extended for other nonlinear fractional delay differential equations of various orders.