Introduction

Wavelets, known as very well-localized functions, a powerful and recognized tool used in image processing, quantum mechanics, signal processing, computer science and many more other areas. Wavelets are greatly useful for solving differential, fractional differential [1,2,3], integral, integro-differential and fractional Volterra integro-differential [4,5,6] equations and give accurate solutions. The wavelet technique allows the development of extremely fast algorithms when it is compared with the algorithms ordinarily used. Gu and Jiang [7] derived the Haar wavelets operational matrix of integration. Burgers and sine–Gordon equations in [8], nonlinear PDEs of fractional order in [9], Fisher’s equation in [10], Fitz Hugh-Nagumo equation in [11], Convection–diffusion equations in [12], film-pore diffusion model in [13], nonlinear parabolic equations in [14] nonlinear boundary value problems in [15], generalized Burgers-Huxley equation in [16] and magnetohydrodynamic flow equations in [17] were solved by Haar wavelet method. In the literature, special attention has been given to the applications of Legendre wavelets [18, 19]. The Legendre and Chebyshev wavelets operational matrixes of integration and product operation matrix have been introduced in [20, 21] and in [22,23,24] respectively. These matrices can be used to solve problems such as identification, analysis and optimal control. Fredholm integral equations of the first kind in [25], fractional differential equations in [26], nonlinear fractional integrodifferential equations in [27], one dimensional heat equation in [28], Bratu’s problem in [29] and water quality assessment model problem in [30] were solved by Chebyshev wavelet method. Çelik [31,32,33] solved differential equations, generalized Burgers-Huxley equation and Free vibration problems of non-uniform Euler–Bernoulli beam by Chebyshev wavelet collocation method. Gegenbauer wavelets have been introduced to solve numerically the Abel’s integral equation in [34]. Fractional-order differential equations in [35,36,37], Lane–Emden type differential equations in [38, 39] and various 2nth-order initial and boundary value problems in [40, 41] were solved by Gegenbauer wavelets.

Operational matrices of rth integration of Gegenbauer wavelets have been presented first time in this study. Proposed method has been applied to the nonlinear partial differential equation called as generalized Kuramoto–Sivashinsky (GKS) equation

$$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + \alpha \frac{{\partial^{2} u}}{{\partial x^{2} }} + \beta \frac{{\partial^{3} u}}{{\partial x^{3} }} + \gamma \frac{{\partial^{4} u}}{{\partial x^{4} }} = 0 $$
(1)

where α, β and γ are nonzero real constants. It is noteworthy that the GKS equation retains the fundamental elements of any nonlinear process that involves wave evolution: the simplest possible nonlinearity uux, instability and energy production uxx, stability and energy dissipation uxxxx and dispersion uxxx. In the context of thin-film flows, the terms uxx, uxxx and uxxxx are due to the interfacial kinematics associated with inertia, viscosity and surface tension, respectively, with the corresponding parameters α, β and γ all positive and measuring the relative importance of these effects [42].

We consider nonlinear partial differential equations of the form

$$ \dot{u}(x,t) = F(u,u^{{\prime }} ,u^{{\prime \prime }} , \ldots ,u^{(r)} ) $$

The quasilinearizations of these equations give a set of recurrence linear differential equations

$$ \dot{u}_{s + 1} (x,t) = F(u_{s} ,u_{s}^{{\prime }} ,u_{s}^{{\prime \prime }} , \ldots ,u_{s}^{(r)} ) + \sum\limits_{i = 0}^{r} {\left( {u_{s + 1}^{(i)} - u_{s}^{(i)} } \right)} \,F_{{u_{s}^{(i)} }} (u_{s} ,u_{s}^{{\prime }} ,u_{s}^{{\prime \prime }} , \ldots ,u_{s}^{(r)} ) $$
(2)

where \( F_{{u_{s}^{(i)} }} (u_{s} ,u_{s}^{{\prime }} ,u_{s}^{{\prime \prime }} , \ldots ,u_{s}^{(r)} ) = \frac{\partial }{{\partial u_{s}^{(i)} }}\left( {F(u_{s} ,u_{s}^{{\prime }} ,u_{s}^{{\prime \prime }} , \ldots ,u_{s}^{(r)} )} \right) \), \( \dot{u}(x,t) = \frac{\partial u(x,t)}{\partial t} \), \( u^{{\prime }} (x,t) = \frac{\partial u(x,t)}{\partial x} \) and \( u_{0} (x,t) \) is taken as a function satisfying initial/boundary conditions [43].

The method is based on the approximation by the truncated Gegenbauer wavelets series. By using the Chebyshev collocation points, algebraic equation system has been obtained. Solving this algebraic equation system, the coefficients of the Gegenbauer wavelet series can be found. Hence, we have the implicit form of the approximate solution of nonlinear partial differential equations. This method is applied to the three generalized Kuramoto–Sivashinsky (GKS) equations using quasilinearization technique. Calculations demonstrated that the accuracy of the Gegenbauer wavelet collocation method is quite high even in the case of a small number of grid points.

Gegenbauer (Ultraspherical) Polynomials

Gegenbauer polynomials [44], or ultra-spherical harmonics polynomials of order \( m \in {\rm Z}^{ + } \) are defined as \( C_{m}^{\lambda } (x) \) for \( \lambda > - \frac{1}{2} \) on the interval \( [ - 1,\,\,1] \) and given by the following recurrence formulae,

$$ C_{0}^{\lambda } (x) = 1,\quad C_{1}^{\lambda } (x) = 2\lambda x, $$
$$ (m + 1)C_{m + 1}^{\lambda } (x) = 2(m + \lambda )xC_{m}^{\lambda } (x) - (m + 2\lambda - 1)C_{m - 1}^{\lambda } (x),\quad m = 1,2,3, \ldots $$

These polynomials are also given by the generating function

$$ \frac{1}{{(1 - 2xt + t^{2} )^{\lambda } }} = \sum\limits_{m = 0}^{\infty } {C_{m}^{\lambda } (x)t^{m} } . $$
(3)

The following relations of Gegenbauer polynomials can be derive by using generating function.

$$ \frac{d}{dx}\left( {C_{m}^{\lambda } (x)} \right) = 2\lambda C_{m - 1}^{\lambda + 1} (x),\quad m \ge 1. $$
(4)
$$ (m + \lambda )C_{m}^{\lambda } (x) = \lambda \left( {C_{m}^{\lambda + 1} (x) - C_{m - 2}^{\lambda + 1} (x)} \right),\quad m \ge 2. $$
(5)
$$ \frac{d}{dx}\left( {C_{m + 1}^{\lambda } (x) - C_{m - 1}^{\lambda } (x)} \right) = 2\lambda \left( {C_{m}^{\lambda + 1} (x) - C_{m - 2}^{\lambda + 1} (x)} \right) = 2(m + \lambda )C_{m}^{\lambda } (x) $$
(6)

By integration of the Eq. (6) from − 1 to x, the following relation can be obtained

$$ \int\limits_{ - 1}^{x} {C_{m}^{\lambda } (t)} dt = \frac{1}{2(m + \lambda )}\left( {C_{m + 1}^{\lambda } (x) - C_{m - 1}^{\lambda } (x) - C_{m + 1}^{\lambda } ( - 1) + C_{m - 1}^{\lambda } ( - 1)} \right),\quad m \ge 1. $$

The equation given as:

$$ \int {(1 - x^{2} )^{{\lambda - \frac{1}{2}}} C_{m}^{\lambda } (x)dx} = - \frac{{2\lambda (1 - x^{2} )^{{\lambda + \frac{1}{2}}} }}{m(m + 2\lambda )}C_{m - 1}^{\lambda + 1} (x),\quad m \ge 1 $$
(7)

can be obtained from the Rodrigues formula. Gegenbauer polynomials satisfy the following relations

$$ C_{m}^{\lambda } ( - 1) = \frac{{( - 1)^{m} \varGamma (m + 2\lambda )}}{m!\varGamma (2\lambda )},\quad C_{m}^{\lambda } (1) = \frac{\varGamma (m + 2\lambda )}{m!\varGamma (2\lambda )} $$

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

$$ \int\limits_{ - 1}^{1} {(1 - x^{2} )^{{\lambda - \frac{1}{2}}} C_{m}^{\lambda } (x)C_{n}^{\lambda } (x)dx = \left\{ {\begin{array}{*{20}l} {L_{m}^{\lambda } ,} \hfill & {m = n} \hfill \\ {0,} \hfill & {m \ne n} \hfill \\ \end{array} } \right.} $$

where

$$ L_{m}^{\lambda } = \left\{ {\begin{array}{*{20}l} {\frac{{\pi 2^{1 - 2\lambda } \varGamma (m + 2\lambda )}}{{m!(m + \lambda )(\varGamma (\lambda ))^{2} }},} \hfill & {\lambda \ne 0} \hfill \\ {\frac{2\pi }{{m^{2} }},} \hfill & {\lambda = 0} \hfill \\ {\pi ,} \hfill & {\lambda = 0,m = 0} \hfill \\ \end{array} } \right. $$
(8)

is the normalizing factor.

Gegenbauer polynomials are generalized forms of the Legendre and Chebyshev polynomials. For λ = 0, λ = 1, and \( \lambda = \frac{1}{2} \), we can get first kind Chebyshev polynomials as:

$$ T_{m} (x) = \frac{m}{2}\mathop {\lim }\limits_{\lambda \to 0} \frac{{C_{m}^{\lambda } (x)}}{\lambda }\quad (m \ge 1), $$

second kind Chebyshev polynomials as:

$$ U_{m} (x) = C_{m}^{1} (x)\,\,\, $$

and Legendre polynomial as:

$$ L_{m} (x) = C_{m}^{{\tfrac{1}{2}}} (x) $$

respectively.

Gegenbauer (Ultraspherical) Wavelet Method

Wavelets consist of a family of functions coming from dilation and translation of a single function named the mother wavelet. If a as a dilation parameter and b as translation parameter vary continuously, the following family of continuous wavelets may be obtained [45].

$$ \psi_{a,b} (x) = \left| a \right|^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \psi \left( {\frac{x - b}{a}} \right),\quad a,b \in R,\quad a \ne 0. $$
(9)

Gegenbauer wavelets are written as

$$ \psi_{nm} (x) = \psi \,(k,n,m,x). $$

where \( k = 0,1,2, \ldots ,\quad n = 1,2, \ldots ,2^{k} \), m is degree of Gegenbauer polynomials, λ is the known ultraspherical parameter and \( x \in [0,1) \). They are defined by:

$$ \psi_{nm} (x) = \left\{ {\begin{array}{*{20}l} {\frac{{2^{{\frac{k + 1}{2}}} }}{{\sqrt {L_{m}^{\lambda } } }}C_{m}^{\lambda } (2^{k + 1} x - 2n + 1),} \hfill &\quad {\frac{n - 1}{{2^{k} }} \le x < \frac{n}{{2^{k} }},} \hfill \\ {0,} \hfill &\quad {otherwise} \hfill \\ \end{array} } \right. $$
(10)

where \( C_{m}^{\lambda } (2^{k + 1} x - 2n + 1) \) are Gegenbauer polynomials of degree m which are orthogonal with respect to the weight function \( w_{n} (x) = w(2^{k + 1} x - 2n + 1) = \left( {1 - (2^{k + 1} x - 2n + 1)^{2} } \right)^{{\lambda - \frac{1}{2}}} \) on \( [ - 1,\,\,1] \).

A function \( f(x) \in L_{w}^{2} [0,1] \) may be expanded as:

$$ f(x) = \sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 0}^{\infty } {f_{nm} \psi_{nm} (x)} } $$
(11)

where

$$ f_{nm} = \left\langle {f(x),\psi_{nm} (x)} \right\rangle $$
(12)

\( \left\langle {\,.\,,\,.\,} \right\rangle \) denotes the inner product with weight function \( w_{n} (x) \) in Eq. (12).

Truncated form of Eq. (11) can be written as:

$$ f(x) \cong \sum\limits_{n = 1}^{{2^{k} }} {\sum\limits_{m = 0}^{M - 1} {f_{nm} \psi_{nm} (x)} } = {\mathbf{C}}^{{\mathbf{T}}} {\varvec{\Psi}}(x) $$
(13)

where \( {\mathbf{C}} \) and \( {\varvec{\Psi}}(x) \) are \( 2^{k} M \times 1 \) columns vectors given by:

$$ {\mathbf{C}}^{{\mathbf{T}}} = [f_{10} ,f_{11} , \ldots ,f_{1M - 1} ,f_{20} , \ldots ,f_{2M - 1} , \ldots ,f_{{2^{k} 0}} , \ldots ,f_{{2^{k} M - 1}} ] $$
$$ {\varvec{\Psi}}(x) = [\psi_{10} ,\psi_{11} , \ldots ,\psi_{1M - 1} ,\psi_{20} , \ldots ,\psi_{2M - 1} , \ldots ,\psi_{{2^{k} 0}} , \ldots ,\psi_{{2^{k} M - 1}} ]^{T} $$
(14)

The integration of the \( \psi_{nm} (x) \) given in Eq. (10) can be shown as

$$ p_{nm} (x) = \int\limits_{0}^{x} {\psi_{nm} (s)ds} $$
(15)

For m = 0, m = 1 and m > 1, \( p_{nm} (x) \) can be obtained as

$$ p_{n0} (x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {0 \le x < \frac{n - 1}{{2^{k} }}} \hfill \\ {\psi_{n0} (u) + \frac{{2^{ - 1} }}{\lambda }\sqrt {\frac{{L_{1}^{\lambda } }}{{L_{0}^{\lambda } }}} \psi_{n1} (u),} \hfill & {\frac{n - 1}{{2^{k} }} \le x < \frac{n}{{2^{k} }}} \hfill \\ {2,} \hfill & {\frac{n}{{2^{k} }} \le x < 1} \hfill \\ \end{array} } \right. $$
$$ p_{n1} (x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {0 \le x < \frac{n - 1}{{2^{k} }}} \hfill \\ { - \frac{\lambda (2\lambda + 1)}{2(\lambda + 1)}\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{1}^{\lambda } }}} \psi_{n0} (u) + \frac{{2^{ - 1} }}{(\lambda + 1)}\sqrt {\frac{{L_{2}^{\lambda } }}{{L_{1}^{\lambda } }}} \psi_{n2} (u),} \hfill & {\frac{n - 1}{{2^{k} }} \le x < \frac{n}{{2^{k} }}} \hfill \\ {0,} \hfill & {\frac{n}{{2^{k} }} \le x < 1} \hfill \\ \end{array} } \right. $$
$$ p_{nm} (x) = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {0 \le x < \frac{n - 1}{{2^{k} }}} \hfill \\ {\left( {\frac{{C_{m - 1}^{\lambda } ( - 1) - C_{m + 1}^{\lambda } ( - 1)}}{2(\lambda + m)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{m}^{\lambda } }}} \psi_{n0} (u) - \frac{{ - 2^{ - 1} }}{(\lambda + m)}\sqrt {\frac{{L_{m - 1}^{\lambda } }}{{L_{m}^{\lambda } }}} \psi_{nm - 1} (u)} \hfill & {} \hfill \\ {\quad \quad + \frac{{2^{ - 1} }}{(\lambda + m)}\sqrt {\frac{{L_{m + 1}^{\lambda } }}{{L_{m}^{\lambda } }}} \psi_{nm + 1} (u),} \hfill & {\frac{n - 1}{{2^{k} }} \le x < \frac{n}{{2^{k} }}} \hfill \\ {\left( {\frac{{C_{m + 1}^{\lambda } (1) - C_{m + 1}^{\lambda } ( - 1) - C_{m - 1}^{\lambda } (1) + C_{m - 1}^{\lambda } ( - 1)}}{2(\lambda + m)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{m}^{\lambda } }}} \psi_{n0} (u),} \hfill & {\frac{n}{{2^{k} }} \le x < 1} \hfill \\ \end{array} } \right. $$

where \( u = 2^{k + 1} x - 2n + 1 \). The integration of the \( {\varvec{\Psi}}(x) \) can be represented as

$$ \int\limits_{0}^{x} {\varvec{\Psi}}(s)ds = [p_{10} ,p_{11} , \ldots ,p_{1M - 1} ,p_{20} , \ldots ,p_{2M - 1} , \ldots ,p_{{2^{k} 0}} , \ldots ,p_{{2^{k} M - 1}} ]^{T} = {\mathbf{P}}_{{\mathbf{1}}} {\varvec{\Psi}}_{{\mathbf{1}}} (x) $$
(16)

where

$$ {\varvec{\Psi}}_{{\mathbf{1}}} (x) = [\psi_{10} ,\psi_{11} , \ldots ,\psi_{1M} ,\psi_{20} , \ldots ,\psi_{2M} , \ldots ,\psi_{{2^{k} 0}} , \ldots ,\psi_{{2^{k} M}} ]^{T} $$
$$ {\mathbf{L}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} 1 & {\frac{{2^{ - 1} }}{\lambda }\sqrt {\frac{{L_{1}^{\lambda } }}{{L_{0}^{\lambda } }}} } & 0 & 0 & \cdots & 0 & 0 & 0 \\ { - \frac{\lambda (2\lambda + 1)}{{2(\lambda + 1)\mathop {}\limits_{{}} }}\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{1}^{\lambda } }}} } & 0 & {\frac{{2^{ - 1} }}{(\lambda + 1)}\sqrt {\frac{{L_{2}^{\lambda } }}{{L_{1}^{\lambda } }}} } & 0 & \cdots & 0 & 0 & 0 \\ {\left( {\frac{{C_{1}^{\lambda } ( - 1) - C_{3}^{\lambda } ( - 1)}}{2(\lambda + 2)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{2}^{\lambda } }}} } & {\frac{{ - 2^{ - 1} }}{{(\lambda + 2)\mathop {}\limits_{{}} }}\sqrt {\frac{{L_{1}^{\lambda } }}{{L_{2}^{\lambda } }}} } & 0 & {\frac{{2^{ - 1} }}{(\lambda + 2)}\sqrt {\frac{{L_{3}^{\lambda } }}{{L_{2}^{\lambda } }}} } & \cdots & 0 & 0 & 0 \\ { \vdots \mathop {}\limits_{{}}^{{}} } & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ {\left( {\frac{{C_{M - 2}^{\lambda } ( - 1) - C_{M}^{\lambda } ( - 1)}}{2(\lambda + M - 1)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{M - 1}^{\lambda } }}} } & 0 & 0 & 0 & \cdots & {\frac{{ - 2^{ - 1} }}{(\lambda + M - 1)}\sqrt {\frac{{L_{M - 2}^{\lambda } }}{{L_{M - 1}^{\lambda } }}} } & 0 & {\frac{{2^{ - 1} }}{(\lambda + M - 1)}\sqrt {\frac{{L_{M}^{\lambda } }}{{L_{M - 1}^{\lambda } }}} } \\ \end{array} } \right] $$
$$ \begin{aligned} & {\mathbf{F}}_{{\mathbf{1}}} = \left[ {\begin{array}{*{20}c} 2 & 0 & \cdots & 0 \\ 0 & {\mathop 0\limits_{{}}^{{\mathop {}\limits_{{}} }} } & \cdots & 0 \\ {\left( {\frac{{C_{3}^{\lambda } (1) - C_{3}^{\lambda } ( - 1) - C_{1}^{\lambda } (1) + C_{1}^{\lambda } ( - 1)}}{2(\lambda + 2)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{2}^{\lambda } }}} } & 0 & \cdots & 0 \\ { \vdots \mathop {}\limits_{{}}^{{}} } & \vdots & \ddots & \vdots \\ {\left( {\frac{{C_{M - 1}^{\lambda } (1) - C_{M - 1}^{\lambda } ( - 1) - C_{M - 3}^{\lambda } (1) + C_{M - 3}^{\lambda } ( - 1)}}{2(\lambda + M - 2)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{M - 2}^{\lambda } }}} } & {\mathop 0\limits_{{\mathop {}\limits_{{}}^{{}} }} } & \cdots & 0 \\ {\left( {\frac{{C_{M}^{\lambda } (1) - C_{M}^{\lambda } ( - 1) - C_{M - 2}^{\lambda } (1) + C_{M - 2}^{\lambda } ( - 1)}}{2(\lambda + M - 1)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{M - 1}^{\lambda } }}} } & 0 & \cdots & 0 \\ \end{array} } \right] \\ & {\mathbf{P}}_{{\mathbf{1}}} = \frac{1}{{2^{k + 1} }}\left[ {\begin{array}{*{20}c} {{\mathbf{L}}_{{\mathbf{1}}} } &\quad {{\mathbf{F}}_{{\mathbf{1}}} } &\quad {{\mathbf{F}}_{{\mathbf{1}}} } &\quad \cdots &\quad {{\mathbf{F}}_{{\mathbf{1}}} } &\quad {{\mathbf{F}}_{{\mathbf{1}}} } \\ {\mathbf{0}} &\quad {{\mathbf{L}}_{{\mathbf{1}}} } &\quad {{\mathbf{F}}_{{\mathbf{1}}} } &\quad \cdots &\quad {{\mathbf{F}}_{{\mathbf{1}}} } &\quad {{\mathbf{F}}_{{\mathbf{1}}} } \\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \vdots \\ {\mathbf{0}} &\quad {\mathbf{0}} &\quad {\mathbf{0}} &\quad \cdots &\quad {{\mathbf{L}}_{{\mathbf{1}}} } &\quad {{\mathbf{F}}_{{\mathbf{1}}} } \\ {\mathbf{0}} &\quad {\mathbf{0}} &\quad {\mathbf{0}} &\quad \cdots &\quad {\mathbf{0}} &\quad {{\mathbf{L}}_{{\mathbf{1}}} } \\ \end{array} } \right] \\ \end{aligned} $$

The second integrations of the \( {\varvec{\Psi}}(x) \) can be represented as

$$ \int\limits_{0}^{x} {\int\limits_{0}^{{x_{1} }} {{\mathbf{\varPsi (}}s{\mathbf{)}}ds} } dx_{1} = \int\limits_{0}^{x} {{\mathbf{P}}_{{\mathbf{1}}} {\varvec{\Psi}}_{{\mathbf{1}}} {\mathbf{(}}x_{{\mathbf{1}}} {\mathbf{)}}} dx_{1} = {\mathbf{P}}_{{\mathbf{1}}} \int\limits_{0}^{x} {{\varvec{\Psi}}_{{\mathbf{1}}} {\mathbf{(}}x_{{\mathbf{1}}} {\mathbf{)}}dx_{1} } = {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\varvec{\Psi}}_{{\mathbf{2}}} (x) \ne {\mathbf{P}}_{{\mathbf{1}}}^{{\mathbf{2}}} {\varvec{\Psi}}(x) $$

The rth integrations of the \( {\varvec{\Psi}}(x) \) can be represented as

$$ \int\limits_{0}^{x} {\int\limits_{0}^{{x_{1} }} {\int\limits_{0}^{{x_{2} }} \cdots } } \int\limits_{0}^{{x_{r - 1} }} {{\mathbf{\varPsi (}}s{\mathbf{)}}} \,dsdx_{r - 1} dx_{r - 2} \cdots dx_{1} = {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} \cdots {\mathbf{P}}_{{\mathbf{r}}} {\varvec{\Psi}}_{{\mathbf{r}}} (x) \ne {\mathbf{P}}_{{\mathbf{1}}}^{{\mathbf{r}}} {\varvec{\Psi}}(x) $$

where

$$ {\mathbf{L}}_{{\mathbf{r}}} = \left[ {\begin{array}{*{20}c} 1 & {\frac{{2^{ - 1} }}{\lambda }\sqrt {\frac{{L_{1}^{\lambda } }}{{L_{0}^{\lambda } }}} } & 0 & 0 & \cdots & 0 & 0 & 0 \\ { - \frac{\lambda (2\lambda + 1)}{{2(\lambda + 1)\mathop {}\limits_{{}} }}\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{1}^{\lambda } }}} } & 0 & {\frac{{2^{ - 1} }}{(\lambda + 1)}\sqrt {\frac{{L_{2}^{\lambda } }}{{L_{1}^{\lambda } }}} } & 0 & \cdots & 0 & 0 & 0 \\ {\left( {\frac{{C_{1}^{\lambda } ( - 1) - C_{3}^{\lambda } ( - 1)}}{2(\lambda + 2)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{2}^{\lambda } }}} } & {\frac{{ - 2^{ - 1} }}{{(\lambda + 2)\mathop {}\limits_{{}} }}\sqrt {\frac{{L_{1}^{\lambda } }}{{L_{2}^{\lambda } }}} } & 0 & {\frac{{2^{ - 1} }}{(\lambda + 2)}\sqrt {\frac{{L_{3}^{\lambda } }}{{L_{2}^{\lambda } }}} } & \cdots & 0 & 0 & 0 \\ { \vdots \mathop {}\limits_{{}}^{{}} } & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ {\left( {\frac{{C_{M + r - 3}^{\lambda } ( - 1) - C_{M + r - 1}^{\lambda } ( - 1)}}{2(\lambda + M + r - 2)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{M + r - 2}^{\lambda } }}} } & 0 & 0 & 0 & \cdots & {\frac{{ - 2^{ - 1} }}{(\lambda + M + r - 2}\sqrt {\frac{{L_{M + r - 3}^{\lambda } }}{{L_{M + r - 2}^{\lambda } }}} } & 0 & {\frac{{2^{ - 1} }}{(\lambda + M + r - 2)}\sqrt {\frac{{L_{M + r - 1}^{\lambda } }}{{L_{M + r - 2}^{\lambda } }}} } \\ \end{array} } \right] $$
$$ \begin{aligned} & {\mathbf{F}}_{{\mathbf{r}}} = \left[ {\begin{array}{*{20}c} 2 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ {\left( {\frac{{C_{3}^{\lambda } (1) - C_{3}^{\lambda } ( - 1) - C_{1}^{\lambda } (1) + C_{1}^{\lambda } ( - 1)}}{2(\lambda + M - 1)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{2}^{\lambda } }}} } & 0 & \cdots & 0 \\ { \vdots \mathop {}\limits_{{}}^{{}} } & \vdots & \ddots & 0 \\ {\left( {\frac{{C_{M - 1}^{\lambda } (1) - C_{M - 1}^{\lambda } ( - 1) - C_{M - 3}^{\lambda } (1) + C_{M - 3}^{\lambda } ( - 1)}}{2(\lambda + M - 1)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{M - 2}^{\lambda } }}} } & 0 & \cdots & 0 \\ { \vdots \mathop {}\limits_{{}}^{{}} } & \vdots & \ddots & \vdots \\ {\left( {\frac{{C_{M + r - 1}^{\lambda } (1) - C_{M + r - 1}^{\lambda } ( - 1) - C_{M + r - 3}^{\lambda } (1) + C_{M + r - 3}^{\lambda } ( - 1)}}{2(\lambda + M + r - 2)}} \right)\sqrt {\frac{{L_{0}^{\lambda } }}{{L_{M + r - 2}^{\lambda } }}} } & 0 & \cdots & 0 \\ \end{array} } \right] \\ & {\mathbf{P}}_{{\mathbf{r}}} = \frac{1}{{2^{k + 1} }}\left[ {\begin{array}{*{20}c} {{\mathbf{L}}_{{\mathbf{r}}} } &\quad {{\mathbf{F}}_{{\mathbf{r}}} } &\quad {{\mathbf{F}}_{{\mathbf{r}}} } &\quad \cdots &\quad {{\mathbf{F}}_{{\mathbf{r}}} } &\quad {{\mathbf{F}}_{{\mathbf{r}}} } \\ {\mathbf{0}} &\quad {{\mathbf{L}}_{{\mathbf{r}}} } &\quad {{\mathbf{F}}_{{\mathbf{r}}} } &\quad \cdots &\quad {{\mathbf{F}}_{{\mathbf{r}}} } &\quad {{\mathbf{F}}_{{\mathbf{r}}} } \\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots &\quad \vdots \\ {\mathbf{0}} &\quad {\mathbf{0}} &\quad {\mathbf{0}} &\quad \cdots &\quad {{\mathbf{L}}_{{\mathbf{r}}} } &\quad {{\mathbf{F}}_{{\mathbf{r}}} } \\ {\mathbf{0}} &\quad {\mathbf{0}} &\quad {\mathbf{0}} &\quad \cdots &\quad {\mathbf{0}} &\quad {{\mathbf{L}}_{{\mathbf{r}}} } \\ \end{array} } \right] \\ \end{aligned} $$

and

$$ {\varvec{\Psi}}_{{\mathbf{r}}} (x) = [\psi_{10} ,\psi_{11} , \ldots ,\psi_{1M + r - 1} ,\psi_{20} , \ldots ,\psi_{2M + r - 1} , \ldots ,\psi_{{2^{k} 0}} , \ldots ,\psi_{{2^{k} M + r - 1}} ]^{T} $$
(17)

The matrices \( {\mathbf{L}}_{{\mathbf{r}}} \,\,{\text{and }}\,{\mathbf{F}}_{{\mathbf{r}}} \,\, \) have the dimension \( (M + r - 1) \times (M + r) \). Hence \( \,{\mathbf{P}}_{{{\mathbf{r}}\,}} \, \) has the dimension \( 2^{k} (M + r - 1) \times 2^{k} (M + r) \).

Gegenbauer Wavelet Collocation Method for the (GKS) Equation

Consider Eq. (1) with initial and boundary conditions

$$ u(x,0) = f(x),\,\,\,\,u(0,t) = g_{1} (t),\,\,\,\,u(1,t) = g_{2} (t),\,\,\,\,u_{x} (0,t) = g_{3} (t),\,\,\,\,u_{xx} (0,t) = g_{4} (t) $$

It is assumed that \( \dot{u}^{(4)} (x,t) \) can be expanded in terms of truncated Gegenbauer wavelet series as

$$ \dot{u}^{(4)} (x,t) = \sum\limits_{n = 1}^{{\,2^{k} }} {\,\sum\limits_{m = 0}^{M - 1} {f_{nm} \psi_{nm} (x)} } = {\mathbf{C}}^{{\mathbf{T}}} {\varvec{\Psi}}(x). $$
(18)

where “˙” and “(4)” means differentiation with respect to t and x.

By integrating Eq. (18) with respect to t from ts to t and four times with respect to x from 0 to x, following equations are obtained

$$ u^{(4)} (x,t) = u^{(4)} (x,t_{s} ) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\varvec{\Psi}}(x) $$
(19)
$$ u^{(3)} (x,t) = u^{(3)} (x,t_{s} ) + u^{(3)} (0,t) - u^{(3)} (0,t_{s} ) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\varvec{\Psi}}_{{\mathbf{1}}} (x) $$
(20)
$$ \begin{aligned} u^{(2)} (x,t) & = u^{(2)} (x,t_{s} ) + u^{(2)} (0,t) - u^{(2)} (0,t_{s} )\, + x(u^{(3)} (0,t) - u^{(3)} (0,t_{s} ))\mathop {}\limits_{{}} \\ & \quad + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\varvec{\Psi}}_{{\mathbf{2}}} (x) \\ \end{aligned} $$
(21)
$$ \begin{aligned} u_{x} (x,t) & = u_{x} (x,t_{s} ) + u_{x} (0,t) - u_{x} (0,t_{s} )\, + x(u^{(2)} (0,t) - u^{(2)} (0,t_{s} )) \\ & \quad + \frac{{x^{2} }}{2}(u^{(3)} (0,t) - u^{(3)} (0,t_{s} )) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\varvec{\Psi}}_{{\mathbf{3}}} (x) \\ \end{aligned} $$
(22)
$$ \begin{aligned} u(x,t) & = u(x,t_{s} ) + u(0,t) - u(0,t_{s} )\, + x(u_{x} (0,t) - u_{x} (0,t_{s} )) + \frac{{x^{2} }}{2}(u^{(2)} (0,t) - u^{(2)} (0,t_{s} )) \\ & \quad + \frac{{x^{3} }}{6}(u^{(3)} (0,t) - u^{(3)} (0,t_{s} )) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} (x) \\ \end{aligned} $$
(23)

From the initial and boundary conditions

$$ u(x,0) = f(x),\,\,\,\,u(0,t) = g_{1} (t),\,\,\,\,u(1,t) = g_{2} (t),\,\,\,\,u_{x} (0,t) = g_{3} (t),\,\,\,\,u_{xx} (0,t) = g_{4} (t) $$
(24)

we have the following equation as:

$$ \begin{aligned} \frac{1}{6}(u^{(3)} (0,t) - u^{(3)} (0,t_{s} )) & = - (g_{1} (t) - g_{1} (t_{s} )) + (g_{2} (t) - g_{2} (t_{s} )) - (g_{3} (t) - g_{3} (t_{s} )) \\ & \quad - \frac{1}{2}(g_{4} (t) - g_{4} (t_{s} )) - (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}} \\ \end{aligned} $$
(25)

If Eqs. (24, 25) are substituted into Eqs. (19–23), the following equations are obtained.

$$ u^{(4)} (x,t) = (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} {\varvec{\Psi}}(x) + u^{(4)} (x,t_{s} ) $$
(26)
$$ \begin{aligned} u^{(3)} (x,t) & = u^{(3)} (x,t_{s} ) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} \left[ {{\mathbf{P}}_{{\mathbf{1}}} {\varvec{\Psi}}_{{\mathbf{1}}} (x) - 6{\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}}} \right] - 6(g_{1} (t) - g_{1} (t_{s} ))\mathop {}\limits_{{}} \\ & \quad + 6(g_{2} (t) - g_{2} (t_{s} )) - 6(g_{3} (t) - g_{3} (t_{s} ))\, - 3(g_{4} (t) - g_{4} (t_{s} )) \\ \end{aligned} $$
(27)
$$ \begin{aligned} u_{xx} (x,t) & = u_{xx} (x,t_{s} ) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} \left[ {{\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\varvec{\Psi}}_{{\mathbf{2}}} (x) - 6x{\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}}} \right] + g_{4} (t) - g_{4} (t_{s} )\,\mathop {}\limits_{{}} \\ & \quad + 6x\left( { - (g_{1} (t) - g_{1} (t_{s} )) + (g_{2} (t) - g_{2} (t_{s} )) - (g_{3} (t) - g_{3} (t_{s} )) - \frac{1}{2}(g_{4} (t) - g_{4} (t_{s} ))} \right) \\ \end{aligned} $$
(28)
$$ \begin{aligned} u_{x} (x,t) & = u_{x} (x,t_{s} ) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} \left[ {{\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\varvec{\Psi}}_{{\mathbf{3}}} (x) - 3x^{2} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}}} \right] - 3x^{2} (g_{1} (t) - g_{1} (t_{s} )) \\ & \quad + 3x^{2} (g_{2} (t) - g_{2} (t_{s} )) + (1 - 3x^{2} )(g_{3} (t) - g_{3} (t_{s} )) + \left( {x - \frac{{3x^{2} }}{2}} \right)(g_{4} (t) - g_{4} (t_{s} )) \\ \end{aligned} $$
(29)
$$ \begin{aligned} u(x,t) & = u(x,t_{s} ) + (t - t_{s} ){\mathbf{C}}^{{\mathbf{T}}} \left[ {{\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}x{\mathbf{)}} - x^{3} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}}} \right] \\ & \quad + (1 - x^{3} )(g_{1} (t) - g_{1} (t_{s} )) + x^{3} (g_{2} (t) - g_{2} (t_{s} )) \\ & \quad + (x - x^{3} )(g_{3} (t) - g_{3} (t_{s} ))\, + (x^{2} - x^{3} )\frac{1}{2}\left( {g_{4} (t) - g_{4} (t_{s} )} \right) \\ \end{aligned} $$
(30)
$$ \begin{aligned} u_{t} (x,t) & = {\mathbf{C}}^{{\mathbf{T}}} \left[ {{\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} (x) - x^{3} {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}}} \right] \\ & \quad + (1 - x^{3} )g_{1}^{{\prime }} (t) + x^{3} g_{2}^{{\prime }} (t) + (x - x^{3} )g_{3}^{{\prime }} (t) + (x^{2} - x^{3} )\frac{1}{2}g_{4}^{{\prime }} (t) \\ \end{aligned} $$
(31)

Nonlinear Eq. (1) is converted into a sequence of linear differential equations by quasilinearization technique. First approximate solution satisfying initial/boundary conditions is taken as

$$ \begin{aligned} u^{0} (x,t) & = f(x) + (1 - x^{3} )(g_{1} (t) - g_{1} (t_{s} )\,)\, + x^{3} (g_{2} (t) - g_{2} (t_{s} )) \\ & \quad + (x - x^{3} )(g_{3} (t) - g_{3} (t_{s} ))\, + (x^{2} - x^{3} )\frac{1}{2}\left( {g_{4} (t) - g_{4} (t_{s} )} \right) \\ \end{aligned} $$
(32)

and

$$ (u(x,t)\,u_{x} (x,t))^{l + 1} \cong \,u^{l + 1} (x,t)\,u_{x}^{l} (x,t) + u^{l} (x,t)\,u_{x}^{l + 1} (x,t) - u^{l} (x,t)\,u_{x}^{l} (x,t) $$
(33)

can be obtained by quasilinearization technique. Hence converted problem is obtained as

$$ \begin{aligned} & u_{t}^{l + 1} (x,t) + u^{l + 1} (x,t)\,u_{x}^{l} (x,t) + u^{l} (x,t)\,u_{x}^{l + 1} (x,t) + \alpha u_{xx}^{l + 1} (x,t) \\ & \quad + \beta u_{xxx}^{l + 1} (x,t) + \gamma u_{xxxx}^{l + 1} (x,t) - u^{l} (x,t)\,u_{x}^{l} (x,t) = 0 \\ \end{aligned} $$
(34)

where l is index of quasilinearization technique and l = 0, 1, 2, …

Replacing Eqs. (26–31) into the Eq. (34), we have the following equation.

$$ \begin{aligned} & {\mathbf{C}}^{{\mathbf{T}}} (t - t_{s} )\left[ {\begin{array}{*{20}l} {\left( {\frac{1}{{t - t_{s} }} + u_{x}^{l} (x,t)} \right){\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}x{\mathbf{)}} + u^{l} (x,t){\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\varvec{\Psi}}_{{\mathbf{3}}} {\mathbf{(}}x{\mathbf{)}} + \alpha {\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\varvec{\Psi}}_{{\mathbf{2}}} (x)} \hfill \\ { + \beta {\mathbf{P}}_{{\mathbf{1}}} {\varvec{\Psi}}_{{\mathbf{1}}} {\mathbf{(}}x{\mathbf{)}} + \delta {\mathbf{\varPsi (}}x{\mathbf{)}} - \left( {\frac{{x^{3} }}{{t - t_{s} }} + x^{3} u_{x}^{l} (x,t) + 3x^{2} u^{l} (x,t) + 6\alpha x + 6\beta } \right){\mathbf{P}}_{{\mathbf{1}}} {\mathbf{P}}_{{\mathbf{2}}} {\mathbf{P}}_{{\mathbf{3}}} {\mathbf{P}}_{{\mathbf{4}}} {\varvec{\Psi}}_{{\mathbf{4}}} {\mathbf{(}}1{\mathbf{)}}} \hfill \\ \end{array} } \right] \\ & \quad = u^{l} (x,t)u_{x}^{l} (x,t) - u_{x}^{l} (x,t)u(x,t_{s} ) - u^{l} (x,t)u_{x} (x,t_{s} ) - \alpha u^{(2)} (x,t_{s} ) - \beta u^{(3)} (x,t_{s} ) - \delta u^{(4)} (x,t_{s} ) \\ & \quad \quad - g^{\prime}_{1} (t) - xg^{\prime}_{3} (t) - \frac{{x^{2} }}{2}g^{\prime}_{4} (t) - x^{3} \left( {g^{\prime}_{2} (t) - g^{\prime}_{1} (t) - g^{\prime}_{3} (t) - \frac{1}{2}g^{\prime}_{4} (t)} \right) \\ & \quad \quad - \left( {x^{3} u_{x}^{l} (x,t) + 3x^{2} u^{l} (x,t) + 6\alpha x + 6\beta } \right)\left( {g_{2} (t) - g_{2} (t_{s} )} \right) \\ & \quad \quad + \left( {(x^{3} - 1)u_{x}^{l} (x,t) + 3x^{2} u^{l} (x,t) + 6\alpha x + 6\beta } \right)\left( {g_{1} (t) - g_{1} (t_{s} )} \right) \\ & \quad \quad + \left( {(x^{3} - x)u_{x}^{l} (x,t) + (3x^{2} - 1)u^{l} (x,t) + 6\alpha x + 6\beta } \right)\left( {g_{3} (t) - g_{3} (t_{s} )\,} \right) \\ & \quad \quad + \left( {(x^{3} - x^{2} )u_{x}^{l} (x,t) + (3x^{2} - 2x)u^{l} (x,t) + 6\alpha x - 2\alpha + 6\beta } \right)\left( {\frac{1}{2}g_{4} (t) - \frac{1}{2}g_{4} (t_{s} )} \right) \\ \end{aligned} $$
(35)

The collocation points can be taken as \( 2^{k + 1} x_{ni} - 2n + 1 = \cos \frac{((M + 1) - i)\pi }{(M + 1)} \) or

$$ x_{ni} = \frac{1}{{2^{k + 1} }}\left( {2n - 1 + \cos \frac{((M + 1) - i)\pi }{(M + 1)}} \right),\quad i = 1,2, \ldots ,M,\quad n = 1,2, \ldots ,2^{k} . $$
(36)

Substituting the collocation points \( x \to x_{ni} \) and time variable \( t \to t_{s + 1} \) into (35), a discretized form of the vectors \( {\mathbf{\varPsi (}}x_{{{\mathbf{ni}}}} {\mathbf{)}} \), \( {\varvec{\Psi}}_{{\mathbf{1}}} {\mathbf{(}}x_{{{\mathbf{ni}}}} {\mathbf{)}} \) and \( {\varvec{\Psi}}_{{\mathbf{r}}} {\mathbf{(}}x_{{{\mathbf{ni}}}} {\mathbf{)}} \) can be obtained. Hence form Eq. (35), we obtain algebraic equation system whose matrix notation is

$$ {\mathbf{C}}^{{\mathbf{T}}} {\mathbf{U}} = {\mathbf{B}} $$
(37)

where \( {\mathbf{U}} \) is a \( 2^{k} M \times 2^{k} M \) matrix. \( {\mathbf{C}} \) and \( {\mathbf{B}} \) are \( 2^{k} M \times 1 \) vectors. Hence, by solving algebraic equation system (37), we can find the coefficients of the Gegenbauer wavelet series that satisfied differential equation and given initial and boundary conditions.

Error Analysis

Theorem 1

Let \( f(x) \in L_{w}^{2} [0,1] \) with bounded second order derivative \( \left| {f^{\prime\prime}(x)} \right| \le N, \) can be expanded as an infinite sum of Gegenbauer wavelets, and the series converges uniformly to f(x) [34]. That is

$$ f(x) = \sum\limits_{n = 1}^{\infty } {\sum\limits_{m = 0}^{\infty } {f_{nm} \psi_{nm} (x)} } ,\,\,\,\,\,\,\forall \lambda > - \frac{1}{2}. $$

Theorem 2

Let \( f(x) \in L_{w}^{2} [0,1] \) with bounded second order derivative \( \left| {f^{{\prime \prime }} \left( x \right)} \right| \le N \), then we have the following accuracy estimation [34]:

$$ \int\limits_{0}^{1} {\left[ {f(x) - \sum\limits_{n = 1}^{{2^{k} }} {\sum\limits_{m = 0}^{M - 1} {f_{nm} } \psi_{nm} (x)} } \right]}^{2} \omega_{n} (x)dx < N^{2} \sum\limits_{{n = 2^{k} + 1}}^{\infty } {\sum\limits_{m = M}^{\infty } {\frac{1}{{n^{5} m(m - 1)(m + 2\lambda )(m + 2\lambda + 1)}}} } . $$

Numerical Results

Example 1

Consider generalized Kuramoto–Sivashinsky Eq. (1) with \( \alpha = \gamma = 1 \) and β = 4. Analytic solution is given in [46, 47] as:

$$ u(x,t) = 2c + 9 + 15\left( {\tanh (ct - \frac{x}{2}) - \tanh^{2} (ct - \frac{x}{2}) - \tanh^{3} (ct - \frac{x}{2})} \right) $$

The required initial and boundary conditions can be obtained from the exact solution. This nonlinear differential equation is converted into a sequence of linear differential equation generated by quasilinearization technique in (34). Replacing initial boundary conditions into the Eq. (35) and solving algebraic equation system in Eq. (37), we have coefficients \( {\mathbf{C}}^{{\mathbf{T}}} \) of the Chebyshev wavelet series. By substituting the Gegenbauer wavelet coefficients into Eq. (30), we have the implicit form of the approximate solution satisfied differential equation and whose boundary conditions. Table 1 shows the absolute errors in Chebyshev and equal collocation points for c = 1 λ = 0, t = 2, M = 4, k = 1, M = 8, k = 1. We can see that if M or k increase; approximate results are converged to the exact solution and Chebyshev collocation points give better results from equal collocation points. Table 2 shows the absolute errors in collocation points for c = 1, M = 8, k = 1, t = 2, δt = (ts+1−ts) = 0.02 and various values of λ. Graphical presentation of the approximate solution is given in Fig. 1 for λ = 0, c = 1, M = 16, k = 1, t = 2 and δt = 0.01. Table 3 shows the absolute errors in collocation points for M = 8, k = 1, t = 1, δt = 0.05 and various values of c. Comparisons of the maximal absolute error of present method [46,47,48] are given in the Table 4. As can be seen in Table 4, it is clear that the results obtained by the presented method are superior respect to [46,47,48].

Table 1 Absolute error of Example 1 with Gegenbauer wavelets collocation method for various collocation points
Full size table
Table 2 Absolute error of Example 1 with Gegenbauer wavelets collocation method for various values of λ
Full size table
Fig. 1
figure 1

Approximate solution of Example 1 for δt = 0.01

Full size image
Table 3 Absolute error of Example 1 with Gegenbauer wavelets collocation method for various values of c
Full size table
Table 4 Comparisons of the maximal absolute error of Example 1 for various values of c
Full size table

Example 2

Consider generalized Kuramoto–Sivashinsky Eq. (1) with α = 1 β = 0 and γ = 0.5. Analytic solution is given in [47] as:

$$ u(x,t) = - \frac{0.1}{K} + \frac{60}{19}K( - 38K^{2} \gamma + \alpha )\tanh (0.1t + Kx) + 120\gamma K^{3} \tanh^{3} (0.1t + Kx) $$

where \( K = \frac{1}{2}\left( {\frac{11\alpha }{19\gamma }} \right)^{{\frac{1}{2}}} \). The required initial and boundary conditions can be obtained from the exact solution. Table 5 shows the absolute errors in collocation points for λ = 0, t = 2 M = 4, k = 0, M = 4, k = 1 and M = 4, k = 2. Table 6 shows the absolute errors in collocation points for M = 8, k = 1, t = 2, δt = 0.02 and various values of λ. Comparisons of the maximal absolute error of present method and [48] are given in the Table 7 for M = 8, k = 0 and δt = 0.01. As can be seen in Table 7, it is clear that the results obtained by the presented method are superior respect to [48].

Table 5 Absolute error of Example 2 with Gegenbauer wavelets collocation method for various collocation points
Full size table
Table 6 Absolute error of Example 2 with Gegenbauer wavelets collocation method for various values of λ
Full size table
Table 7 Comparisons of the maximal absolute error of Example 2 for various values of t
Full size table

Example 3

Consider generalized Kuramoto–Sivashinsky Eq. (1) with α = 1 \( \beta = \frac{12}{{\sqrt {47} }} \) and γ = 1. Analytic solution is given in [46] as:

$$ u(x,t) = - \frac{396.8}{{47\sqrt {47} }} + \frac{15}{{47\sqrt {47} }}\left( {3\tanh (\theta ) - 3\tanh^{2} (\theta ) + \tanh^{3} (\theta )} \right) $$

where \( \theta = 0.1\,t + \frac{1}{{2\sqrt {47} }}x \). The required initial and boundary conditions can be obtained from the exact solution. Table 8 shows the absolute errors in collocation points for λ = 0, t = 2, M = 4, k = 0, M = 4, k = 1 and M = 4, k = 2. Table 9 shows the absolute errors in collocation points for M = 8, k = 1, t = 2, δt = 0.02 and various values of λ. Comparisons of the maximal absolute error of present method and [46, 48] are given in the Table 10 for M = 8, k = 0 and δt = 0.01. As can be seen in Table 10, it is clear that the results obtained by the presented method are superior respect to [46, 48].

Table 8 Absolute error of Example 3 with Gegenbauer wavelets collocation method for various collocation points
Full size table
Table 9 Absolute error of Example 3 with Gegenbauer wavelets collocation method for various values of λ
Full size table
Table 10 Comparisons of the maximal absolute error of Example 3 for various values of t
Full size table

Conclusion

Gegenbauer wavelet collocation method is proposed to obtain approximate solution of generalized Kuramoto–Sivashinsky equation. The method has been applied to the three nonlinear differential equations by using quasilinearization technique. Approximate and exact solutions of examples are correspondingly compared. For all examples, comparisons of the maximal absolute errors given in Tables 4,7 and 10 show that the results obtained by the proposed method are better than the represented in [46,47,48]. As can be seen from all tables, the present method is highly efficient and accurate. All of the calculations have been made by Maple program with 15 digits. These calculations demonstrated that the accuracy of the Gegenbauer wavelet collocation method is quite high even in the case of a small number of grid points. Application of proposed method is very simple because there are no complex integrals or methodology. Moreover, the this method is reliable, simple, fast, minimal computation costs, flexible, and convenient alternative method.