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Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method
Advances in Difference Equations volume 2019, Article number: 293 (2019)
Abstract
In this study the method which was obtained from a combination of the conformable fractional double Laplace transform method and the Adomian decomposition method has been successfully applied to solve linear and nonlinear singular conformable fractional Boussinesq equations. Two examples are given to illustrate our method. Furthermore, the results show that the proposed method is effective and is easy to use for certain problems in physics and engineering.
1 Introduction
There are numerous scientific, engineering, and technological processes that can be mathematically modeled by linear and nonlinear Boussinesq equations such as model flows of water in unconfined aquifers. The fractional Boussinesq equations are appropriate for discussing the water propagation through heterogeneous porous media. Many powerful methods have been modified and developed to obtain numerical and analytical solutions of fractional linear differential equations [1]. In [2] conformable fractional derivative was used to obtain the exact analytical solutions for the time fractional variant Boussinesq equations. In [3] the authors discussed the one- and two-dimensional heat diffusion models involving fractional order derivative in time and also considered the fractional orders that include Caputo’s and the new fractional conformable derivatives. The conformable Laplace transform was initiated in [4] and studied and modified in [5]. The conformable Laplace transform is not only useful to solve local conformable fractional dynamical systems but also it can be employed to solve systems within nonlocal conformable fractional derivatives that were defined in [6] and used in [7]. Very recently, the authors in [8] defined and studied a more general version of generalized Laplace transforms with its corresponding convolution theory, which can be applied to solve systems of generalized fractional derivatives with a kernel depending on a certain function \(g(t)\). In case \(g(t)=\frac{ ( t-a ) ^{ \rho }}{\rho }\) we can treat the fractional derivatives in [6, 7], and if \(g(t)=\frac{t^{\rho }}{\rho }\) we can treat the so-called Katugampola fractional derivative [9]. Fractional double Laplace transform and its properties [10] and the fractional variational principles beside the semi-inverse technique are applied to derive the space-time fractional Boussinesq equation [11]. The space-time fractional Boussinesq equations in Caputo sense derivatives are studied by using the homotopy perturbation method [12]. The authors in [13] discussed the nonlinear conformable problem by using the \(\exp (-\phi ( \varepsilon ) )\)-expansion and modified Kudryashov methods. In the present study, the new conformable fractional double Laplace transform decomposition method is recommended for developing the solutions of singular Boussinesq equation. In [14] the authors introduced a new definition of fractional calculus, which is called conformable fractional derivative of order α, \(0<\alpha <1\), as follows:
Here we briefly recall some definitions from the conformable Laplace transform which are used further in this work.
Definition 1
Let \(f:[a,\infty )\rightarrow R\) and \(0<\beta \leq 1\). Then the fractional Laplace transform of order β is defined by
Definition 2
([17])
Let \(u(x,t)\) be a piecewise continuous function on the interval \([0,\infty )\times {}[ 0,\infty )\) of exponential order, consider for some \(a,b\in \mathbb{R} \) \(\sup x,t>0\), \(\frac{ \vert u ( x,t ) \vert }{e^{\frac{ax^{\alpha }}{\alpha }+\frac{bt ^{\beta }}{\beta }}}\). Under these conditions a conformable double Laplace transform is defined by
where the symbol \(L_{x}^{\alpha }L_{t}^{\beta }\) indicates the conformable double Laplace transform and \(p,s\in \mathbb{C} \), \(0<\alpha ,\beta \leq 1\).
2 Properties of conformable fractional Laplace transform
The main objective of this section is to study the conformable double Laplace transform using three examples. In addition, we discuss the existence condition of the conformable double Laplace transform.
Example 1
Conformable double Laplace transform of the function \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })=e^{i ( a\frac{x^{\alpha }}{\alpha }+b\frac{t^{\beta }}{\beta } ) } \) is denoted by
Hence,
and
Example 2
([17])
By applying the conformable double Laplace transform for the function \(f(\frac{x^{\alpha }}{\alpha },\frac{t ^{\beta }}{\beta })= ( \frac{x^{\alpha }}{\alpha }\frac{t^{ \beta }}{\beta } ) ^{n}\), we have
where n is a positive integer. If the conditions of \(a(>-1)\) and \(b(>-1)\) are real numbers, then
then it follows from Eq. (1.2) that
Let \(p\frac{x^{\alpha }}{\alpha }=\frac{r^{\alpha }}{\alpha }\) and \(s\frac{t^{\beta }}{\beta }=\frac{q^{\beta }}{\beta }\), we get
where \(p,s\in \mathbb{C} \), \(0<\alpha ,\beta \leq 1\), gamma functions of a and b are given by
Example 3
The conformable double Laplace transform for the function
is as follows:
where \(H(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })=H(\frac{x ^{\alpha }}{\alpha })\otimes H(\frac{t^{\beta }}{\beta })\) is a Heaviside function and the symbol ⊗ indicates the tensor product, see [18]. By substituting \(\frac{\zeta ^{\alpha }}{ \alpha }=p\frac{x^{\alpha }}{\alpha }\) and \(\frac{\eta ^{\beta }}{ \beta }=s\frac{t^{\beta }}{\beta }\) in Eq. (2.1), we have
where the symbol \(\gamma =0.5772\cdot\ldots\) is Euler’s constant.
Existence condition for the conformable double Laplace transform
If \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) is an exponential order a and b as \(\frac{x^{\alpha }}{\alpha }\rightarrow \infty \), \(\frac{t^{\beta }}{\beta }\rightarrow \infty \), if there exists a constant \(K>0\), then for all \(\frac{x^{\alpha }}{\alpha }>X\) and \(\frac{t^{\beta }}{\beta }>T\)
one can get
which is equivalent to
where \(\mu >a\) and \(\eta >a\). The function \(f(\frac{x^{\alpha }}{ \alpha },\frac{t^{\beta }}{\beta })\) does not grow faster than \(Ke^{a\frac{x^{\alpha }}{\alpha }+b\frac{t^{\beta }}{\beta }}\) as \(\frac{x^{\alpha }}{\alpha }\rightarrow \infty \), \(\frac{t^{\beta }}{ \beta }\rightarrow \infty \).
Theorem 1
The function \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) is defined on \((0,X)\) and \((0,T)\) and of exponential order \(e^{a\frac{x ^{\alpha }}{\alpha }+b\frac{t^{\beta }}{\beta }}\), then the conformable Laplace transform of \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{ \beta })\) exists for all \(\Re (p)>\mu \), \(\Re (s)>\eta \).
Proof
From Eq. (1.2), we obtain
For \(\Re (p)>\mu \), \(\Re (s)>\eta \), from Eq. (2.4), we get
 □
Theorem 2
If \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) is a periodic function of the periods \(\lambda >0\) and \(\mu >0\) such that \(f ( \frac{x^{\alpha }}{\alpha }+\frac{\lambda ^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta }+\frac{\mu ^{\beta }}{\beta } ) =f ( \frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta } ) \) for all \(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta } \in {}[ 0,\infty )\) if conformable fractional double Laplace of \(L_{x}^{\alpha }L_{t}^{\beta } ( f ( \frac{x^{\alpha }}{ \alpha },\frac{t^{\beta }}{\beta } ) ) =F_{\alpha , \beta } ( p,s ) \) exists, then
Proof
By using the definition of conformable fractional double Laplace transform, we have
Let \(\frac{x^{\alpha }}{\alpha }=\frac{u^{\alpha }}{\alpha }+\frac{ \lambda ^{\alpha }}{\alpha }\) and \(\frac{t^{\beta }}{\beta }=\frac{v ^{\beta }}{\beta }+\frac{\mu ^{\beta }}{\beta }\) in the second double integral, we get
Consequently,
we have
 □
3 The conformable Laplace transform of the convolution product
Theorem 3
Let \(f(\frac{t^{\beta }}{\beta })\) and \(g(\frac{t^{\beta }}{\beta })\) be integrable functions. If the convolution of \(f(\frac{t^{\beta }}{ \beta })\) and \(g(\frac{t^{\beta }}{\beta })\) is denoted by
then the conformable fractional Laplace transform of the convolution product is defined as follows:
where symbols \(F_{\beta } ( s ) \) and \(G_{\beta } ( s ) \) indicate the conformable fractional Laplace transforms of \(f ( \frac{t^{\beta }}{\beta } ) \) and \(g ( \frac{t ^{\beta }}{\beta } ) \) respectively.
Theorem 4
Let \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) and \(g(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) have a conformable fractional double Laplace transform. Then the conformable fractional double Laplace transform of the double convolution of \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) and \(g(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) is
Therefore one has the equality
where \(F_{\alpha ,\beta } ( p,s ) \) and \(G_{\alpha ,\beta } ( p,s ) \) indicate the conformable fractional double Laplace transforms of \(f ( \frac{x^{\alpha }}{\alpha },\frac{t ^{\beta }}{\beta } ) \) and \(g ( \frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta } ) \) respectively.
Proof
If we apply the definition of conformable fractional double Laplace transform and fractional double convolution above, then we obtain
Let \(\frac{\mu ^{\alpha }}{\alpha }=\frac{x^{\alpha }}{\alpha }-\frac{ \zeta ^{\alpha }}{\alpha }\) and \(\frac{\nu ^{\beta }}{\beta }=\frac{t ^{\beta }}{\beta }-\frac{\eta ^{\beta }}{\beta }\), then \(x^{\alpha -1}\,dx= \mu ^{\alpha -1}\,d\mu \), \(t^{\beta -1}\,dt=\nu ^{\beta -1}\,d\nu \), and applying the valid extension of upper bound of integrals to \(\frac{x^{\alpha }}{\alpha }\rightarrow \infty \) and \(\frac{t^{\beta }}{\beta }\rightarrow \infty \), we get
Both functions \(f ( \frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{ \beta } ) \) and \(g ( \frac{x^{\alpha }}{\alpha },\frac{t ^{\beta }}{\beta } ) \) have zero value for \(\frac{t^{\beta }}{ \beta } <0\) and \(\frac{x^{\alpha }}{\alpha } <0\), it follows with respect to the lower limit of integrations that
Therefore,
 □
If the conformable fractional double Laplace transform of the function \(f(\frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta })\) is given by \(L_{x}^{\alpha }L_{t}^{\beta } [ u ( \frac{x^{\alpha }}{ \alpha },\frac{t^{\beta }}{\beta } ) ] =U_{\alpha , \beta }(p,s)\), then the conformable fractional double Laplace transforms of \(\frac{\partial ^{\alpha }u}{\partial x^{\alpha }}\), \(\frac{\partial ^{2\alpha }u}{\partial x^{2\alpha }}\), \(\frac{ \partial ^{\beta }u ( x,t ) }{\partial t^{\beta }}\), and \(\frac{\partial ^{2\beta }u ( x,t ) }{\partial t^{2\beta }}\) are given by
and
Next, we generalize the conformable fractional double Laplace transform for m, n times conformable fractional derivatives.
Theorem 5
Let \(0<\alpha ,\beta \leq 1\) and \(m,n\in \mathbb{N} \) such that \(u ( \frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta } ) \in C^{l} ( \mathbb{R} ^{+}\times \mathbb{R} ^{+} ) \), \(l=\max ( m,n ) \). Also, let the conformable fractional Laplace transforms of the functions \(u ( \frac{x^{\alpha }}{ \alpha },\frac{t^{\beta }}{\beta } )\), \(\frac{\partial ^{m\alpha }u}{ \partial x^{m\alpha }}\), and \(\frac{\partial ^{n\beta }u}{\partial t ^{n\beta }}\). Then
where \(\frac{\partial ^{m\alpha }u}{\partial x^{m\alpha }}\) and \(\frac{\partial ^{n\beta }v}{\partial t^{n\beta }}\) denote m, n times conformable fractional derivatives of function \(u(\frac{x^{\alpha }}{ \alpha },\frac{t^{\beta }}{\beta })\) with order b and a respectively.
The conformable fractional double Laplace transforms of functions \(\frac{x^{\alpha }}{\alpha }f ( \frac{x^{\alpha }}{\alpha },\frac{t ^{\beta }}{\beta } ) \) and \(( \frac{x^{\alpha }}{\alpha } ) ^{n} \frac{\partial ^{\beta }f}{\partial t^{\beta }}\) are given by
4 Conformable double Laplace transform decomposition method applied to singular Boussinesq equation
The aim of this section is to discuss the use of the conformable double Laplace transform decomposition method (CFDLDM) for the linear and nonlinear singular one-dimensional Boussinesq equations. In this work, we define the conformable double Laplace transform of the function \(u ( \frac{x^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta } ) \) by \(U_{\alpha ,\beta }(p,s)\). We suggest here two important problems.
First problem: Consider the linear Boussinesq equation with initial conditions in the following form:
subject to
where the functions \(f ( \frac{x^{\alpha }}{\alpha },\frac{t^{ \beta }}{\beta } ) \), \(f_{1} ( \frac{x^{\alpha }}{\alpha } )\), \(f_{2} ( \frac{x^{\alpha }}{ \alpha } )\), \(a ( \frac{x^{\alpha }}{\alpha } )\), and \(b ( \frac{x^{\alpha }}{\alpha } ) \) are given. Firstly, multiply Eq. (4.1) by \(\frac{x^{\alpha }}{\alpha }\), and then using the properties of partial derivative of the conformable fractional double Laplace transform and single conformable fractional transform for Eq. (4.1) and Eq. (4.2) respectively and using Eq. (3.9) and Eq. (3.10), we obtain
where the conformable Laplace transforms of \(u ( \frac{x^{\alpha }}{\alpha },0 ) \) and \(\frac{\partial ^{\beta }u ( \frac{x ^{\alpha }}{\alpha },0 ) }{\partial t^{\beta }}\) are denoted by \(U_{\alpha } ( p,0 ) =F_{1} ( p,0 ) \), \(\frac{ \partial ^{\beta }u ( p,0 ) }{\partial t^{\beta }}=F_{2} ( p,0 ) \) respectively and
Arranging Eq. (4.3), we get
By applying the integral for both sides of Eq. (4.4), from 0 to p with respect to p, we have
where \(F_{\alpha ,\beta } ( p,s )\), \(F_{1} ( p,0 ) \), and \(F_{2} ( p,0 ) \) are conformable fractional Laplace transforms of the functions \(f ( \frac{x^{ \alpha }}{\alpha },\frac{t^{\beta }}{\beta } )\), \(f_{1} ( \frac{x^{\alpha }}{\alpha } ) \), and \(f_{2} ( \frac{x^{ \alpha }}{\alpha } ) \) respectively. The solution is obtained by taking the inverse conformable fractional double Laplace transform for Eq. (4.5)
where \(L_{p}^{-1}L_{s}^{-1}\) indicates double inverse conformable fractional double Laplace transform.
The conformable fractional double Laplace transform decomposition method (CFDLDM) defines the solutions \(u(\frac{x^{\alpha }}{\alpha },\frac{t ^{\beta }}{\beta })\) with the help of infinite series as follows:
By substituting Eq. (4.7) into Eq. (4.6), we obtain
By comparing both sides of Eq. (4.6), we get
In general, the remaining terms are given by
where the inverse conformable fractional double Laplace transform is given by \(L_{p}^{-1}L_{s}^{-1}\). To explain this method for solving the conformable fractional Boussinesq equation, we let \(a ( \frac{x^{\alpha }}{\alpha } ) =1\), \(b ( \frac{x^{\alpha }}{\alpha } ) =-1\), and \(f ( \frac{x ^{\alpha }}{\alpha },\frac{t^{\beta }}{\beta } ) =- ( \frac{x ^{\alpha }}{\alpha } ) ^{2}\sin \frac{t^{\beta }}{\beta }-2 \sin \frac{t^{\beta }}{\beta }\), in Eq. (4.1), we obtain the following example.
Example 4
Consider a singular conformable fractional Boussinesq equation in one dimension
subject to the initial condition
Multiplying both sides of Eq. (4.11) by \(\frac{x^{\alpha }}{ \alpha }\) and applying the definition of partial derivatives of the conformable fractional double Laplace transform, single conformable fractional transform for Eq. (4.12) respectively, we have
Applying the integral for Eq. (4.13), from 0 to p with respect to p, we get
By implementing the inverse Laplace transform on both sides of Eq. (4.14), we have
Substituting Eq. (4.7) into Eq. (4.15), we have
By applying the conformable fractional double Laplace transform decomposition method, we obtain
and
hence
and
By applying Eq. (4.7), we get
Therefore, the solution is denoted by
Second problem: Consider the following general form of the nonlinear singular Boussinesq equation in one dimension:
with the initial condition
where the functions \(a(\frac{x^{\alpha }}{\alpha })\), \(b ( \frac{x ^{\alpha }}{\alpha } )\), \(c ( \frac{x^{\alpha }}{\alpha } ) \), and \(d ( \frac{x^{\alpha }}{\alpha } ) \) are arbitrary. In order to obtain the solution of Eq. (4.18), first, by multiplying Eq. (4.18) by \(\frac{x^{\alpha }}{\alpha }\) and taking conformable fractional double Laplace transform, we have
Second, applying Eq. (3.9), Eq. (3.10), Eq. (3.8) and the initial condition given in Eq. (4.19), one can get that
where
By taking the integral for Eq. (4.21), from 0 to p with respect to p, we have
The third step, using (CFDLDM), the solution can be written in infinite series as in Eq. (4.7). Applying the inverse transform for Eq. (4.22), we obtain
Furthermore, the nonlinear terms \(\frac{\partial ^{\beta }u}{\partial t^{\beta }}\frac{\partial ^{2\alpha }u}{\partial x^{2\alpha }}\) and \(\frac{\partial ^{\alpha }u}{\partial x^{\alpha }}\frac{ \partial ^{\alpha +\beta }u}{\partial x^{\alpha }\partial t^{\beta }}\) can be defined by
We have a few terms of the Adomian polynomials for \(A_{n}\) and \(B_{n}\) that are denoted by
and
where \(n=0,1,2,\ldots \) . By putting Eq. (4.25), Eq. (4.26), and Eq. (4.24) into Eq. (4.23), we get
where \(A_{n}\) and \(B_{n}\) are defined as
and
Hence, from Eq. (4.27) above, we have
and
In the next example, the proposed method is applied in order to obtain the solution of the nonlinear singular Boussinesq equation.
Example 5
Consider that nonlinear singular Boussinesq equation in one dimension is governed by
subject to the initial condition
In order to implement our method for Eq. (4.31), we have
and
The first iteration is given by
where
Therefore,
In the same way,
hence,
Similarly,
Therefore,
Then we have
By using Eq. (4.7) the series solutions is denoted by
and hence the conformable solution is given by
5 Conclusion
In the present research, we proposed a combination of conformable double Laplace transform and decomposition methods to solve the singular linear and nonlinear Boussinesq equations. The new method, developed in the current work, was tested on two examples. In addition, if we let \(\alpha =1\) and \(\beta =1\) in two examples, we obtain the solutions which are studied in [19].
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The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this research group No (RG-1440-030).
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Eltayeb, H., Bachar, I. & Gad-Allah, M. Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method. Adv Differ Equ 2019, 293 (2019). https://doi.org/10.1186/s13662-019-2230-1
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DOI: https://doi.org/10.1186/s13662-019-2230-1