1. Introduction
The beam ranks among the most fundamental and practical mechanical structures in engineering issue modeling. Beams are used to create models of real-life structures like bridges, machine components, wind turbines, aircraft wings, and helicopter blades. Scientific and engineering professionals can simplify three-dimensional difficulties using beam theories. One can find several beam theories in many sources. Popular theories include the classical beam theory (Euler–Bernoulli), the Reddy beam theory, and the Timoshenko beam theory [
1]. Many authors are interested in investigating beam problems from various aspects. For example, the authors of [
2] developed analytic solutions for Euler–Bernoulli beams. Other analytic solutions were developed in [
3] to nonlocal Euler–Bernoulli beam equations. In [
4], A closed-form solution for non-uniform Euler–Bernoulli beams was presented. Other types of beam equations were given in [
5]. From a numerical perspective, the beam types of equations were handled by many methods. For example, the authors of [
6] applied a finite difference method for the Euler–Bernoulli beam equation, two collocation approaches were followed in [
7,
8], and the Lagrange interpolation method was used in [
9]. One can refer to [
10,
11,
12,
13] for other contributions.
In recent times, spectral methods have become increasingly popular in computational mechanics and applied mathematics due to their effectiveness and accuracy in solving various differential equations. These methods provide substantial accuracy and efficiency benefits for numerous applications in scientific computing, engineering, and physics, particularly for issues with smooth solutions; see, for example, [
14,
15,
16]. There are different methods of spectral methods and their variants. The three main methods are the tau, collocation, and Galerkin methods. To apply the different spectral methods, one should choose two families of functions, namely, test and trial functions. The choice of these functions varies from one method to another. To apply the Galerkin method, we must select the basis functions to fulfill the underlying boundary conditions; see, for example, [
17,
18]. This restriction is unnecessary when applying the tau method; see, for example, [
19,
20]. The collocation method can be applied without any restrictions on the basis functions. In addition, it can be applied to all types of differential equations; see, for example, [
21,
22,
23].
As a modification of the standard Galerkin method, the Petrov–Galerkin scheme is notable for its proficiency in handling intricate boundary conditions. This paper presents an application of the spectral Petrov–Galerkin method to solve the fourth-order uniform Euler–Bernoulli beam equation, which is very important in structural engineering and biomechanics. The fourth-order uniform Euler–Bernoulli Beam Equation encapsulates the dynamic response of slender beams under diverse loading conditions. Its fractional derivative term accounts for non-local effects, making it a powerful tool for modeling viscoelastic materials and systems with memory. Considering both the clamped–clamped and pinned–pinned cases, we encompass a broad spectrum of practical scenarios encountered in engineering applications. For recent advances in Euler–Bernoulli beam theory, see [
24,
25,
26,
27].
At the core of our methodology lies the utilization of second-kind Chebyshev polynomials, with [
28] as a foundation in spatial variables. These polynomials possess distinct characteristics that render them highly suitable for spectral methods, notably their capability to accommodate multiple types of boundary conditions simultaneously. This adaptability is crucial in tackling the Euler–Bernoulli beam equation, which commonly involves clamped–clamped and pinned–pinned boundary conditions in practical settings.
The Petrov-Galerkin method [
29,
30,
31,
32,
33] is a mathematical approach employed to approximate solutions of partial differential equations that involve terms with odd orders and where the test function and solution function reside in distinct function spaces. It can be considered an expansion of the Bubnov–Galerkin method, wherein the bases of test and solution functions coincide. From an operator standpoint of the differential equation, the Petrov–Galerkin method involves applying a projection that may not be orthogonal, unlike the Bubnov–Galerkin method.
To simulate memory and hereditary characteristics in real-life phenomena, fractional calculus and differential equations are important in this regard. Applications include simulating blood flow, drug distribution, and nerve impulses in physiology, as well as tackling viscoelasticity, heat conduction, and fluid dynamics problems in engineering physics and civil engineering. See the seminal textbook on fractional differential equations ([
34,
35]) for more in-depth information.
The organization of this paper is as follows:
Section 2 provides an overview of the preliminaries underlying our work, including relevant properties of the Chebyshev polynomials of the second kind.
Section 3 details the spectral Petrov–Galerkin method employed to solve the Euler–Bernoulli beam equation, elucidating the explicit formulas derived for inner products and the resulting system of algebraic equations. In
Section 5, we delve into the convergence analysis of the method.
Section 6 accounts for the computational complexity of the resulting matrix system.
Section 7 presents numerical examples demonstrating the efficacy and applicability of our approach. Finally,
Section 8 offers concluding remarks and avenues for future research, consolidating the contributions and implications of our work in advancing the field of computational mechanics.
4. An Extension to Fractional Case
In this section, we consider the following time-fractional fourth-order partial differential equations (TFFPDE):
subject to the initial and boundary conditions (
20) and (
21), where
is the Caputo’s fractional derivative defined as follows ([
34]):
The residual
of Equation (
37) has the following form:
The application of the PGM implies that
which can be rewritten as follows:
where
are those defined in Theorem 2, and
In addition, the conditions in (
20) lead to the following equations:
Now, Equations (
41) and (
42) yield a linear system of algebraic equations of dimension
in the unknown expansion coefficients
, which can be solved using the Gaussian elimination procedure.
Theorem 3. The elements are given by the following:wherewhere is the regularized hypergeometric function. Proof. The application of Caputo’s fractional derivative (
38) enables us to write
Using the previous property along with relation (
1), one can write
Inserting the following identity [
41] in the previous equation can obtain the desired result:
This completes the proof of this theorem. □
6. Computational Complexity
In numerical linear algebra, the computational expense of computing a matrix’s inverse significantly influences the viability and effectiveness of different algorithms. For pentadiagonal matrices, distinguished by their sparse nature featuring only five nonzero diagonals, the computational workload is notably diminished compared to denser matrices. As highlighted in “Matrix Computations” by Gene H. Golub and Charles F. Van Loan [
42], sparse matrices, such as pentadiagonal matrices, offer significant computational advantages due to their sparsity. Specialized algorithms tailored for solving systems involving pentadiagonal matrices, while not as well-known as those for tridiagonal systems, exhibit computational complexities on the order of
, where
n represents the size of the matrix. This contrasts sharply with the
complexity typically associated with dense matrix inversions. Thus, utilizing efficient algorithms tailored for pentadiagonal matrices underscores a marked reduction in computational cost, making them a preferred choice for various numerical computations in engineering, physics, and other scientific domains.
As a direct conclusion from the system of Equation (
30), we ended up with the following comment on the computational complexity of the method. The efficient structure of the system of algebraic equations derived from the Euler–Bernoulli beam problem using the Petrov–Galerkin approach is notable. The system becomes a combination of tridiagonal, pentadiagonal, and diagonal matrices through temporal and spatial discretization. Interestingly, this combination results in a final pentadiagonal matrix that captures the essence of the situation. With the help of the Thomas method designed specifically for pentadiagonal systems, the inversion procedure exhibits a computational complexity of
, which is significantly less than that of denser matrices. This tactical approach guarantees both numerical stability and efficient solution procedures, making it especially appropriate for solving complex structural dynamics in engineering and related fields.
For example, the matrices
and
take the following forms for
: