An Effective Projection Method for Solving a Coupled System of Fractional-Order Bagley–Torvik Equations via Fractional Shifted Legendre Polynomials

Abstract

This work proposes a valuable and successful strategy for approximating the solutions to the Bagley–Torvik system, which plays an essential role in fractional calculus. The Caputo sense is used to derive the basic conformable fractional. The Bagley–Torvik problem is numerically solved in this study using an effective symmetric projection method. From this symmetry, there are some interesting original results. The proposed approach has two key benefits. We began by converting the connected fractional Bagley–Torvik equations into two fractional-order Bagley–Torvik equations, which we then solved using the current method. Second, two linear equation systems are solved to obtain approximate solutions.

1. Introduction

Because of its applications, fractional calculus has received a lot of attention from mathematicians and scientists in recent decades. Numerous investigators have shown the ramifications of fractional order derivative as a different physical phenomenon. Nonlinear dynamics, Maclaurin theory, theoretical physics, biomedicine, polymeric materials, dynamical mechanics, electron microscopy, transportation system, discharge and viscoelasticity, confusion and geometrisc, identification and modeling, filtration, financial system, and finance are all implementations of fractional differential problems.

The Bagley–Torvik equation plays a critical part in fractional differential equations, because it appears in the mathematical model of the movement of a big thin plate immersed in a Newtonian fluid. Torvik and Bagley introduced this fractional differential equation for the first time in 1984. The researchers stated that the fractional derivative is usually noticed in the behavior of real materials (see [1]). The authors of [2] investigated the stability conditions for linear matrix inequality in fractional order systems. In [3], a novel approach for fractional-order inputted systems is presented that is deduced from the guardian map and described as a family of monoparametric pseudo-polynomials. Reference [4] discussed a numerical approximation to a class of variable-order fractional functional boundary value problems.

The Bagley–Torvik equation has been investigated by a number of researchers using various approximate numerical methods. The goal of [5] is to propose and study a novel operational Tau approach with a high order to achieve the numerical solution to a Bagley–Torvik problem that uses Chebyshev polynomials as basis functions. Some derivatives of these equations’ solutions have a singularity at their origin, according to the author. The authors of [6] proposed a new numerical method for approximating the solution to the Bagley–Torvik problem. The fundamental goal of their strategy is to obtain a more general approximation solution in the form of Müntz–Legendre polynomials. The authors of [7] proposed an effective numerical technique founded on shifted Chebyshev polynomials for deriving numerical solutions to the Bagley–Torvik equation. The authors of [8] considered restructuring the Bagley–Torvik equation as a system of fractional differential equations of the order $1/2$. An approximate method for solving the Bagley–Torvik problem over the range $[0,1]$ was shown by the authors of [9], via the second-order Chebyshev wavelet.

Projection approximation methods are at the heart of approximation theory and can be used for a wide range of things, including solving integral and integro-differential equations (see [10,11]).

This paper serves two main purposes. On one hand, we extend projection methods to numerically solve a system of Bagley–Torvik equations. We offer a new symmetric projection approach based on the fractional shifted Legendre polynomials. This symmetry leads to some interesting original results. On the other hand, we use a new procedure to solve the system of Bagley–Torvik equations.

There are two main advantages to the suggested method. We started by converting the problem into a system of two separable fractional-order Bagley–Torvik equations, which we then solved with the current method. Second, two systems of linear equations are solved to achieve approximate solutions.

2. Preliminaries

We will start by reviewing certain basic fractional calculus vocabulary and mathematical notations in this part.

Let $\Gamma$ denote the Euler Gamma function, which is fundamental in fractional calculus theory.

Definition 1.

The left sided Riemann–Liouville fractional integral of order $\mu >0$ of an integrable function $\phi :(0,\infty )\to \mathbb{R}$ is described by

$$J_{0_{+}}^{\mu }\phi \left(s\right):=\frac{1}{\Gamma \left(\mu \right)}{\int }_0^s\frac{\phi \left(\tau \right)}{{(s-\tau )}^{1-\mu }}d\tau ,s>0.$$

Remark 1.

The above integral can be represented in convolution form as follows:

$$J_{0_{+}}^{\mu }\phi \left(s\right)=({\psi }_{\mu }\star \phi )\left(s\right),$$

where

$${\psi }_{\mu }\left(s\right):=\left\{\begin{array}{cc}{s}^{\mu -1}/\Gamma \left(\mu \right),\hfill & s>0,\hfill \\ 0\hfill & s\le 0.\hfill \end{array}\right.$$

Definition 2.

The left sided Riemann–Liouville fractional derivative of order $0<\mu <1$ of a continuous function $\phi :(0,\infty )\to \mathbb{R}$ is defined by

$$D_{0_{+}}^{\mu }\phi \left(s\right):=\frac{1}{\Gamma (1-\mu )}\frac{d}{ds}{\int }_0^s\frac{\phi \left(\tau \right)}{{(s-\tau )}^{\mu }}d\tau .$$

Remark 2.

For $0<\mu <1$, we have

$$D_{0_{+}}^{\mu }\phi \left(s\right)=\frac{d}{ds}{J}_{0_{+}}^{1-\mu }\phi \left(s\right).$$

$$J_{0_{+}}^{\mu }{D}_{0_{+}}^{\mu }\left(\phi \left(s\right)-\phi \left(0\right)\right)=\phi \left(s\right)-\phi \left(0\right).$$

Definition 3.

For an absolutely continuous function $\varphi (.)$, the Caputo fractional derivative of order $0<\alpha <1$ is defined by

$${}^c{D}_{0_{+}}^{\alpha }\varphi \left(s\right)=J_{0_{+}}^{1-\alpha }\frac{d}{ds}\varphi \left(s\right)=\frac{1}{\Gamma (1-\alpha )}{\int }_0^s{(s-t)}^{-\alpha }{\varphi }^{\prime }\left(t\right)dt.$$

Remark 3.

1. 

We have

$${}^c{D}_{0_{+}}^{\mu }\phi \left(s\right)=J_{0_{+}}^{1-\mu }\frac{d}{ds}\phi \left(s\right),$$

2. 

For a continuous function φ, the relationship between the Caputo and Riemann–Liouville fractional derivatives is provided by

$${}^c{D}_{0_{+}}^{\mu }\phi \left(s\right)=D_{0_{+}}^{\mu }(\phi \left(s\right)-\phi \left(0\right)),0<\mu <1,$$

and

$${}^c{D}_{0_{+}}^{\mu }\phi \left(s\right)=D_{0_{+}}^{\mu }\left(\phi \left(s\right)-\sum _{k=0}^{r-1}\frac{s^k}{k!}{\phi }^{\left(k\right)}\left(0\right)\right),s>0,r-1<\mu <r.$$

Reference [12] demonstrates the fundamentals of fractional differential equations.

3. System of Coupled Fractional Bagley–Torvik Equations

Let us consider the following system of coupled fractional Bagley–Torvik equations:

$$\left\{\begin{array}{c}{A}_1{\mathcal{D}}^{2}u\left(s\right)+A_2{\mathcal{D}}^{\nu }u\left(s\right)+A_{3}v\left(s\right)=f\left(s\right),0\le s\le R,1<\nu <2,\hfill \\ A_1{\mathcal{D}}^{2}v\left(s\right)+A_2{\mathcal{D}}^{\nu }v\left(s\right)+A_{3}u\left(s\right)=g\left(s\right),\hfill \end{array}\right.$$

(1)

depending on the initial conditions (ICs):

$$\begin{array}{ccc}\hfill u\left(0\right)& ={\alpha }_0,u^{\prime }\left(0\right)\hfill & \hfill ={\alpha }_1,\\ \hfill v\left(0\right)& ={\delta }_0,v^{\prime }\left(0\right)\hfill & \hfill ={\delta }_1,\end{array}$$

or the boundary conditions (BCs):

$$\begin{array}{ccc}\hfill u\left(0\right)& ={\beta }_0,u\left(R\right)\hfill & \hfill ={\beta }_1,\\ \hfill v\left(0\right)& ={\gamma }_0,v\left(R\right)\hfill & \hfill ={\gamma }_1.\end{array}$$

Our first goal is to convert the above system of connected fractional Bagley–Torvik equations into two fractional-order Bagley–Torvik equations. As in [13], let

$$\zeta :=u+v,F:=f-g;$$

$$\xi :=u-v,G:=f+g.$$

Theorem 1.

The problem (1) can be represented as follows

$$\left\{\begin{array}{c}{A}_1{\mathcal{D}}^2\zeta \left(s\right)+A_2{\mathcal{D}}^{\nu }\zeta \left(s\right)+A_3\zeta \left(s\right)=G\left(s\right),\hfill \\ A_1{\mathcal{D}}^2\xi \left(s\right)+A_2{\mathcal{D}}^{\nu }\xi \left(s\right)-A_3\xi \left(s\right)=F\left(s\right),\hfill \end{array}\right.$$

(2)

depending on the initial conditions (ICs):

$$\begin{array}{ccc}\hfill \xi \left(0\right)& ={\rho }_0,{\xi }^{\prime }\left(0\right)\hfill & \hfill ={\rho }_1,\\ \hfill \zeta \left(0\right)& ={\eta }_0,{\zeta }^{\prime }\left(0\right)\hfill & \hfill ={\eta }_1,\end{array}$$

or the boundary conditions (BCs):

$$\begin{array}{ccc}\hfill \xi \left(0\right)& ={\lambda }_0,\xi \left(R\right)\hfill & \hfill ={\lambda }_1,\\ \hfill \zeta \left(0\right)& ={\sigma }_0,\zeta \left(R\right)\hfill & \hfill ={\sigma }_1.\end{array}$$

Proof. 

It is obvious that

$$u=\frac{\zeta +\xi }{2},f=\frac{G+F}{2};$$

$$v=\frac{\zeta -\xi }{2},g=\frac{G-F}{2}.$$

Substituting this into (1) produces

$$\begin{array}{cc}\hfill A_1{\mathcal{D}}^2\left[\zeta \left(s\right)+\xi \left(s\right)\right]+A_2{\mathcal{D}}^{\nu }\left[\zeta \left(s\right)+\xi \left(s\right)\right]+A_3\left[\zeta \left(s\right)-\xi \left(s\right)\right]& =\left[G\left(s\right)+F\left(s\right)\right],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill A_1{\mathcal{D}}^2\left[\zeta \left(s\right)-\xi \left(s\right)\right]+A_2{\mathcal{D}}^{\nu }\left[\zeta \left(s\right)-\xi \left(s\right)\right]+A_3\left[\zeta \left(s\right)+\xi \left(s\right)\right]& =\left[G\left(s\right)-F\left(s\right)\right].\hfill \end{array}$$

(4)

By adding the two Equations (3) and (4) together and by subtracting (4) from (3) we obtain (2). Moreover, for the initial conditions, we have

$$\begin{array}{ccc}\hfill {\rho }_0& :={\alpha }_0-{\delta }_0,{\rho }_1\hfill & \hfill :={\alpha }_1-{\delta }_1,\\ \hfill {\eta }_0& :={\alpha }_0+{\delta }_0,{\eta }_1\hfill & \hfill :={\alpha }_1+{\delta }_1.\end{array}$$

For the boundary conditions, we have

$$\begin{array}{ccc}\hfill {\lambda }_0& :={\beta }_0-{\gamma }_0,{\lambda }_1\hfill & \hfill :={\beta }_1-{\gamma }_1,\\ \hfill {\sigma }_0& :={\beta }_0+{\gamma }_0,{\sigma }_1\hfill & \hfill :={\beta }_1+{\gamma }_1.\end{array}$$

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4. The Fractional-Order Legendre Polynomials

Legendre polynomials are one of physics the most important functions in physics. They are crucial in the theory of approximation, specifically in the solution of integro-differential and integral equations.

The classic Legendre polynomials defined in $[-1,1]$ are famously defined by the following recurrence relation:

$$\begin{array}{cc}\hfill L_0\left(s\right)=& 1,L_1\left(s\right)=s,\hfill \\ \hfill L_{j+1}\left(s\right)=& \frac{2j+1}{j+1}s{L}_j\left(s\right)-\frac{j}{j+1}{L}_{j-1}\left(s\right),j=1,2,3,\dots \hfill \end{array}$$

The following are the first Legendre polynomials:

$${\displaystyle \begin{array}{c}\hfill \begin{array}{ccc}\hfill L_0\left(s\right)& =& {\displaystyle 1;}\hfill \\ \hfill L_1\left(s\right)& =& {\displaystyle s;}\hfill \\ \hfill L_2\left(s\right)& =& {\displaystyle \frac{1}{2}\left(3s^2-1\right);}\hfill \\ \hfill L_3\left(s\right)& =& {\displaystyle \frac{1}{2}\left(5s^3-3s\right);}\hfill \\ \hfill L_4\left(s\right)& =& {\displaystyle \frac{1}{8}\left(35s^4-30s^2+3\right);}\hfill \\ \hfill L_5\left(s\right)& =& {\displaystyle \frac{1}{8}\left(63s^5-70s^3+15s\right).}\hfill \end{array}\end{array}}$$

Let the shifted Legendre polynomials $L_i(\frac{2\tau }{T}-1)$ be denoted by $L_i^S\left(\tau \right)$, $\tau \in \left[0,T\right]$, from which the following recurrence relation can be achieved.

$$\begin{array}{cc}\hfill L_0^S\left(\tau \right)=& 1,L_1^S\left(\tau \right)=\frac{2\tau }{T}-1,\hfill \\ \hfill L_{j+1}^S\left(\tau \right)=& \frac{2j+1}{j+1}\left(\frac{2\tau }{T}-1\right)L_j^S\left(\tau \right)-\frac{j}{j+1}{L}_{j-1}^S\left(\tau \right),j=1,2,3,\dots \hfill \end{array}$$

We recall that

$$\begin{array}{c}\hfill {\int }_0^T{L}_n^S\left(\tau \right)L_m^S\left(\tau \right)d\tau =\left\{\begin{array}{cc}\hfill \frac{T}{2n+1}& \hfill n=m,\\ 0\hfill & \hfill n\ne m.\end{array}\right.\end{array}$$

(5)

Regarding the formulation of the shifted Legendre polynomials, the fractional shifted Legendre polynomials (FSLPs) can be described by shifting the variable $\tau =t^{\alpha }$ for $\alpha >0$. Let $L_{n,\alpha }^S(.)$ signifies the $nth$ FSLP; that is,

$$L_{n,\alpha }^S\left(t\right)=L_n^S\left(t^{\alpha }\right),\mathrm{for} \mathrm{all}t\in \left[0,T\right].$$

It can be provided using the fractional shifted Legendre polynomials recurrence formula as follows:

$$\begin{array}{cc}\hfill L_{0,\alpha }^S\left(t\right)=& 1,L_{1,\alpha }^S\left(t\right)=\frac{2t^{\alpha }}{T}-1,\hfill \\ \hfill L_{j+1,\alpha }^S\left(t\right)=& \frac{2j+1}{j+1}\left(\frac{2t^{\alpha }}{T}-1\right)L_{j,\alpha }^S\left(t\right)-\frac{j}{j+1}{L}_{j-1,\alpha }^S\left(t\right),j=1,2,3,\dots \hfill \end{array}$$

Furthermore, the analytical version of the FSLP $L_{n,\alpha }^S(.)$ is as follows:

$$\begin{array}{c}\hfill L_{n,\alpha }^S\left(t\right)=\sum _{j=0}^n{c}_{n,j}{t}^{\alpha j},n=0,1,2,3,\dots ,\end{array}$$

where

$$\begin{array}{c}\hfill c_{n,p}:={(-1)}^{n+p}\frac{(n+p)!}{(n-p)!{(p!)}^2{T}^p},p=0,\cdots ,n.\end{array}$$

Letting ${\omega }_{\alpha }\left(t\right)=t^{\alpha -1}$. Define the following weight inner product

$${〈\vartheta ,\psi 〉}_{{\omega }_{\alpha }}:={\int }_0^{T^{\frac{1}{\alpha }}}\vartheta \left(t\right)\psi \left(t\right){\omega }_{\alpha }\left(t\right)dt.$$

Via the orthogonality property (5), the sequence of FSLPs is orthogonal with regard to the weight function $\omega (.)$ in the interval $[0,T^{\frac{1}{\alpha }}]$.

$$\begin{array}{c}\hfill {\int }_0^{T^{\frac{1}{\alpha }}}{L}_{m,\alpha }^S\left(\tau \right)L_{n,\alpha }^S\left(\tau \right){\omega }_{\alpha }\left(\tau \right)d\tau =\frac{{\delta }_{mn}T}{\alpha (2n+1)}.\end{array}$$

Denoting the corresponding normalized sequence by $N_{n,\alpha }^S$. That is,

$$N_{n,\alpha }^S\left(t\right):=\frac{L_{m,\alpha }^S\left(t\right)}{\sqrt{\frac{T}{\alpha (2n+1)}}}.$$

Hence

$$\begin{array}{c}\hfill {\int }_0^{T^{\frac{1}{\alpha }}}{N}_{m,\alpha }^S\left(t\right)N_{n,\alpha }^S\left(t\right){\omega }_{\alpha }\left(t\right)dt={\delta }_{mn}.\end{array}$$

5. Galerkin Projection Method

Let ${\left({\pi }_n^S\right)}_{n\ge 0}$ denote the sequence of bounded finite rank orthogonal projections described by

$${\pi }_n^{S}f:=\sum _{j=0}^{n-1}{〈f,N_{j,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{j,\alpha }^S.$$

Any function $\phi (.)$ defined over the range $[0,1]$ can be expanded in terms of normalized fractional shifted Legendre polynomials (NFSLPs) as follows:

$$\begin{array}{c}\hfill \phi \left(s\right)\simeq {\phi }_n\left(s\right)={\pi }_n^S\phi \left(s\right),\end{array}$$

or equivalently

$$\begin{array}{c}\hfill \phi \left(s\right)\simeq {\phi }_n\left(s\right)=C^t{\Psi }_{\alpha }\left(s\right),\end{array}$$

where

$$\begin{array}{c}\hfill C:={\left[c_0,c_2,\dots ,c_{n-1}\right]}^t,{\Psi }_{\alpha }\left(s\right)=\left[N_{0,\alpha }^S\left(s\right),N_{1,\alpha }^S\left(s\right),\dots ,N_{n-1,\alpha }^S\left(s\right)\right].\end{array}$$

Several results for solving compact operator equations using the Galerkin method have been established over the last two decades. The Galerkin method is used in this section to approximate the solution of the fractional Bagley–Torvik equations. For ${{\mathcal{D}}^2}_n:={\pi }_n^S{\mathcal{D}}^2{\pi }_n^S$, ${{\mathcal{D}}^{\nu }}_n:={\pi }_n^S{\mathcal{D}}^{\nu }{\pi }_n^S$, we obtain the following system of Legendre–Galerkin approximate equations

$$\left\{\begin{array}{c}{A}_1{\pi }_n^S{\mathcal{D}}^2{\zeta }_n\left(s\right)+A_2{\pi }_n^S{\mathcal{D}}^{\nu }{\zeta }_n\left(s\right)+A_3{\zeta }_n\left(s\right)={\pi }_n^{S}G\left(s\right),\hfill \\ A_1{\pi }_n^S{\mathcal{D}}^2{\xi }_n\left(s\right)+A_2{\pi }_n^S{\mathcal{D}}^{\nu }{\xi }_n\left(s\right)-A_3{\xi }_n\left(s\right)={\pi }_n^{S}F\left(s\right)\hfill \end{array}\right.$$

(6)

which has a unique solution $({\zeta }_n,{\xi }_n)$, given by

$$\begin{array}{ccc}\hfill {\zeta }_n& =& \sum _{j=0}^n{a}_{n,j}{N}_{j,\alpha }^S,\hfill \\ \hfill {\xi }_n& =& \sum _{j=0}^n{b}_{n,j}{N}_{j,\alpha }^S,\hfill \end{array}$$

for some scalars $a_{n,j}$ and $b_{n,j}$. Equation (6) reads as

$$\left\{\begin{array}{c}{\displaystyle \sum _{j=0}^n}{a}_{n,j}\left[A_1{\pi }_n^S{\mathcal{D}}^2{N}_{j,\alpha }^S+A_2{\pi }_n^S{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S+A_3{N}_{j,\alpha }^S\right]={\pi }_n^{S}G,\hfill \\ {\displaystyle \sum _{j=0}^n}{b}_{n,j}\left[A_1{\pi }_n^S{\mathcal{D}}^2{N}_{j,\alpha }^S+A_2{\pi }_n^S{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S-A_3{N}_{j,\alpha }^S\right]={\pi }_n^{S}F,\hfill \end{array}\right.$$

so that

$$\begin{array}{ccc}\hfill \sum _{j=0}^n{a}_{n,j}[A_1\sum _{i=0}^n{〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{i,\alpha }^S& +& A_2\sum _{i=0}^n{a}_{n,j}{〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{i,\alpha }^S+A_3{N}_{j,\alpha }^S]\hfill \\ & =& \sum _{i=0}^n{〈G,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{i,\alpha }^S,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \sum _{j=0}^n{b}_{n,j}[A_1\sum _{i=0}^n{〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{i,\alpha }^S& +& A_2\sum _{i=0}^n{a}_{n,j}{〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{i,\alpha }^S-A_3{N}_{j,\alpha }^S]\hfill \\ & =& \sum _{i=0}^n{〈F,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{N}_{i,\alpha }^S.\hfill \end{array}$$

Hence

$$\begin{array}{ccc}\hfill \sum _{j=0}^n{a}_{n,j}& [& A_1\sum _{i=0}^n{〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{〈N_{i,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}\hfill \\ & +& A_2\sum _{i=0}^n{a}_{n,j}{〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{〈N_{i,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}\hfill \\ & +& A_3{〈N_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}]=\sum _{i=0}^n{〈G,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{〈N_{i,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }},\hfill \\ \hfill \sum _{j=0}^n{b}_{n,j}& [& A_1\sum _{i=0}^n{〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{〈N_{i,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}\hfill \\ & +& A_2\sum _{i=0}^n{a}_{n,j}{〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{〈N_{i,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}\hfill \\ & -& A_3{〈N_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}]=\sum _{i=0}^n{〈F,N_{i,\alpha }^S〉}_{{\omega }_{\alpha }}{〈N_{i,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \sum _{j=0}^n{a}_{n,j}& [& A_1{〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}+A_2{〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}]+A_3{a}_{n,k}\hfill \\ & =& {〈G,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }},\hfill \\ \hfill \sum _{j=0}^n{b}_{n,j}& [& A_1{〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}+A_2{〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}]-A_3{b}_{n,k}\hfill \\ & =& {〈F,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}.\hfill \end{array}$$

To put it differently, the coefficients $a_{n,j}$ and $b_{n,j}$ are found by solving the linear systems below:

$$\begin{array}{c}\hfill (A_1{\Theta }_n+A_2{\Lambda }_n+A_{3}I)a_n=P_n,\\ \hfill (A_1{\Theta }_n+A_2{\Lambda }_n-A_{3}I)b_n=M_n,\end{array}$$

(7)

where

$$\begin{array}{ccc}\hfill {\Theta }_n(k,j)& :=& {〈{\mathcal{D}}^2{N}_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }},\hfill \\ \hfill {\Lambda }_n(k,j)& :=& {〈{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }},\hfill \\ \hfill P_n\left(k\right)& :=& {〈G,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }},\hfill \\ \hfill M_n\left(k\right)& :=& {〈F,N_{k,\alpha }^S〉}_{{\omega }_{\alpha }}.\hfill \end{array}$$

or equivalently,

$$\begin{array}{ccc}\hfill {\Theta }_n(k,j)& :=& {\int }_0^{T^{\frac{1}{\alpha }}}{\mathcal{D}}^2{N}_{j,\alpha }^S\left(\tau \right)N_{k,\alpha }^S\left(\tau \right){\omega }_{\alpha }\left(\tau \right)d\tau ,\hfill \\ \hfill {\Lambda }_n(k,j)& :=& {\int }_0^{T^{\frac{1}{\alpha }}}{\mathcal{D}}^{\nu }{N}_{j,\alpha }^S\left(\tau \right)N_{k,\alpha }^S\left(\tau \right){\omega }_{\alpha }\left(\tau \right)d\tau ,\hfill \\ \hfill P_n\left(k\right)& :=& {\int }_0^{T^{\frac{1}{\alpha }}}G\left(\tau \right)N_{k,\alpha }^S\left(\tau \right){\omega }_{\alpha }\left(\tau \right)d\tau ,\hfill \\ \hfill M_n\left(k\right)& :=& {\int }_0^{T^{\frac{1}{\alpha }}}F\left(\tau \right)N_{k,\alpha }^S\left(\tau \right){\omega }_{\alpha }\left(\tau \right)d\tau .\hfill \end{array}$$

6. Convergence Analysis

Assume that

$$\begin{array}{ccc}\hfill ∥{(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}∥& \le & {\gamma }_1\mathrm{for} \mathrm{some} \mathrm{positive} \mathrm{constant}{\gamma }_1,\hfill \\ \hfill ∥{(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n-A_{3}I)}^{-1}∥& \le & {\gamma }_2\mathrm{for} \mathrm{some} \mathrm{positive} \mathrm{constant}{\gamma }_2.\hfill \end{array}$$

Theorem 2.

The following estimates hold:

$$\begin{array}{ccc}\hfill {∥{\zeta }_n-\zeta ∥}_{{\omega }_{\alpha }}& \le & \frac{\gamma }{(n+1)!2^{2n+1}}for some positive constant\gamma ,\hfill \\ \hfill {∥{\xi }_n-\xi ∥}_{{\omega }_{\alpha }}& \le & \frac{{\gamma }^{{}^{\prime }}}{(n+1)!2^{2n+1}}for some positive constant{\gamma }^{{}^{\prime }}.\hfill \end{array}$$

Proof. 

We have

$$\begin{array}{ccc}\hfill {\zeta }_n-\zeta & =& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}{\pi }_n^{S}G-{(A_1{\mathcal{D}}^2+A_2{\mathcal{D}}^{\nu }+A_{3}I)}^{-1}G\hfill \\ & =& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}{\pi }_n^{S}G-{(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}G\hfill \\ & +& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}G-{(A_1{\mathcal{D}}^2+A_2{\mathcal{D}}^{\nu }+A_{3}I)}^{-1}G\hfill \\ & =& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}({\pi }_n^S-I)G\hfill \\ & +& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}[(A_1{\mathcal{D}}^2+A_2{\mathcal{D}}^{\nu }+A_{3}I)\hfill \\ & -& (A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)]{(A_1{\mathcal{D}}^2+A_2{\mathcal{D}}^{\nu }+A_{3}I)}^{-1}G\hfill \\ & =& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}\left[({\pi }_n^S-I)G+A_1({\mathcal{D}}^2-{{\mathcal{D}}^2}_n)\zeta +A_2({\mathcal{D}}^{\nu }-{{\mathcal{D}}^{\nu }}_n)\zeta \right].\hfill \end{array}$$

But

$$\begin{array}{ccc}\hfill ({\mathcal{D}}^2-{{\mathcal{D}}^2}_n)\zeta & =& ({\mathcal{D}}^2-{\pi }_n^S{\mathcal{D}}^2{\pi }_n^S)\zeta \hfill \\ & =& (I-{\pi }_n^S){\mathcal{D}}^2\zeta +{\pi }_n^S{\mathcal{D}}^2(I-{\pi }_n^S)\zeta ,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill ({\mathcal{D}}^{\nu }-{{\mathcal{D}}^{\nu }}_n)\zeta & =& ({\mathcal{D}}^{\nu }-{\pi }_n^S{\mathcal{D}}^{\nu }{\pi }_n^S)\zeta \hfill \\ & =& (I-{\pi }_n^S){\mathcal{D}}^{\nu }\zeta +{\pi }_n^S{\mathcal{D}}^{\nu }(I-{\pi }_n^S)\zeta .\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill {\zeta }_n-\zeta & =& {(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}[({\pi }_n^S-I)G+A_1\left[(I-{\pi }_n^S){\mathcal{D}}^2\zeta +{\pi }_n^S{\mathcal{D}}^2(I-{\pi }_n^S)\zeta \right]\hfill \\ & +& A_2\left[(I-{\pi }_n^S){\mathcal{D}}^{\nu }\zeta +{\pi }_n^S{\mathcal{D}}^{\nu }(I-{\pi }_n^S)\zeta \right]].\hfill \end{array}$$

Following [14], it is straightforward to show that for all continuous function $\phi$, we have

$$\begin{array}{c}\hfill \parallel \phi -{\pi }_n^S\phi {\parallel }_{\alpha }\le \frac{c_{\phi }}{(n+1)!2^{2n+1}},\mathrm{for} \mathrm{some} \mathrm{positive} \mathrm{constant}{c}_{\phi }.\end{array}$$

Since $\parallel {\pi }_n^S\parallel =1$, and since

$$\begin{array}{ccc}\hfill {∥{\zeta }_n-\zeta ∥}_{{\omega }_{\alpha }}& \le & ∥{(A_1{{\mathcal{D}}^2}_n+A_2{{\mathcal{D}}^{\nu }}_n+A_{3}I)}^{-1}∥[{∥({\pi }_n^S-I)G∥}_{{\omega }_{\alpha }}\hfill \\ & +& A_1\left[{∥(I-{\pi }_n^S){\mathcal{D}}^2\zeta ∥}_{{\omega }_{\alpha }}+∥{\mathcal{D}}^2∥{∥(I-{\pi }_n^S)\zeta ∥}_{{\omega }_{\alpha }}\right]\hfill \\ & +& A_2\left[{∥(I-{\pi }_n^S){\mathcal{D}}^{\nu }\zeta ∥}_{{\omega }_{\alpha }}+∥{\mathcal{D}}^{\nu }∥{∥(I-{\pi }_n^S)\zeta ∥}_{{\omega }_{\alpha }}\right]].\hfill \end{array}$$

Hence, there exist some positive constants $c_G,c_{\zeta },c_{\zeta }^{{}^{\prime }}\mathrm{and}{c}_{\zeta }^{{}^{″}}$ such that

$$\begin{array}{c}\hfill {∥{\zeta }_n-\zeta ∥}_{{\omega }_{\alpha }}\le \frac{{\gamma }_1}{(n+1)!2^{2n+1}}\left[c_G+A_1\left(c_{\zeta }+∥{\mathcal{D}}^2∥c_{\zeta }^{{}^{\prime }}\right)+A_2\left(c_{\zeta }^{{}^{″}}+∥{\mathcal{D}}^{\nu }∥c_{\zeta }\right)\right].\end{array}$$

Letting

$$\gamma :={\gamma }_1\left[c_G+A_1\left(c_{\zeta }+∥{\mathcal{D}}^2∥c_{\zeta }^{{}^{\prime }}\right)+A_2\left(c_{\zeta }^{{}^{″}}+∥{\mathcal{D}}^{\nu }∥c_{\zeta }\right)\right],$$

we achieve the desired result.

The second estimate can be proven in the same way as the first one provided. □

Theorem 3.

The following estimates hold:

$$\begin{array}{ccc}\hfill {∥u_n-u∥}_{{\omega }_{\alpha }}& \le & \frac{\gamma +{\gamma }^{{}^{\prime }}}{(n+1)!2^{2n+2}},\hfill \\ \hfill {∥v_n-v∥}_{{\omega }_{\alpha }}& \le & \frac{\gamma +{\gamma }^{{}^{\prime }}}{(n+1)!2^{2n+2}}.\hfill \end{array}$$

Proof. 

Since

$$u_n=\frac{{\zeta }_n+{\xi }_n}{2},v_n=\frac{{\zeta }_n-{\xi }_n}{2},$$

we obtain

$$u_n-u=\frac{\left({\zeta }_n-\zeta \right)+\left({\xi }_n-\xi \right)}{2},v_n-v=\frac{\left({\zeta }_n-\zeta \right)+\left(\xi -{\xi }_n\right)}{2}.$$

We achieve the intended outcome by applying the preceding theorem. □

7. Numerical Example

The system of Bagley–Torvik equations can be used to simulate the motion of a plate in a Newtonian fluid. In this section, we consider the Bagley–Torvik equation system (1), with the following exact solution

$$\begin{array}{ccc}\hfill u\left(s\right)& =& \frac{1}{2}\left[s^2+s^5\right],v\left(s\right)=\frac{1}{2}\left[s^2-s^5\right],\hfill \end{array}$$

and the following initial conditions:

$$\begin{array}{cc}\hfill {\rho }_0& =0,{\rho }_1=0,\hfill \\ \hfill {\eta }_0& =0,{\eta }_1=0,\hfill \end{array}$$

where

$$A_1=1,A_2=2,A_3=1,\nu =1.5.$$

To demonstrate our method, consider the following results for $n=5$ obtained by solving linear systems (7), for the unknowns $a_{5,0},\cdots ,a_{5,5}$ and $b_{5,0},\cdots ,b_{5,5}$, respectively.

$$\begin{array}{ccc}\hfill a_{5,0}& =& 0.34994688584237489950,a_{5,1}=0.24242181863767043857,\hfill \\ \hfill a_{5,2}& =& 0.24082750802218830743\times {10}^{-1},a_{5,3}=-0.35772806140104335114\times {10}^{-2},\hfill \\ \hfill a_{5,4}& =& 0.10641147349488199344\times {10}^{-2},a_{5,5}=-0.47311441812443250151\times {10}^{-3}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill b_{5,0}& =& 0.18842256637045510705,b_{5,1}=0.20397238982845960939,\hfill \\ \hfill b_{5,2}& =& 0.97015810181886046239\times {10}^{-1},b_{5,3}=0.20870155082626365279\times {10}^{-1},\hfill \\ \hfill b_{5,4}& =& 0.94639740185993567875\times {10}^{-3},b_{5,5}=-0.77426268074446688191\times {10}^{-4}.\hfill \end{array}$$

In this case, the approximate solution $({\zeta }_5,{\xi }_5)$ is

$$\begin{array}{ccc}\hfill {\zeta }_5\left(s\right)& =& -0.228203152181668994\times {10}^{-2}+0.35783798758255339381s^{\frac{3}{2}}\hfill \\ & +& 1.4989320129175203445s^3-1.8554152043491273285s^{\frac{9}{2}}\hfill \\ & +& 1.4844204098105792136s^6-0.48429356253136554893s^{\frac{15}{2}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\xi }_5\left(s\right)& =& -0.6770852731039787\times {10}^{-4}+0.68959666818352111152s^{\frac{9}{2}}\hfill \\ & -& 0.079255760895919951972s^{\frac{15}{2}}+0.0037925790697463120s^{\frac{3}{2}}\hfill \\ & -& 0.0556763054987918832s^3+0.44154942872731887767s^6.\hfill \end{array}$$

The ways of connecting errors for this example are shown in

Table 1. Numerical Example.

n	${∥\mathit{\zeta }-{\mathit{\zeta }}_{\mathit{n}}∥}_2$	${∥\mathit{\xi }-{\mathit{\xi }}_{\mathit{n}}∥}_2$
3	1.2108 × 10−3	9.7890 × 10−4
5	2.5271 × 10−4	1.3877 × 10−4
7	8.5470 × 10−5	115295 × 10−6
9	3.7174 × 10−5	1.8479 × 10−7
11	1.8893 × 10−5	4.2876 × 10−8
13	1.0679 × 10−5	1.2666 × 10−8
15	6.5213 × 10−6	4.4412 × 10−9
17	4.2231 × 10−6	1.7705 × 10−9
19	2.8642 × 10−6	7.8008 × 10−10
21	2.0163 × 10−6	3.7235 × 10−10
23	1.4638 × 10−6	1.8982 × 10−10
25	1.0904 × 10−6	1.0221 × 10−10

.

Figure 1 and Figure 2 compare the exact and approximate solutions at n = 25.

8. Conclusions

The purpose of this work was to develop an efficient projection method and to analyze its convergence for a fractional-order Bagley–Torvik system. We employed fractional shifted Legendre polynomials as the basis for our development. Two significant advantages exist with the proposed method. We began by converting the problem into a system of two separable fractional-order Bagley–Torvik equations, which we then solved using the present method. Second, approximate solutions were obtained by solving two linear equation systems. To demonstrate the method’s accuracy and performance, numerical results are provided. The current method can be updated to use polynomials other than Legendre’s polynomials. In the future, we will investigate this problem using a family of fractional Chebyshev polynomials.