1 Introduction

Stiff systems of ordinary differential equations are a significant special case of the systems taken up in Initial Value Problems. There is no universally accepted definition of stiffness. Some attempts to understand stiffness examine the behaviour of fixed step size solutions of linear ordinary differential equations systems with constant coefficients [1]. Nonlinear ordinary stiff differential equations are generated by various physical systems, including nuclear reactors, laser oscillators, and others, and their magnitude of eigenvalues can fluctuate significantly. Additionally, the solutions to some differential equations can vary on different time scales, distinguishing them from other equations. These differential equations are often referred to as stiff. It is common for stiff problems to occur in fields like chemical kinetics, climatology, fluid dynamics, ionospheric physics, biochemistry, control theory, electronics, etc., when the equilibrium solution slowly changes while the transients rapidly decrease. Stability and accuracy are the two major issues associated with stiff systems. In some circumstances, the traditional single-step approaches are ineffective because a small step size can result in adequate round-off error to create instability. Despite equations may take longer time to compute. The essence of the difficulty is that when solving non-stiff problems, a step size small enough to provide the desired accuracy is small enough that the stability of the numerical method is qualitatively the same as that of the differential equations. For a stiff problem, step sizes that would provide an accurate solution with an explicit Runge–Kutta method must be reduced significantly to keep the computation stable. An equally important practical matter is that strategies with superior stability properties cannot be evaluated by simple iteration for non-stiff problems because the step size must be very small for the iteration to converge. In practice, stiffness of an initial value problem is depends on the stability of the problem, the length of the interval of integration, the stability of the numerical method, and how the method is implemented. Often the best way to proceed is to try one of the solvers intended for non-stiff systems. If it is unsatisfactory, the problem may be stiff [1].

Consider the general form of the system of stiff ODEs:

$$\left.\begin{array}{c}{y}^{\mathrm{^{\prime}}}\left(\xi \right)=F\left(y\left(\xi \right),z\left(\xi \right),\xi \right) \\ {z}^{\mathrm{^{\prime}}}\left(\xi \right)=G\left(y\left(\xi \right),z\left(\xi \right),\xi \right)\end{array}\right\}$$
(1.1)

where \(y\) and \(z\) are dependent variables, and \(\xi\) is an independent variable with primary constraints: \(y\left(0\right)=\alpha\) and\(z\left(0\right)=\beta\), where\(\alpha\),\(\beta \in {\mathbb{R}}\). A significant amount of research has been done on Stiff systems arising in the chemical field, including a solution to the stiff system of IVPs that converge quickly provided by Carrol [2]. Hojjati et al. insisted adaptive method called EBDF for the numerical solution of stiff systems [3], Cash JR implemented an extended backward differentiation formula for the stiff problems [4], Hosseini proposed matrix-free MEBDF method for the solution of stiff systems of ODEs [5], Hsiao insisted Haar wavelet approach to stiff linear systems [6], Bujurke NM proposed the numerical solution of stiff systems using single term Haar wavelet series [7]. In [8], Darvishi MT et al. illustrated the variational iteration method for the linear and nonlinear stiff problems. Badar G et al. implemented a semi-implicit mid-point rule for stiff systems [9], and Prothero A proposed the stability and accuracy of the one-step method for solving stiff systems [10]. But neither the computation effort nor the stability requirement has been significantly reduced by these methods. Here, we imposed the wavelet-based numerical method called the Taylor wavelet collocation scheme to overcome issues like the stability and accuracy of the systems.

Wavelets are built on Joseph Fourier's fundamental theory of superpositioning, which states that a collection of self-similar functions can express a complex function. With the help of wavelets, which are mathematical processes, data may be divided into different frequency components, and each component can then be analyzed with a resolution that matches its scale. Wavelet theory has recently attracted much interest because of its numerous valuable applications in system analysis, numerical analysis, and optimal control. A wavelet is an oscillation resembling a wave with an amplitude that begins at zero, increases or decreases, and then repeats one or more times. Wavelets have several characteristics that support their application in numerically solving differential equations. The orthogonal, compactly supported wavelet basis precisely approximates an increasingly higher-order polynomial. This wavelet-based representation of differential operations can be precise and stable even in areas with significant gradients or oscillations. Numerous dynamical system problems have extensively used approximate solutions using an orthogonal family of functions. Using truncated orthogonal functions to approximate the various signals in the equation, one can approximate the underlying differential equation using orthogonal functions. The functional integration matrix can then be used to eliminate the integral operations. Although the Taylor series and Fibonacci polynomials are not built on orthogonal functions, they have the operational integration matrix. Orthogonal wavelet basis also has the advantage of multi-resolution analysis over conventional techniques. For some of the common mathematical problems, various wavelet collocation techniques have been used, such as the Chebyshev wavelet collocation method [11], collocation method based on Bernoulli and Gegenbauer wavelets [12], Hermite wavelet collocation method [13, 14], Laguerre wavelet collocation method [15]. Several wavelet collocation techniques are typically employed to solve fractional differential equations, which include Haar wavelets [16], Chelyshkov wavelets [17], Fibonacci wavelets [18], Cubic B Spline [19], Chebyshev wavelets [20], Genocchi wavelets [21], Bernoulli wavelets [22,23,24], Legendre wavelet tau method [25], Hermite wavelets [26, 27], Legendre wavelets and Gegenbaur wavelets [28].

The current work's objective is to create a Taylor wavelets collocation method that is fast and simple. A recent addition to the wavelet families is the Taylor wavelets formed by Taylor polynomials. It guarantees the required accuracy for relatively small grid points to solve stiff systems. It is challenging to find the necessary approximations using a new numerical design.

The significance of the proposed method (TWCM) is as follows:

  • The number of terms of the Taylor polynomials \({T}_{m}(x)\) is less than the number of the terms of the other polynomials, say Bernoulli polynomials \({B}_{m}(x)\), Fibonacci polynomials \({F}_{m}(x)\), and Legendre polynomials \({L}_{m}\left(x\right).\) It helps to reduce the CPU time.

  • Error components in the FMI representing Taylor polynomials are less than that of other polynomials.

  • Taylor polynomials have less coefficient of individual terms than corresponding ones in other polynomials \(.\) Computational errors can be reduced using this property.

  • Taylor wavelets are superior wavelets not based on orthogonal polynomials, but we can express the Taylor polynomials in approximating orthogonal polynomials.

  • The Taylor wavelet method is suitable for solutions with sharp edge/ jump discontinuities.

  • Fractional differential equations, delay differential equations, and stiff systems can be solved using this method directly without using any control parameters.

  • The Taylor wavelet method can solve the higher-order system of ordinary differential equations by slightly modifying the method.

  • This method can be extended to PDEs and other mathematical models with different physical conditions.

  • It is used to obtain the solution of the differential equation in the universal domain by taking the suitable transformation.

Wavelet approaches for resolving differential and integral equations have recently received greater attention. Researchers start using this package to solve mathematical problems such as Bratu- type equations [29], fractional delay differential equations [30], systems of nonlinear fractional differential equations with application to human respiratory syncytial virus infection [31], Benjamin-Bona-Mohany PDEs [32], Burger’s equation [33], Biological models [34], linear and nonlinear Lane-Emden equations [35], Chemistry problems [36], reaction–diffusion problems in science and engineering [37, 38], Fisher’s equations [39], and nonlinear Parabolic equations [40]. Here, the Taylor wavelet collocation technique was successfully applied to the system, and significant approximation was obtained in the system’s solution. To our knowledge, no one solved these stiff systems by Taylor wavelets which motivates us to study this by the developed strategy. The absolute error with the Exact solution and ND Solver solution for numerous values of M and k are computed to validate the efficacy and precision of the developed strategy.

This article's structure is as follows: Sect. 2, named "Preliminaries" provides the definitions of wavelets. FMI of Taylor wavelets carried out in Sect. 3. The method of solution and application of the newly developed strategy is described in Sect. 4 and Sect. 5, respectively. Finally, Sect. 6 provides the conclusion of the article.

2 Preliminaries of Taylor wavelets and some theorems

2.1 Taylor wavelets

On the interval \([\mathrm{0,1}]\), the Taylor wavelets are defined as [29],

$$\k{S}_{n, m}\left(\xi \right)=\left\{\begin{array}{c}{2 }^{\frac{k -1}{2}} {T}_{m}\left({2}^{k-1} \xi -n+1\right), \quad \frac{n-1}{{2}^{ k-1}}\le \xi <\frac{n}{ {2}^{ k-1}}\\ \qquad \quad \qquad \quad 0, \qquad \qquad Otherwise\end{array}\right.$$

where

$${T}_{m}\left(\xi \right)=\sqrt{2m+1}\,\,{\xi }^{m},$$

where \({T}_{m}\left(\xi \right)\) is the normal Taylor polynomial of degree-\(m\), translation parameter \(n=1, 2, \dots , {2}^{k-1}\) and, \(k\) represents the level of resolution \(k=1, 2, \dots\) and respectively. The quantity \(\sqrt{{2m+1}}\) is a normalization factor. Taylor wavelets are compactly supported wavelets formed by Taylor polynomials over the interval [0,1].

Theorem 1.

Let \(H\) be a Hilbert space and \(W\) be a closed subspace of \(H\) such that \(\dim W < \infty\) and \(\{ w_{1} ,w_{2} ,...,w_{n} \}\) is any basis for \(W\). Let \(g\) be an arbitrary element in \(H\) and \(g_{0}\) be the unique best approximation to \(g\) out of \(W\). Then. [29]

\(\left\| {g - g_{0} } \right\|_{2} = G_{g}\), Where \(G_{g} = \left( {\frac{{Z(g,w_{1} ,w_{2} ,...w_{n} )}}{{Z(g,w_{1} ,w_{2} ,...w_{n} )}}} \right)^{\frac{1}{2}}\) and \(Z\) is introduced [29] as follows:

$$Z\,(g,\,w_{1} ,\,w_{2} ,...\,w_{n} )\,\, = \,\,\,\left| {\,\,\begin{array}{*{20}c} {\, < g,g > } & { < g,w_{1} > } & {...} & { < g,w_{n} > } \\ { < w_{1} ,g > } & { < w_{1} ,w_{1} > } & {...} & { < w_{1} ,w_{n} > } \\ {...} & {...} & {...} & {...} \\ { < w_{n} ,g > } & { < w_{n} ,w_{1} > } & {...} & { < w_{n} ,w_{n} > } \\ \end{array} \,\,} \right|$$

Theorem 2: Let \({L}^{2}[\mathrm{0,1}]\) be the Hilbert space generated by the Taylor wavelet basis. Let \(\eta \left(\xi \right)\) be the continuous bounded function in \({L}^{2}[\mathrm{0,1}]\). Then the Taylor wavelet expansion of \(\eta (\xi )\) converges with it.

Proof:

Let \(\eta :[\mathrm{0,1}]\to R\) be a continuous function and \(\left|\eta (\xi )\right|\le \mu\), where \(\mu\) be any real number. Then Taylor wavelet dilation of \(y(\xi )\) can be expressed as,

\(\eta \left(\xi \right)={\sum }_{n=1}^{{2}^{\frac{k -1}{2}}}{\sum }_{m=0}^{M-1}{ a}_{n,m}\,\k{S}_{n,m} (\xi )\).

\({a}_{n,m}-=\langle \eta ( \xi ),\,\k{S}_{n,m} ( \xi )\rangle\) Denotes inner product.

\({a}_{n,m}-={\int }_{0}^{1}\eta (\xi )\,\k{S}_{n,m}\left(\xi \right) d\xi\),

Since \(\k{S}_{n,m}\) are the orthogonal basis.

\({a}_{n,m}={\int }_{I}\eta \left(\xi \right){T}_{m}\left({2}^{k-1}\xi -n+ 1\right)d\xi\) where \(I=\left[\frac{n -1}{{2}^{k-1}}\right., \left.\frac{n}{{2}^{k-1}}\right)\)

Since \({T}_{m}\left(\xi \right)=\sqrt{2m+1} \, {\xi }^{m},\) we obtain.

\({a}_{n,m}={\int }_{I}\eta \left(\xi \right) \sqrt{2m+1} {\left({2}^{k-1}\xi -n+ 1\right)}^{m}d\xi\) where \(I=\left[\frac{n -1}{{2}^{k-1}}\right., \left.\frac{n}{{2}^{k-1}}\right)\)

Then substitute \({2}^{k-1}\xi -n+1=y\) then we get,

$${a}_{n,m}=\frac{{2}^{\frac{k-1}{2}}}{\sqrt{2m+1}}{\int }_{0}^{1}\eta \left(\frac{y + n-1}{{2}^{k-1}}\right) {y}^{m}\frac{dy}{{2}^{k-1}}$$
$${a}_{n,m} =\frac{{ 2 }^{\frac{1-k}{2}}}{\sqrt{2m+1}}{\int }_{.0}^{1}\eta \left(\frac{y + n-1}{{2 }^{k-1}}\right) {y}^{m} dy$$

By generalized mean value theorem,

\({a}_{n,m} =\frac{{2 }^{\frac{-k+1}{2}}}{\sqrt{2m+1}} \eta \left(\frac{\delta +n-1}{{2 }^{k- 1}}\right){\int }_{0}^{1} {y}^{m} dy\) for some \(\delta \in (\mathrm{0,1})\)

Since \({y}^{m}\) is a bounded continuous function. Put \({\int }_{0}^{1} {y}^{m} dy=h\)

$$\left|{a}_{n,m}\right| =\left|\frac{{2 }^{\frac{-k+1}{2}}}{\sqrt{2m+1}}\right|\left|\eta \left(\frac{\delta +n- 1}{{2}^{k-1}}\right)\right| h$$

Since \(\eta\) remains bounded.

Hence, \(\left|{a}_{n,m}\right| =\left|\frac{{2 }^{\frac{-k+1}{2}}}{\sqrt{2m+1}} \mu h\right|\) where \(\mu =\eta \left(\frac{\delta +n- 1}{{2}^{k-1}}\right)\)

Therefore, \({\sum }_{n,m=0}^{\infty }{a}_{n,m}\) is absolutely convergent. Hence the Taylor wavelet series expansion \(\eta (\xi )\) converges uniformly to it.

Remarks: Error estimation for continuous bounded function \(\eta (\xi )\) by using the above theorem 2.

\(\eta (\xi )\) is the exact solution and \({\eta }_{app}(\xi )\) is the Taylor wavelet approximation.

Where, \(\eta \left(\xi \right)={\sum }_{n=1}^{\infty }{\sum }_{m=0}^{\infty }{ a}_{n,m}\,\k{S}_{n,m} \left(\xi \right)\) and \({\eta }_{app}\left(\xi \right)={\sum }_{n=1}^{{2}^{\frac{k -1}{2}}}{\sum }_{m=0}^{M-1}{ a}_{n,m}\,\k{S}_{n,m} (\xi )\).

$$\eta \left(\xi \right)-{\eta }_{app}\left(\xi \right)={\sum }_{n=1}^{\infty }{\sum }_{m=0}^{\infty }{ a}_{n,m}\,\k{S}_{n,m} \left(\xi \right)-{\sum }_{n=1}^{{2}^{\frac{k -1}{2}}}{\sum }_{m=0}^{M-1}{ a}_{n,m}\,\k{S}_{n,m} \left(\xi \right)={\sum }_{n={2}^{k}}^{\infty }{\sum }_{m=M}^{\infty }{ a}_{n,m}\,\k{S}_{n,m} \left(\xi \right)$$

Now, \({\Vert {E}_{n}\Vert }^{2}={\Vert \eta \left(\xi \right)-{\eta }_{app}\left(\xi \right)\Vert }^{2}=\langle \eta \left(\xi \right)-{\eta }_{app}\left(\xi \right),\eta \left(\xi \right)-{\eta }_{app}\left(\xi \right)\rangle\)

$$\Rightarrow {\Vert {E}_{n}\Vert }^{2}=\underset{0}{\overset{1}{\int }}{\sum }_{n={2}^{k}}^{\infty }{\sum }_{m=M}^{\infty }{a}_{n,m }^{2}\,\k{S}_{n,m}^{2} \left(\xi \right)$$

But from the above theorem, we have, \(\left|{a}_{n,m}\right| \le \left|\frac{{2 }^{\frac{-k+1}{2}}}{\sqrt{2m+1}} \mu h\right|\)

$$\Rightarrow {\Vert {E}_{n}\Vert }^{2}\le \underset{0}{\overset{1}{\int }}{\sum }_{n={2}^{k}}^{\infty }{\sum }_{m=M}^{\infty }\frac{{2}^{-k+1}}{2m+1} {\mu }^{2} {h}^{2}\,\k{S}_{n,m}^{2} \left(\xi \right).$$

Theorem 3

[41]: Let the Taylor wavelet sequence \({\{{\lambda }_{n,m}^{k}(\xi )\}}_{k=1}^{\infty }\) which are continuous functions defined in \({L}^{2}({\mathbb{R}})\) in \(\xi\) on \([a,b]\) converges to the function \(\k{S}(\xi )\) in \({L}^{2}({\mathbb{R}})\) uniformly in \(\xi\) on \([a,b]\). Then \(\k{S}(\xi )\) is continuous in \({L}^{2}({\mathbb{R}})\) in \(\xi\) on \([a,b]\).

Theorem 4

[41]: Let the Taylor wavelet sequence \({\{{\lambda }_{n,m}^{k}(\xi )\}}_{k=1}^{\infty }\) converges itself in \({L}_{2}({\mathbb{R}})\) uniformly in \(\xi\) on \(\left[a,b\right]\). Then there is a function \(\k{S}(\xi )\) is continuous in \({L}_{2}({\mathbb{R}})\) in \(\xi\) on \(\left[a,b\right]\) and \(\underset{k\to \infty }{\mathit{lim}}{\lambda }_{n,m}^{k}\left(\xi \right)={\lambda }_{n,m}\left(\xi \right) \forall t\in \left[a,b\right]\).

Theorem 2

[23]: Let \(I\subset R\) be a finite interval with length \(m(I)\). Furthermore, \(f(\xi )\) is an integrable function defined on \(I\) and \(\sum_{i=0}^{M-1}\sum_{j=1}^{{2}^{k-1}}{a}_{i,j}\,{ \k{S}}_{i,j}(\xi )\) be a good Taylor wavelet approximation of \(f\) on \(I\) with for some \(\epsilon >0\), \(\left|f\left(\xi \right)-\sum_{i=0}^{M-1}\sum_{j=1}^{{2}^{k-1}}{a}_{i,j}\,{ \k{S}}_{i,j}(\xi )\right|\le \epsilon , \forall x \in I\). Then \(-\epsilon m\left(I\right)+{\int }_{I}\sum \sum {a}_{i,j} \,{\k{S}}_{i,j}(\xi ) d\xi \le\) \({\int }_{I}f\left(\xi \right)d\xi \le \epsilon m\left(I\right)+{\int }_{I}\sum \sum {a}_{i,j} {\,\k{S}}_{i,j}(\xi ) d\xi\).

This theorem says that when an integral of a complicated function is impossible, it can be evaluated by approximately the \(f(\xi )\) by wavelet functions in a given interval.

3 Functional matrix of integration (FMI)

At \(k=1\) and \(M=6\), the Taylor wavelet basis is obtained as below:

$$\k{S}_{\mathrm{1,0}}\left(\xi \right)=1$$
$$\k{S}_{\mathrm{1,1}}\left(\xi \right)=\sqrt{3}\xi$$
$$\k{S}_{\mathrm{1,2}}\left(\xi \right)=\sqrt{5}{\xi }^{2}$$
$$\k{S}_{\mathrm{1,3}}\left(\xi \right)=\sqrt{7}{\xi }^{3}$$
$$\k{S}_{\mathrm{1,4}}\left(\xi \right)=3{\xi }^{4}$$
$$\k{S}_{\mathrm{1,5}}\left(\xi \right)=\sqrt{11}{\xi }^{5}$$
$$\k{S}_{\mathrm{1,6}}\left(\xi \right)=\sqrt{13}{\xi }^{6}$$
$$\k{S}_{\mathrm{1,7}}\left(\xi \right)=\sqrt{15}{\xi }^{7}$$

Integrating the above first six basis regarding \(\xi\) limit from \(0\) to \(\xi\), and the Taylor wavelet bases are then stated as a linear combination as;

$$\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,0}}\left(\xi \right)d\xi =\left[\begin{array}{ccc}0& \frac{1}{\sqrt{3}}& \begin{array}{ccc}0& 0& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\right] \,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,1}}\left(\xi \right)d\xi =\left[\begin{array}{cccccc}0& 0& \frac{\sqrt{\frac{3}{5}} }{2}& 0& 0& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,2}}\left(\xi \right)d\xi =\left[\begin{array}{cccccc}0& 0& 0& \frac{\sqrt{\frac{5}{7}} }{3}& 0& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,3}}\left(\xi \right)d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& \frac{\sqrt{7}}{12}& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,4}}\left(\xi \right)d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& \frac{3}{5\sqrt{11}}\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,5}}\left(\xi \right)d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\mathop \smallint \limits_{0}^{{\xi ~}} \,\k{S}\left( \xi \right)~d\xi = {\hbox{B}\!\!\!|}_{{6 \times 6}} \,\k{S}_{6} \left( \xi \right) +\, \bar{\k{S}}_{6} \left( \xi \right)$$
(3.1)

where,

$$\k{S}_{6}\left(\xi \right)={[\k{S}_{\mathrm{1,0}}\left(\xi \right),\,{\mathrm{ \k{S}}}_{\mathrm{1,1}}\left(\xi \right), \,\k{S}_{\mathrm{1,2}}\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{1,3}}\left(\xi \right), \,\k{S}_{\mathrm{1,4}}\left(\xi \right),\,{\mathrm{ \k{S}}}_{\mathrm{1,5}}\left(\xi \right)]}^{T}$$
$${\hbox{B}\!\!\!|}_{6\times 6}=\left[\begin{array}{cccccc}0& \frac{1}{\sqrt{3}}& 0& 0& 0& 0\\ 0& 0& \frac{\sqrt{\frac{3}{5}} }{2}& 0& 0& 0\\ 0& 0& 0& \frac{\sqrt{\frac{5}{7}} }{3}& 0& 0\\ 0& 0& 0& 0& \frac{\sqrt{7}}{12}& 0\\ 0& 0& 0& 0& 0& \frac{3}{5\sqrt{11}}\\ 0& 0& 0& 0& 0& 0\end{array}\right] ,\,\bar{\k{S}}_{6}\left(\xi \right)=\left[\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ \frac{\sqrt{\frac{11}{13}} }{6}\,\k{S}_{\mathrm{1,6}}\left(\xi \right)\end{array}\end{array}\right]$$

The generalized functional matrix of integration of \(n\)-wavelet basis at \(k=1\) is defined as:

$$\underset{0}{\overset{\xi }{\int }}\,\k{S}\left(\xi \right)d\xi ={\hbox{B}\!\!\!|}_{n\times n}\,\k{S}\left(\xi \right)+\,\bar{\k{S}}_{n}\left(\xi \right),$$

where,

\({\hbox{B}\!\!\!|}_{{n \times n}} = \left[ {\begin{array}{*{20}c} 0 & {\frac{1}{{\sqrt 3 }}} & 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & {\frac{{\sqrt 3 }}{{2\sqrt 5 }}} & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & {\frac{{\sqrt 5 }}{{3\sqrt 7 }}} & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{\sqrt 7 }}{{12}}} & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{\sqrt {2\left( {n - 2} \right) + 1} }}{{\left( {n - 1} \right)\sqrt {2\left( {n - 2} \right) + 3} }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & {\frac{{\sqrt {2\left( {n - 1} \right) + 1} }}{{n\sqrt {2\left( {n - 1} \right) + 3} }}} \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 0 \\ \end{array} } \right]\) and \(\bar{\k{S}}_{n} \left( \xi \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \\ {\frac{{\sqrt {2n - 1} }}{{n\sqrt {2n + 1} }}\,\k{S}_{{1,{\text{n}}}} \left( \xi \right)} \\ \end{array} } \\ \end{array} } \right]\).

Integrating the basis stated above again, we attain the following;

$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,0}}\left(\xi \right) d\xi =\left[\begin{array}{cccccc}0& 0& \frac{1}{2\sqrt{5}}& 0& 0& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,1}}\left(\xi \right) d\xi =\left[\begin{array}{cccccc}0& 0& 0& \frac{1}{2\sqrt{21}}& 0& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,2}}\left(\xi \right) d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& \frac{\sqrt{5}}{36}& 0\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,3}}\left(\xi \right) d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& \frac{\sqrt{\frac{7}{11}} }{20}\end{array}\right]\,\k{S}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,4}}\left(\xi \right) d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0 \end{array}\right]\,{\mathrm{ \k{S}}}_{6}\left(\xi \right)$$
$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\,\k{S}_{\mathrm{1,5}}\left(\xi \right) d\xi =\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]{\mathrm{\k{S}}}_{6}\left(\xi \right)$$

Hence,

$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}\mathrm{ \k{S}}\left(\xi \right) d\xi ={\hbox{B}\!\!\!|}^{\prime}_{6\times 6} \k{S}_{6}\left(\xi \right)+{{{\bar {\k{S}}}}}^{\prime}_{6}\left(\xi \right)$$
(3.2)

where \({\hbox{B}\!\!\!|}^{\prime}_{6\times 6}=\left[\begin{array}{cccccc}0& 0& \frac{1}{2\sqrt{5}}& 0& 0& 0\\ 0& 0& 0& \frac{1}{2\sqrt{21}}& 0& 0\\ 0& 0& 0& 0& \frac{\sqrt{5}}{36}& 0\\ 0& 0& 0& 0& 0& \frac{\sqrt{\frac{7}{11}} }{20}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\), \(\bar{\k{S}}^{\prime}_{6} \left( \xi \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ {\frac{1}{{10\sqrt {13} }}~\k{S}_{{1,6}} \left( \xi \right)} \\ {\frac{{\sqrt {\frac{{11}}{{15}}} }}{{42}}~~\k{S}_{{1,7}} \left( \xi \right)} \\ \end{array} } \\ \end{array} } \right]\)

The Taylor wavelet basis is examined at \(k=2\) and \(M=6\) as follows:

$$\left.\begin{array}{c}\k{S}_{\mathrm{1,0}}\left(\xi \right)=\sqrt{2}\\ \k{S}_{\mathrm{1,1}}\left(\xi \right)=2\sqrt{6} \xi \\ {\mathrm{ \k{S}}}_{\mathrm{1,2}}\left(\xi \right)=4\sqrt{10}{ \xi }^{2}\\ {\mathrm{ \k{S}}}_{\mathrm{1,3}}\left(\xi \right)=8\sqrt{14}{ \xi }^{3}\\ \k{S}_{\mathrm{1,4}}\left(\xi \right)=48\sqrt{2}{ \xi }^{4}\\ \k{S}_{\mathrm{1,5}}\left(\xi \right)=32\sqrt{22} {\xi }^{5}\end{array}\right\}0\le \xi <\frac{1}{2}$$
$$\left.\begin{array}{c}\k{S}_{\mathrm{2,0}}\left(\xi \right)=\sqrt{2}\\ \k{S}_{\mathrm{2,1}}\left(\xi \right)=\sqrt{6}(-1+2 \xi )\\ \k{S}_{\mathrm{2,2}}\left(\xi \right)=\sqrt{10}{(1-2 \xi )}^{2}\\ \k{S}_{\mathrm{2,3}}\left(\xi \right)=\sqrt{14}{(-1+2 \xi )}^{3}\\ \k{S}_{\mathrm{2,4}}\left(\xi \right)=3\sqrt{2}{(1-2 \xi )}^{4}\\ \k{S}_{\mathrm{2,5}}\left(\xi \right)=\sqrt{22}{(-1+2 \xi )}^{5}\end{array}\right\} \frac{1}{2}~ \le \xi ~ < 1$$
$$\underset{0}{\overset{\xi }{\int }}\k{S}\left(\xi \right) d\xi ={\hbox{B}\!\!\!|}_{12\times 12}{\mathrm{ \k{S}}}_{12}\left(\xi \right)+\bar{\k{S}}_{12}\left(\xi \right)$$
(3.3)

where \(\k{S}_{12}\left(\xi \right)={[\k{S}_{\mathrm{1,0}}\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{1,1}}\left(\xi \right),\k{S}_{\mathrm{1,2}}\left(\xi \right),\k{S}_{\mathrm{1,3}}\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{1,4}}\left(\xi \right),\k{S}_{\mathrm{1,5}}\left(\xi \right), {\mathrm{ \k{S}}}_{\mathrm{2,0}}\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{2,1}}\left(\xi \right),\k{S}_{\mathrm{2,2}}\left(\xi \right),\k{S}_{\mathrm{2,3}}\left(\xi \right),\k{S}_{\mathrm{2,4}}\left(\xi \right),\k{S}_{\mathrm{2,5}}\left(\xi \right)]}^{T}\)

\({\hbox{B}\!\!\!|}_{12\times 12} =\left[\begin{array}{cccccccccccc}0& \frac{1}{2\sqrt{3}}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{\sqrt{\frac{3}{5}} }{4}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \frac{\sqrt{\frac{5}{7}} }{6}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\sqrt{7}}{24}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{3}{10\sqrt{11}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \frac{1}{2\surd 3}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{\frac{3}{5}} }{4}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{\frac{5}{7}} }{6}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{7}}{24}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{3}{10\sqrt{11}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]\), and \(\bar{\k{S}}_{12}\left(\xi \right)=\left[\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ \frac{\sqrt{\frac{11}{13}} }{12} \k{S}_{\mathrm{1,6}}\left(\xi \right)\\ 0\\ 0\\ 0\\ 0\\ 0\\ \frac{\sqrt{\frac{11}{13}} }{12}\k{S}_{\mathrm{2,6}}\left(\xi \right)\end{array}\end{array}\right]\)

The generalized first integration of \(n\)-wavelet basis at \(k=2\) is defined as:

$$\underset{0}{\overset{\xi }{\int }}\k{S}\left(\xi \right)d\xi ={\hbox{B}\!\!\!|}_{2n\times 2n} \k{S}\left(\xi \right)+\bar{\k{S}}_{2n}\left(\xi \right),$$

where

$${\hbox{B}\!\!\!|}_{2n\times 2n}=\left[\begin{array}{cccccccccccccccc}0& \frac{1}{2\sqrt{3}}& 0& 0& 0& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{\sqrt{\frac{3}{5}} }{4}& 0& 0& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \frac{\sqrt{\frac{5}{7}} }{6}& 0& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\sqrt{7}}{24}& \cdots & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \frac{\sqrt{2\left(n-2\right)+1}}{\left(2n-1\right)\sqrt{2\left(n-2\right)+3}}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{2\left(n-1\right)+1}}{2n\sqrt{2\left(n-1\right)+3}}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{2\sqrt{3}}& 0& 0& 0& \dots & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{\frac{3}{5}} }{4}& 0& 0& \dots & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{\frac{5}{7}} }{6}& 0& \dots & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{7}}{24}& \dots & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \frac{\sqrt{2\left(n-2\right)+1}}{\left(2n-1\right)\sqrt{2\left(n-2\right)+3}}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{2\left(n-1\right)+1}}{2n\sqrt{2\left(n-1\right)+3}}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$
$$\bar{\k{S}}_{2n}\left(\xi \right)=\left[\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ \vdots \\ \frac{\sqrt{2n-1}}{2n\sqrt{2n+1}} \k{S}_{1,n}\left(\xi \right)\\ 0\\ 0\\ 0\\ 0\\ 0\\ \vdots \\ \frac{\sqrt{2n-1}}{2n\sqrt{2n+1}}\k{S}_{2,n}\left(\xi \right)\end{array}\end{array}\right]$$

Similarly, the second integration can be written as:

$$\underset{0}{\overset{\xi }{\int }}\underset{0}{\overset{\xi }{\int }}{ \k{S}}\left(\xi \right) d\xi d\xi ={\hbox{B}\!\!\!|}^{\prime}_{12\times 12}{{ \k{S}}}_{12}\left(\xi \right)+k{{\bar {\k{S}}}}^{\prime}_{12}\left(\xi \right)$$
(3.4)

where \(\k{S}_{12}\left(\xi \right)={ [\k{S}_{\mathrm{1,0}}\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{1,1}}\left(\xi \right),\k{S}_{\mathrm{1,2}}\left(\xi \right),\k{S}_{\mathrm{1,3} }\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{1,4}}\left(\xi \right),\k{S}_{\mathrm{1,5}}\left(\xi \right), {\mathrm{ \k{S}}}_{\mathrm{2,0}}\left(\xi \right),{\mathrm{ \k{S}}}_{\mathrm{2,1}}\left(\xi \right),\k{S}_{\mathrm{2,2}}\left(\xi \right),\k{S}_{\mathrm{2,3}}\left(\xi \right),\k{S}_{\mathrm{2,4}}\left(\xi \right),\k{S}_{\mathrm{2,5}}\left(\xi \right)]}^{T}\)

$${\text{B}\!\!\!|}^{\prime}_{12\times 12} =\left[\begin{array}{cccccccccccc}0& 0& \frac{1}{8\sqrt{5}}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{8\sqrt{21}}& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\sqrt{5}}{144}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{\sqrt{\frac{7}{11}} }{80}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{8\sqrt{5}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{1}{8\sqrt{21}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{5}}{144}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& \frac{\sqrt{\frac{7}{11}} }{80}\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right] {{{\bar {\k{S}}}}{^{\prime}}}_{12}\left(\xi \right)=\left[\begin{array}{c}\begin{array}{c}0\\ 0\\ 0\\ 0\\ \frac{1}{40\sqrt{13}}\k{S}_{{1,6}}\left(\xi \right)\\ {\frac{\sqrt{\frac{11}{15}} }{168}\k{S}}_{{1,7}}\left(\xi \right)\\ 0\\ 0\\ 0\\ 0\\ \frac{1}{40\sqrt{13}}\k{S}_{{2,6}}\left(\xi \right)\\ \frac{\sqrt{\frac{11}{15}} }{168}\k{S}_{{2,7}}\left(\xi \right)\end{array}\end{array}\right],$$

Similarly, we can generate matrices for our convenience.

4 Taylor wavelet collocation method

In this part, the TWCM is used to solve a stiff system numerically as a SCODE (1.1).

Consider the subsequent nonlinear stiff system involving two equations of the state:

$$\left.\begin{array}{c} {y}^{\prime}\left(\xi \right)=F\left(y\left(\xi \right),z\left(\xi \right),\xi \right) \\ { z}^{\prime}\left(\xi \right)=G\left(y\left(\xi \right),z\left(\xi \right),\xi \right)\end{array}\right\}$$
(4.1)

With the primary constraints given by \(y\left(0\right)=\alpha\), \(z\left(0\right)=\beta\).

Assume that,

$$\left.\begin{array}{c}{y}^{\prime}\left(\xi \right) ={A}^{T}\k{S}\left(\xi \right)\\ {z}^{\prime}\left(\xi \right)={B}^{T}\k{S}\left(\xi \right)\end{array}\right\}$$
(4.2)

where,\({A}^{T}=[ {a}_{{1,0}},...{a}_{1, {\prime}M-1},{ a}_{{2,0}},\dots {a}_{2,M-1},{a}_{{2}^{k-1},0},\dots .{a}_{{2}^{k-1},M-1}],\)

$${B}^{T}=[{ b}_{{1,0}},\dots {b}_{1,M-1}, {b}_{{2,0}},\dots {b}_{2,M-1},{b}_{{2}^{k-1},0},\dots {b}_{{2}^{k-1},M-1}],$$

\(\k{S}\left(\xi \right)=[\k{S}(\xi {)}_{{1,0}},...\k{S}(\xi {)}_{1,M-1},\k{S}(\xi {)}_{{2,0}},...\k{S}(\xi {)}_{2,M-1},\k{S}(\xi {)}_{{2}^{k-1},0},...\k{S}(\xi {)}_{{2}^{k-1},M-1}]\)

Integrating the Eq. (4.2) regarding '\(\xi\)' from '\(0\)' to '\(\xi\).' We get.

\(y(\xi )\) =\(y(0)\) +\(\underset{0}{\overset{\xi }{\int }}A\) T \(\k{S}\left(\xi \right)d\xi\)

\(z(\xi )\) =\(z(0)\) +\(\underset{0}{\overset{\xi }{\int }}B\) T \(\k{S}\left(\xi \right)d\xi\)

Using Eq. (3.1) and primary constraints \(y\left(0\right)=\alpha\) and \(z\left(0\right)=\beta\) expressed in terms of \(\k{S}(\xi )\) by using the linear combination of Taylor wavelets we obtain,

$$\left. {\begin{array}{*{20}c} {y\left( \xi \right) = C^{T} ~\k{S}\left( \xi \right){\text{~}} + ~A^{T} \left[{\hbox{B}\!\!\!|}{~\k{S}\left( \xi \right)~ + ~\overline{\k{S}} \left( {\xi _{i} } \right)} \right]~~} \\ {z\left( \xi \right) = E^{T} \k{S}\left( \xi \right)~~~ + ~B^{T} \left[{\hbox{B}\!\!\!|}{~\k{S}\left( \xi \right) + \overline{\k{S}} \left( {\xi _{i} } \right)} \right]} \\ \end{array} } \right\}$$
(4.3)

where C and E are the known vectors. Now, substitute (4.2) and (4.3) in (4.1), we obtain

$$\left. {\begin{array}{*{20}c} {~~~A^{T} \k{S}\left( \xi \right) = F\left( {C^{T} ~\k{S}\left( \xi \right){\text{~}} + ~A^{T} \left[ {\k{S}{\hbox{B}\!\!\!|}\left( \xi \right)~ + ~\overline{\k{S}} \left( \xi \right)} \right]~,~~~E^{T} ~\k{S}\left( \xi \right)~~~ + ~B^{T} \left[ {\k{S}{\hbox{B}\!\!\!|}\left( \xi \right) + ~\overline{\k{S}} \left( \xi \right)~} \right]~,\xi } \right)~} \\ {B^{T} \k{S}\left( \xi \right) = G\left( {C^{T} ~\k{S}\left( \xi \right){\text{~}} + ~A^{T} \left[ {\k{S}{\hbox{B}\!\!\!|}\left( \xi \right)~ + ~~\overline{\k{S}} \left( \xi \right)} \right]~,~~~~~~E^{T} ~\k{S}\left( \xi \right)~~~ + ~B^{T} \left[ {\k{S}{\hbox{B}\!\!\!|}\left( \xi \right) + ~\overline{\k{S}} \left( \xi \right)~} \right]~,\xi } \right)} \\ \end{array} } \right\}$$
(4.4)

and collocate the obtained equations by the following collocation points \({\xi }_{i}=\frac{2i-1}{{2}^{k}M}, i=1, 2\dots M\).

$$\left.\begin{array}{c} {A}^{T}\k{S}\left({\xi }_{i}\right)=F\left({{C}^{T} \k{S}({\xi }_{i}) + A}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right) + \bar{\k{S}}({\xi }_{i})\right] , {{E}^{T} \k{S}\left({\xi }_{i}\right) + B}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right)+\bar{\k{S}}({\xi }_{i}) \right] ,{\xi }_{i}\right) \\ {B}^{T}\k{S}\left({\xi }_{i}\right)=G\left({{C}^{T} \k{S}\left({\xi }_{i}\right)+ A}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right) + \bar{\k{S}}({\xi }_{i})\right] ,{{ E}^{T} \k{S}\left({\xi }_{i}\right) + B}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right)+\bar{\k{S}}({\xi }_{i}) \right] ,{\xi }_{i}\right)\end{array}\right\}$$
(4.5)

Let \({f}_{i}\left({\xi }_{1},{\xi }_{2},\dots ,{\xi }_{i}\right)={A}^{T}\k{S}\left({\xi }_{i}\right)-F\left({{C}^{T} \k{S}({\xi }_{i}) + A}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right) + \bar{\k{S}}({\xi }_{i})\right] , {{E}^{T} \k{S}\left({\xi }_{i}\right) + B}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right)+\bar{\k{S}}({\xi }_{i}) \right] ,{\xi }_{i}\right)\) and

$${g}_{i}\left({\xi }_{1},{\xi }_{2},\dots ,{\xi }_{i}\right)={B}^{T}\k{S}\left({\xi }_{i}\right)-G\left({{C}^{T} \k{S}\left({\xi }_{i}\right)+ A}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right) + \bar{\k{S}}({\xi }_{i})\right] ,{{ E}^{T} \k{S}\left({\xi }_{i}\right) + B}^{T}\left[{\hbox{B}\!\!\!|} \k{S}\left({\xi }_{i}\right)+\bar{\k{S}}({\xi }_{i}) \right] ,{\xi }_{i}\right)$$

If the components of one iteration \({\xi }^{(l)}\in {\mathbb{R}}\) are known as: \({\xi }_{1}^{(l)},{\xi }_{2}^{(l)},\dots ,{\xi }_{i}^{(l)}\), then, the Taylor expansion of the first equations around these components is given by:

$${f}_{1}\left({\xi }_{1}^{\left(l+1\right)},{\xi }_{2}^{\left(l+1\right)},\dots ,{\xi }_{i}^{\left(l+1\right)}\right)=f\left({\xi }_{1}^{\left(l\right)},{\xi }_{2}^{\left(l\right)},\dots ,{\xi }_{i}^{\left(l\right)}\right)+{\left.\frac{\partial {f}_{1}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right)+{\left.\frac{\partial {f}_{1}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}\left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right)+\left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right)$$

and

$${g}_{1}\left({\xi }_{1}^{\left(l+1\right)},{\xi }_{2}^{\left(l+1\right)},\dots ,{\xi }_{i}^{\left(l+1\right)}\right)=g\left({\xi }_{1}^{\left(l\right)},{\xi }_{2}^{\left(l\right)},\dots ,{\xi }_{i}^{\left(l\right)}\right)+{\left.\frac{\partial {g}_{1}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right)+{\left.\frac{\partial {g}_{1}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}\left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right)+{\left.\frac{\partial g}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right)$$

Applying the Taylor expansion in the same manner for \({f}_{2},{f}_{3},\dots ,{f}_{i}\) and \({g}_{1},{g}_{3},\dots ,{g}_{i}\), we obtained the following system of linear equations, with the unknown being the components of the vector \({\xi }^{\left(l+1\right)}\):

\(\left(\genfrac{}{}{0pt}{}{\begin{array}{c}{f}_{1}\left({\xi }^{(l+1)}\right)\\ {f}_{2}\left({\xi }^{(l+1)}\right)\end{array}}{\begin{array}{c}\vdots \\ {f}_{i}\left({\xi }^{(l+1)}\right)\end{array}}\right)=\left(\genfrac{}{}{0pt}{}{{f}_{1}\left({\xi }^{(i)}\right)}{\begin{array}{c}{f}_{2}\left({\xi }^{\left(i\right)}\right)\\ \vdots \\ {f}_{i}\left({\xi }^{(i)}\right)\end{array}}\right)+\left(\begin{array}{cccc}{\left.\frac{\partial {f}_{1}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {f}_{1}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {f}_{1}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ {\left.\frac{\partial {f}_{2}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {f}_{2}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {f}_{2}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ \vdots & \vdots & \vdots & \vdots \\ {\left.\frac{\partial {f}_{i}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {f}_{i}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {f}_{i}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\end{array}\right) \left(\begin{array}{c}\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right)\\ \left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right)\\ \vdots \\ \left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right) \end{array}\right)\) and

$$\left(\genfrac{}{}{0pt}{}{\begin{array}{c}{g}_{1}\left({\xi }^{(l+1)}\right)\\ {g}_{2}\left({\xi }^{(l+1)}\right)\end{array}}{\begin{array}{c}\vdots \\ {g}_{i}\left({\xi }^{(l+1)}\right)\end{array}}\right)=\left(\genfrac{}{}{0pt}{}{{g}_{1}\left({\xi }^{(i)}\right)}{\begin{array}{c}{g}_{2}\left({\xi }^{\left(i\right)}\right)\\ \vdots \\ {g}_{i}\left({\xi }^{(i)}\right)\end{array}}\right)+\left(\begin{array}{cccc}{\left.\frac{\partial {g}_{1}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {g}_{1}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {g}_{1}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ {\left.\frac{\partial {g}_{2}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {g}_{2}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {g}_{2}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ \vdots & \vdots & \vdots & \vdots \\ {\left.\frac{\partial {g}_{i}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {g}_{i}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {g}_{i}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\end{array}\right) \left(\begin{array}{c}\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right)\\ \left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right)\\ \vdots \\ \left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right)\end{array}\right)$$

By setting the left-hand side to zero (which is desired value for the functions \({f}_{2},{f}_{3},\dots ,{f}_{i}\) and \({g}_{1},{g}_{3},\dots ,{g}_{i}\)) then, the system can write as:

$$\left(\begin{array}{cccc}{\left.\frac{\partial {f}_{1}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {f}_{1}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {f}_{1}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ {\left.\frac{\partial {f}_{2}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {f}_{2}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {f}_{2}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ \vdots & \vdots & \vdots & \vdots \\ {\left.\frac{\partial {f}_{i}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {f}_{i}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {f}_{i}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\end{array}\right) \left(\begin{array}{c}\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right)\\ \left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right)\\ \vdots \\ \left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right) \end{array}\right)=-\left(\genfrac{}{}{0pt}{}{{f}_{1}\left({\xi }^{(i)}\right)}{\begin{array}{c}{f}_{2}\left({\xi }^{\left(i\right)}\right)\\ \vdots \\ {f}_{i}\left({\xi }^{(i)}\right)\end{array}}\right)$$
(4.6)
$$\left(\begin{array}{cccc}{\left.\frac{\partial {g}_{1}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {g}_{1}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {g}_{1}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ {\left.\frac{\partial {g}_{2}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {g}_{2}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {g}_{2}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\\ \vdots & \vdots & \vdots & \vdots \\ {\left.\frac{\partial {g}_{i}}{\partial {\xi }_{1}}\right|}_{{\xi }^{\left(l\right)}}& {\left.\frac{\partial {g}_{i}}{\partial {\xi }_{2}}\right|}_{{\xi }^{\left(l\right)}}& \dots & {\left.\frac{\partial {g}_{i}}{\partial {\xi }_{i}}\right|}_{{\xi }^{\left(l\right)}}\end{array}\right) \left(\begin{array}{c}\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right)\\ \left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right)\\ \vdots \\ \left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right) \end{array}\right)=-\left(\genfrac{}{}{0pt}{}{{g}_{1}\left({\xi }^{(i)}\right)}{\begin{array}{c}{g}_{2}\left({\xi }^{\left(i\right)}\right)\\ \vdots \\ {g}_{i}\left({\xi }^{(i)}\right)\end{array}}\right)$$
(4.7)

Setting \(K={\left.\frac{\partial {f}_{i}}{\partial {\xi }_{j}}\right|}_{{\xi }^{\left(l\right)}}\) and \(J={\left.\frac{\partial {g}_{i}}{\partial {\xi }_{j}}\right|}_{{\xi }^{\left(l\right)}}\) , where \(K\) and \(J\) are Jacobian matrices for the above systems. Then, the above system can be written in matrix form as follows:

$$\left.\begin{array}{c} K \Delta x=-f \\ J \Delta x=-g \end{array}\right\}$$

where K and J are \(i\times i\) matrices, \(-f\) and \(-g\) are vectors of \(n\) components, and \(\Delta x\) is an \(n\)- dimensional vector with the components \(\left({\xi }_{1}^{\left(l+1\right)}-{\xi }_{1}^{\left(l\right)}\right), \left({\xi }_{2}^{\left(l+1\right)}-{\xi }_{2}^{\left(l\right)}\right), \left({\xi }_{i}^{(l+1)}-{\xi }_{i}^{(l)}\right)\).

Combining the above two systems of equations, we obtain \(\mathrm{Q \Delta }x=-p\), Where \(Q=K+L\) and \(p=f+g\). If \(Q\) is invertible, then the above system can be solved as follows:

$$\Delta x=-{Q}^{-1} p \Rightarrow {\xi }^{\left(l+1\right)}={\xi }^{\left(l\right)}-{K}^{-1}p$$

which provides the values of unknown Taylor wavelet coefficients. Substitute these coefficient values in (4.3) yields the TWCM numerical solution for the system.

5 Numerical results and discussion

Here, we considered a few examples to demonstrate the proposed method and outcomes are computed using the symbolic calculus programming language Mathematica.

Example 1:

Let us consider the nonlinear stiff system of the form: [42]

$$\left.\begin{array}{c} \frac{d{Y}_{1}\left(\xi \right)}{d\xi }=-1002 {Y}_{1}\left(\xi \right)+1000 {Y}_{2}^{2}(\xi )\\ \frac{d{Y}_{2}\left(\xi \right)}{d\xi }={Y}_{1}\left(\xi \right)-{Y}_{2}(\xi )(1+{Y}_{2}\left(\xi \right))\end{array}\right\}$$
(5.1)

The given initial condition are: \({Y}_{1}\left(0\right)=1, {Y}_{2}\left(0\right)=1.\) The exact solutions to the stiff chemical problem are \({Y}_{1}\left(\xi \right)={e}^{-2\xi }\) and \({Y}_{2}\left(\xi \right)= {e}^{\xi }\). The Taylor wavelet collocation method (TWCM) solutions, for example, 1, are shown in Figs. 1, 2, revealing that the proposed method solutions are reasonably close to the exact solution compared to existing methods such as Chebyshev polynomials (CMCP), Fibonacci wavelet collocation method (FWCM), Haar wavelet method and ND Solver. Numerical approximations obtained by the developed technique called TWCM and other current methods are compared with the exact solution. They are tabulated in Tables 1, 2, 3, and 4, and absolute errors of the developed approach with the exact solution are tabulated in Tables 1 and 3. It reveals that the errors obtained by the proposed method are better than those obtained using other existing techniques. TWCM solutions are calculated at diverse values of M and k. Also, by increasing the M and k, we get further precision in the result, which can be seen in Tables 1 and 3. It shows that increasing M and k can obtain a higher-order accuracy. Figures 1, 2, 3, and 4 depict all the graphical representations of numerical simulations and absolute error analysis. From the tables and graphs, it is clear that the TWCM method dominates all the other techniques in obtaining the numerical approximation and yields a satisfactory result for the desired system.

Fig. 1
figure 1

Visual interpretation of the result \({Y}_{1}(\xi )\) of example 1

Full size image
Fig. 2
figure 2

Visual interpretation of the result \({Y}_{2}(\xi )\) of example 1

Full size image
Table 1 Comparison of the solution \({Y}_{1}(\xi )\) obtained by TWCM at diverse values of \(M\) and \(k,\) for example, 1
Full size table
Table 2 Comparison of absolute error for the solution \({Y}_{1}(\xi )\) with different methods
Full size table
Table 3 \(\text{Comparison of the solution} \,\,{Y}_{2}(\xi )\,\,\text{obtained by TWCM at diverse values of} \,M \,\text{and}\,\,k,\) for example, 1
Full size table
Table 4 Comparison of absolute error for the solution \({Y}_{2}(\xi )\) with different methods
Full size table
Fig. 3
figure 3

Comparison of AE of the result \({Y}_{1}(\xi )\) with different techniques, for example, 1

Full size image
Fig. 4
figure 4

Comparison of AE of the result \({Y}_{2}(\xi )\) with different techniques, for example, 1

Full size image

Example 2:

Let us consider the following nonlinear stiff chemical problem of the form: [42]

$$\left.\begin{array}{c} {P}^{\prime}\left(\xi \right)=\delta P\left(\xi \right)+{Q}^{2}(\xi )\\ {Q}^{\prime}\left(\xi \right)=-Q(\xi )\end{array}\right\}$$
(5.2)

with the given initial condition are: \(P\left(0\right)=-\frac{1}{\delta +2}, Q\left(0\right)=1.\) The exact solutions to the stiff chemical problem are \(P\left(\xi \right)=\frac{{e}^{-2\xi }}{\delta +2}\) and \(Q\left(\xi \right)= {e}^{-\xi }\). The TWCM solutions are shown in Tables 5, 6, 7, and 8 and Figs. 5 and 6, revealing that the proposed method solutions are reasonably close to the exact solution compared to the current scheme, such as the Haar wavelet collocation method and ND Solver. Numerical approximations and absolute errors of the developed approach with the exact solution are tabulated in Tables 5, 6, 7, and 8. It is evident from Figs. 5, 6, 7 and 8 that the approximations obtained from TWCM come closer and closer to the exact solution. The errors obtained using the proposed method are better than other existing techniques. TWCM solutions are calculated at diverse values of M and k. Also, by increasing the values of M and k, we get more accuracy in the solution, which can be seen in Tables 5 and 7. It shows that increasing M and k can obtain a higher-order accuracy. Figures 5, 6, 7, and 8 depict all the graphical representations of numerical simulations and absolute error analysis. From the tables and graphs, it is clear that the TWCM dominates all the other techniques in obtaining the numerical approximation and yields a satisfactory result for the desired system.

Table 5 Comparison of the solution \(P(\xi )\) obtained by TWCM at diverse values of \(M\) and \(k,\) for example, 2
Full size table
Table 6 Comparison of absolute error for the solution \(P(\xi )\) with different methods
Full size table
Table 7 Comparison of the solution \(Q(\xi )\) obtained by TWCM at diverse values of \(M\) and \(k,\) for example, 2
Full size table
Table 8 Comparison of absolute error for the solution \(Q(\xi )\) with different methods
Full size table
Fig. 5
figure 5

Plot of the result \(P(\xi )\) of example 2

Full size image
Fig. 6
figure 6

Plot of the result \(Q(\xi )\) of example 2

Full size image
Fig. 7
figure 7

AE comparison of the result \(P(\xi )\) with different techniques, for example, 2

Full size image
Fig. 8
figure 8

Judgement of AE of the result \(Q(\xi )\) with different techniques, for example, 2

Full size image

Example 3:

Let us consider the following nonlinear stiff chemical problem of the form: [42]

$$\left.\begin{array}{c} {Z}_{1}^{\prime}\left(\xi \right)=-\alpha {Z}_{1}\left(\xi \right)-\beta {Z}_{2}\left(\xi \right)+\left(\alpha +\beta -1\right) {e}^{-\xi }\\ { Z}_{2}^{\prime}\left(\xi \right)=-\beta {Z}_{1}\left(\xi \right)-\alpha {Z}_{2}\left(\xi \right)+\left(\alpha -\beta -1\right) {e}^{-\xi }\end{array}\right\}$$
(5.3)

The given initial condition are: \({Z}_{1}\left(0\right)=1, {Z}_{2}\left(0\right)=1.\) The numerical and exact solutions are compared for various values of \(\xi\), and the absolute error between the analytical and approximative solutions at different resolutions (various k and M) is presented in Tables 9 and 11. Tables 10 and 12 provide the comparison of absolute errors of the TWCM with other methods, such as the Haar wavelet collocation method (HWCM) and the ND Solve method. Figures 9 and 10 provide the visual representation of the obtained solution by TWCM compared with other methods. The graphical representation of the absolute error of our suggested approach is carried out in Fig. 11 and 12. Tables and graphs confirm that by simply raising the k and M values, our suggested approach can generate more precise results. The findings shown in the tables and figures demonstrate the suggested approach's superior accuracy over the current numerical models. The tables and graphs show that the proposed approach converges more rapidly with the exact values than other methods, and the proposed method is a suitable and powerful tool to solve the above type of stiff chemical problem.

Table 9 Comparison of the solution \({Z}_{1}(\xi )\) obtained by TWCM at diverse values of \(M\) and \(k,\) for example, 3
Full size table
Table 10 Comparison of absolute error for the solution \({Z}_{1}(\xi )\) with different methods
Full size table
Table 11 Comparison of the solution \({Z}_{2}(\xi )\) obtained by TWCM at diverse values of \(M\) and \(k,\) for example, 3
Full size table
Table 12 Comparison of absolute error for the solution \({{\varvec{Z}}}_{2}({\varvec{\xi}})\) with different methods
Full size table
Fig. 9
figure 9

Plot of the solution \({Z}_{1}(\xi )\) of example 3

Full size image
Fig. 10
figure 10

Plot of the solution \({Z}_{2}(\xi )\) of example 3

Full size image
Fig. 11
figure 11

Comparison of AE of the solution \({Z}_{1}(\xi )\) with different techniques, for example 3

Full size image
Fig. 12
figure 12

Comparison of AE of the solution \({Z}_{2}(\xi )\) with different techniques, for example, 3

Full size image

Example 4:

Let us consider the following nonlinear stiff nuclear reactor model: [10]

$$\left.\begin{array}{c} {Z}_{1}^{\mathrm{^{\prime}}}\left(\xi \right)=0.01-\left(0.01+{Z}_{1}\left(\xi \right)+{Z}_{2}\left(\xi \right)\right) \left({1+(Z}_{1}\left(\xi \right)+1000)\left( {Z}_{1}\left(\xi \right)+1\right)\right) \\ {Z}_{2}^{\mathrm{^{\prime}}}\left(\xi \right)=0.01-\left(0.01+{Z}_{1}\left(\xi \right)+{Z}_{2}\left(\xi \right)\right)\left(1+{Z}_{1}{\left(\xi \right)}^{2}\right)\end{array}\right\}\left(5.4\right)$$

The given initial condition are: \({Z}_{1}\left(0\right)=0, {Z}_{2}\left(0\right)=0.\) This problem has no exact solution. We have solved this model with the Taylor wavelet collocation method at M = 6,7 and k = 1,2. Numerical approximations obtained from the TWCM and ND solve are tabulated in Tables 13 and 14 for \({Z}_{1}\left(\xi \right)\) and \({Z}_{2}\left(\xi \right)\), respectively. Also, the absolute error computed at different values of k and with the ND solve method (due to the non-availability of the exact solution for the model) are tabulated in the same tables. The numerical approximations are computed with the Haar wavelet collocation method, and the solutions are compared with the ND Solve method and the absolute error of the Haar method is also tabulated in Tables 13 and 14. The graphical representation of the solution \({Z}_{1}\left(\xi \right)\) and \({Z}_{2}\left(\xi \right)\) are carried out in Figs. 13 and 14. Also, a graphical representation of the absolute errors is drawn in Figs. 15 and 16. It is easy to see that the errors obtained using the proposed method are lesser than those obtained using the Haar wavelet method. Observing all the above-listed tables and graphs, one can conclude that the developed technique solutions are reasonably close to the ND Solve solution compared to the Haar wavelet collocation method. TWCM solutions are calculated at diverse values of M and k. Tables 13 and 14 reveal that after some stage (M) solution will remain the same; that is, the solution remains stable (never changes by increasing the values of M). The ND Solve solution and the TWCM solution are incredibly similar. Tables and Graphs confirm the above statement. It suggests that TWCM is a suitable method to solve the stiff models.

Table 13 Comparison of the solution \({Z}_{1}(\xi )\) obtained by TWCM at different values of \(M\) and \(k,\) for example 4
Full size table
Table 14 Comparison of the solution \({Z}_{2}(\xi )\) obtained by TWCM at diverse values of \(M\) and \(k,\) for example, 4
Full size table
Fig. 13
figure 13

Plot of the solution \({Z}_{1}(\xi )\) of example 4

Full size image
Fig. 14
figure 14

Plot of the solution \({Z}_{2}(\xi )\) of example 4

Full size image
Fig. 15
figure 15

Comparison of AE of the solution \({Z}_{1}(\xi )\) with different techniques, for example, 4

Full size image
Fig. 16
figure 16

Comparison of AE of the solution \({Z}_{2}(\xi )\) with different techniques, for example, 4

Full size image

6 Conclusion

In the present study, we used the Taylor wavelet collocation method (TWCM), which is not stiffness-sensitive and is used to analyze typical nonlinear systems originating in nonlinear dynamics with variable degrees/orders of stiffness. Here we have produced the Taylor wavelets' functional matrix of integration (FMI). The TWCM is implemented to attain the numerical approximation of the stiff chemical problems. The proposed method provides accurate solutions for stiff problems, whereas most numerical methods fail to give numerical solutions for stiff chemical problems. To enchase the developed method's stability and efficiency, familiar, complicated stiff systems are considered for illustration. Numerical values and absolute error of the technique in the tables and graphs strongly suggest that the proposed method is perfectly suitable to obtain the numerical solution of the stiff chemical systems. Also, tables and graphs reveal that the TWCM converges rapidly compared to other existing techniques, such as the Haar wavelet method (HWM), Chebyshev polynomials (CMCP), Fibonacci wavelet collocation method (FWCM), and ND Solver in the literature. The method is simple and takes less computational time.