Abstract
The major goal of this research is to develop and test a numerical technique for solving a linear onedimensional telegraph problem. The generalized polynomials, namely, the generalized Lucas polynomials are selected as basis functions. To solve the linear onedimensional telegraph type equation, we solve instead its corresponding integral equation via the application of the spectral Galerkin method that serves to convert the equation with its underlying conditions into a system of linear algebraic equations that may be solved by a suitable numerical solver. The convergence and error analysis of the generalized Lucas expansion are discussed in depth. The current analysis is based on the assumption that the problem’s solution is separable. Finally, some explanatory numerical examples are displayed together with comparisons to some other articles, to demonstrate the suggested method’s validity, applicability, and accuracy.
Introduction
There are many methods that have vital roles in numerical analysis, see for example [23, 24, 33]. Of these methods are the spectral methods that have effective roles in dealing with partial differential equations, ordinary differential equations and fractional differential equations (see [18, 40]). The principal idea of spectral methods is based on the assumption that the approximate solution can be written as linear combinations of certain basis functions which may be orthogonal or otherwise. Common spectral methods include collocation, tau, and Galerkin methods. The latter methods have been widely used by many authors. For example, the typical collocation method is utilized in Atta et al. [8] to treat multiterm fractional differential equations (FDEs). The spectral tau method is applied in Bhrawy et al. [11] along with the Jacobi operational matrix to treat the timefractional diffusionwave equations. Also, a spectral solution of the nonlinear onedimensional Burgers’ equation was developed based on introducing new derivatives formulas of the Chebyshev polynomials of the sixthkind along with the application of the spectral tau method in [1]. The Galerkin approach was followed by Youssri and AbdElhameed [39] to solve the timefractional telegraph equations. For some other articles employ different spectral methods, see for example Refs. [10, 14, 29].
Because of their relevance in a variety of disciplines, several types of special functions have been intensively researched. The Fibonacci and Lucas polynomials, as well as their variations and extensions, are examples of these important special functions (see [27]). These polynomials have been investigated theoretically and practically. For example, the authors in [6] established new connection formulas between the Fibonacci polynomials and some Chebyshev polynomials. Supersymmetric Fibonacci polynomials were discussed in [38]. Numerically, these polynomials were utilized to treat several types of differential equations. For examples, the authors in [2, 5], used respectively, the Fibonacci polynomials and their generalizations to treat some types of FDEs, while the same authors in [3, 4] employed respectively, the Lucas polynomials and their generalized polynomials to treat some types of FDEs. Mixed types of FDEs were treated in [31] via modified Lucas polynomials. In [21], the authors used Fibonacci polynomials for proposing numerical solutions for the variableorder spacetime fractional Burgers–Huxley equation. Fibonacci wavelets were utilized in [35] along with the Galerkin method to treat some types of fractional optimal control problems. In [9], the authors used the generalized Fibonacci operational tau method to treat the fractional Bagley–Torvik Equation. An approximate solution of the twodimensional Sobolev equation was proposed by employing mixed Lucas and Fibonacci polynomials in [20].
It is known that the telegraph equation is one of the most important problems because it describes many phenomena in different fields. For example, it describes energetic particle transport in the interplanetary medium [19]. Recently, the telegraph type equation has been discussed by many authors. For example, in [16], the authors have proposed a numerical algorithm to solve the onedimensional hyperbolic telegraph equation using the Galerkin algorithm. In [13], the authors have discussed the onedimensional hyperbolic telegraph equation using the collocation method. For further studies, one can see [25, 26, 32, 36].
Our main objectives in the current paper can be listed in the following items:

Stating and proving some new theorems concerned with the generalized Lucas polynomials (GLPs) and their modified ones.

Employing the modified GLPs to obtain a numerical solution of the onedimensional hyperbolic telegraph equation.

Investigate carefully the convergence analysis arising from the proposed modified generalized Lucas expansion.

Performing some comparisons of the proposed technique with other methods to clarify the efficiency and accuracy of the presented technique.
To the best of our knowledge, there are some advantages of the proposed technique can be mentioned as follows:

Selecting the basis functions in terms of the modified GLPs enables one to obtain approximate solutions with high accuracy by taking a few numbers of the retained modes. This leads to less computational time and computational errors.

The presence of two parameters in the GLPs enables one to obtain various approximate solutions for different choices of the two involved parameters.
The above advantages motivate our interest to employ the GLPs. In addition, the numerical investigations based on the GLPs are few. This also gives us another motivation to utilize numerically this kind of polynomials.
The following is how the paper is structured: Section 2 is devoted to presenting some preliminary information and relationships of the GLPs that will be utilized throughout the work. Section 3 is interested in transforming the linear onedimensional telegraph type equation into its corresponding integral equation. Section 4 focuses on the selection of the basis function and the utilization of the Galerkin method to solve the linear onedimensional telegraph problem. Section 5 presents a description of the convergence and error analysis of the proposed expansion. Section 6 displays the numerical results accompanied by some comparisons. Finally, in Sect. 7, a conclusion is displayed.
Some fundamental properties of the GLPs
The following recurrence relation
generates the sequence of GLPs with the initial values:
where a and b be any nonzero real constants.
The power form representation of the GLPs is given by
which can be expressed alternatively as
where
The Binet’s form for GLPs can be written in the following form:
Integral form of linear onedimensional telegraph type equation
In this section, we focus on transforming the linear onedimensional telegraph type equation into its corresponding integral equation, and after that, we handle the integral equation numerically using our presented technique in the next section.
Consider the following linear onedimensional telegraph type equation [13]:
subject to the initial conditions:
and the nonhomogeneous boundary conditions:
where f(x, t) represents the source term and \(\alpha \), \(\beta \) are real constants.
By integrating Eq. (3.1) twice with respect to the variable t to reduce the number of conditions, one gets
subject to the following nonhomogeneous boundary conditions:
with
Therefore, Eq. (3.4) may be solved under condition (3.5) instead of solving Eq. (3.1) under the conditions (3.2) and (3.3).
To further develop our spectral algorithm for treating (3.4)–(3.5), the following transformation is suitable to transform the nonhomogeneous boundary conditions into the homogeneous ones [15]:
Now, Eqs. (3.4)–(3.6) can be combined to give the following integral equation:
subject to the homogeneous boundary conditions:
where
Numerical spectral treatment for linear onedimensional telegraph type equation
This section focuses on analyzing a spectral Galerkin algorithm for numerically solving the linear onedimensional telegraph type equation. First, we consider the following two kinds of basis functions:
Consider the following two spaces:
where \(\Omega =(0,\ell )\times (0,\tau ]\).
Now, two important theorems related to the basis functions \(\psi _i(x)\) and \(\gamma _j(t)\) are stated and proved. The first theorem gives an expression for the ntimes repeated integrations of \(\gamma _j(t)\), while the second introduces an expression for nth derivative of \(\psi _i(x)\).
Lemma 4.1
The general formula that converts the ntimes repeated integrals to a single integral is given by [37]
Theorem 4.2
The ntimes repeated integrations of \(\gamma _j(x)\) are given by
where
and
Proof
Making use of relation (4.3), one finds
For \(j=0\), with the aid of \(\gamma _0(t)=2\), one gets
After integrating the previous equation, we get the desired result.
For \(j\ge 1\), one has
Based on substituting by the formula (2.2) into (4.4) yields
and accordingly, the result for \(j\ge 1\) can be obtained after integrating Eq. (4.5). \(\square \)
As a direct consequence of Theorem 4.2, two specific integrals formulas can be deduced. The following corollary exhibits these formulas.
Corollary 4.3
The following two integrals formulas hold:
Theorem 4.4
For all positive integers i,n and \(i\ge {n}\), the nth derivative of \(\psi _i(x)\) can be expressed in the form
where
\(\delta _{i+n2}\) is defined in (2.3) and
Proof
Equation (2.2) enables one to expand \(\psi _{i}(x)\) in the following form:
Based on the wellknown identity:
the following relation can be obtained
After performing some rather lengthy manipulations, the last formula turns into
Replace \(k\rightarrow k+n1\), the desired formula (4.6) can be obtained. \(\square \)
As a direct consequence of Theorem 4.4, the following special result holds:
Corollary 4.5
For all \(i\ge 2\), one has
Galerkin technique for handling Eq. (3.7)
If we assume that \(v(x,t)\in P(\Omega )\), then v(x, t) can be written as
Now, and for the sake of applying the Galerkin method, we first compute the residual of Eq. (3.7). It can be written in the form
In virtue of Eqs. (4.1), (4.2) and Corollaries 4.3, 4.5, the residual \({\varvec{R}}(x,t)\) may be obtained. And hence the application of the Galerkin method leads to
Now, Eq. (4.7) generates a linear system of equations in the unknown expansion coefficients \(c_{ij}\) of dimension \((M +1)^2\), they may be solved via Gaussian elimination procedure.
Convergence and error analysis
In this section, the convergence and error analysis of the proposed double generalized Lucas expansion are discussed. Several required lemmas are employed in this discussion.
Lemma 5.1
As shown in Abramowitz and Stegun [7], the following inequality holds:
where \(I_n(x)\) is the modified Bessel function of order n of the first kind.
Lemma 5.2
As shown in Luke [28], the following inequality holds:
Lemma 5.3
As shown in AbdElhameed and Youssri [4], let f(x) be an infinitely differentiable function at the origin. Then
Lemma 5.4
The following inequality holds for GLPs:
where \(\epsilon =\sqrt{a^2\,\ell ^2+2\,b}\).
Proof
We proceed by induction on i. Suppose that the inequality (5.1) is true at \((i1)\) and \((i2)\), one gets
recurrence relation (2.1) and inequalities (5.2) enable us to write
With the aid of the following identity
one finds
inserting the last inequality into Eq. (5.3) yields
We obtain the desired result. \(\square \)
Lemma 5.5
The following inequality holds:
Proof
Eq. (4.1) enables one to write
by making use of Lemma 5.4, the desired inequality (5.4) can be obtained. \(\square \)
Theorem 5.6
If a function \(u(x,t)=x\,(\ell x)\,f_{1}(x)\,f_{2}(t)\in L^{2}(\Omega )\), with \(f_{k}^{(i)}(0)\le Q_{k}^i\), \(k=1,2\), \(i\ge 0\), \(Q_{k}\) is a positive constant. And if u(x, t) has the expansion \(u(x,t)=\displaystyle \sum _{i=0}^{\infty }\displaystyle \sum _{j=0}^{\infty }c_{ij}\,\psi _i(x)\,\gamma _j(t),\) the following conclusions are obtained:

1.
\(c_{ij}\le \frac{a^{ij}\,Q_{1}^{i}\,Q_{2}^{j}\,A}{i!\,j!}\), where \(A=\cosh (2\,a^{1}\,b^{\frac{1}{2}}\,Q_{1})\,\cosh (2\,a^{1}\,b^{\frac{1}{2}}\,Q_{2})\).

2.
The series converges absolutely.
Proof
With the aid of Lemma 5.3 and according to the assumption \(u(x,t)=x\,(\ell x)\,f_{1}(x)\,f_{2}(t)\), we have
using the assumption \(f_{1,2}^{(i)}(0)\le Q_{1,2}^i\), one can write
By making use of Lemma 5.1, one gets
Now, the application of Lemma 5.2 leads to
which proves the first part of Theorem 5.6.
To prove the second part of Theorem 5.6, using the inequality of the first part, we have
with the aid of Lemmas 5.4 and 5.5, one gets
Applying the comparison test implies that the series \(\displaystyle \sum _{i=0}^{\infty }\displaystyle \sum _{j=0}^{\infty }\left c_{ij}\psi _i(x)\,\gamma _j(t)\right \) converges absolutely. \(\square \)
Theorem 5.7
If \(u(x,t)\in L^{2}(\Omega )\), satisfy the assumptions of Theorem 5.6, one gets
where \(C_{\xi }=2\,A\,\ell ^2\,e^{2\,\xi }\).
Proof
It is clear that
Theorem 5.6 enables us to write
where \(\xi _{1}={a^{1}\,Q_{1}\,\epsilon }\) and \(\xi _{2}={a^{1}\,Q_{2}\,\epsilon }\).
Now, Inequality (5.5) may be formulated as
where \(\Gamma (.)\), \(\Gamma (.,.)\) and \(\gamma (.,.)\) denote, respectively, gamma, upper incomplete gamma, and lower incomplete gamma functions [22].
In virtue of simple inequality: \(e^{t}\le 1,\quad \forall \ t\ \ge 0\), one gets
Take \(\xi =\max \{\xi _{1},\xi _{2}\}\), the desired result can be obtained. \(\square \)
Illustrative examples
In this section, the generalized Lucas Galerkin method (GLGM) is applied for obtaining a numerical solution to the linear onedimensional telegraph type equation with different conditions. The accuracy of the numerical results is measured using \(L_\infty \), \(L_2\) and root mean square error (RMSE).
Remark 6.1
All results of Examples 1, 2, 3 and 4 are calculated at \(a=b=1\).
Example 6.2
As given in [16, 34], consider Eqs. (3.1)–(3.3) with the following choices:
where the exact solution is \(u(x,t)=x^2+t\). Now, applying the technique described in Section 3, one gets
subject to the homogeneous boundary conditions:
and the exact solution is: \(v(x,t)=x^2x\).
The application of GLGM described in Sect. 4 for \(M=1\) yields the following system of equations:
which yield \(\{c_{00}=\frac{1}{4},\,c_{01}=0,\,c_{10}=0,\,c_{11}=0\}\), and therefore \(v_M(x,t)=x^2x\), that is \(u_M(x,t)=x^2+t\), which is the exact solution.
Example 6.3
As given in [30], consider Eqs. (3.1)–(3.3) with the following choices:
where the exact solution is \(u(x,t)=(xx^2)\,t^2\,e^{t}\).
In Table 1, a comparison between the numerical solutions with the exact solution is presented for the case corresponding to \(M=8\) at different values of time t. Table 2 presents the best \(L_2\) and \(L_\infty \) errors compared with those obtained in [30]. Also, Table 3 shows the computational time (CPU time) for different values of M. In addition, Figure 1 shows the maximum absolute errors at different values of t for the case \(M=8\). We can see from Tables 1, 2 and Fig. 1 that the proposed method is appropriate and effective.
Example 6.4
As given in [12], consider Eqs. (3.1)–(3.3) with the following choices:
where the exact solution is: \(u(x,t)=\cos (t)\,\sin (x)\).
The RMSEs for different values of \(\alpha \), \(\beta \) and time t at \(M=3,5,7\) are shown in Table 4. Table 5 compares the RMSEs with those obtained in [12] at \(M=7\) and \(t=0.5\). In Table 6, we present the absolute error for \(M=7\) at different values of \(\alpha \), \(\beta \) and t. Furthermore, Fig. 2 shows the graphs of the approximate solution and absolute error for \(\alpha =40\), \(\beta =100\) at \(M=8\). The results of Tables 4, 5, 6 and Fig. 2 show that our numerical results by taking few terms of the proposed Generalized Lucas expansion are more accurate.
Example 6.5
As given in [17], consider Eqs. (3.1)–(3.3) with the following choices:
where the exact solution is \(u(x,t)=e^{2\,t}\,\sinh (x)\).
In Table 7, the absolute errors that obtained by GLGM in solving the problem for different values of \(\alpha \), \(\beta \) at different values of time t are listed at \(M=8\). In Table 8, we give a comparison between the maximum absolute errors obtained from the application of the numerical scheme presented in [17] and our method for the two cases corresponding to \(\alpha =20,\, \beta =25\) at \(M=7\) and \(\alpha =40,\, \beta =100\) at \(M=8\), respectively. Figure 3 shows the absolute error for the case \(\alpha =20,\, \beta =25\) at \(M=7\). We can see from the tabulated absolute errors of Tables 7, 8 and Fig. 3 that the proposed method is suitable and powerful for solving the linear onedimensional telegraph equation.
Concluding remarks
In this paper, a new numerical technique to solve the onedimensional linear hyperbolic telegraph type equation using the Galerkin method was analyzed in detail. Two new basis functions of the generalized Lucas polynomials were employed as basis functions, and the spectral Galerkin method is applied to reduce the equation governed by its conditions to a linear system of equations that may be solved with the aid of a suitable numerical solver. The convergence and error analysis of the generalized Lucas expansion were deeply investigated. In addition, our numerical findings are compared with exact solutions and with the solutions obtained by some other approximate methods. These results demonstrate the good accuracy and applicability of this technique.
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Youssri, Y.H., AbdElhameed, W.M. & Atta, A.G. Spectral Galerkin treatment of linear onedimensional telegraph type problem via the generalized Lucas polynomials. Arab. J. Math. (2022). https://doi.org/10.1007/s40065022003740
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DOI: https://doi.org/10.1007/s40065022003740
Mathematics Subject Classification
 11B39
 65M70
 35L52