A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations

Abstract

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory.

1. Introduction

Fractional calculus is now utilized to simulate issues in many other fields, including physics, hydrodynamics, nature, finance, etc. Numerous disciplines employ fractional analysis. The fractional differential operators Riemann–Liouville, Grunwald–Letnikov, Riesz, Hadamard, Caputo, Caputo–Fabrizio, etc., represent only some examples, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14].

A lot of issues in the fields of engineering, mathematics, science, and related disciplines have recently been expressed in fractional integrodifferential equations.

The study in [15] examined whether a set of nonlinear Hadamard fractional differential equations has any positive solutions. These equations have non-negative nonlinear components and are defined on an infinite interval. Additionally, they are sensitive to nonlocal coupled boundary conditions that include Hadamard fractional derivatives and Riemann–Stieltjes integrals. The Guo–Krasnosel’skii fixed-point theorem and the Leggett-Williams fixed-point theorem were both used by the authors to establish the key theorems.

In [16] The authors established the existence and uniqueness of solutions for a system of Caputo fractional differential equations with sequential derivatives, integral terms, two positive factors, and universal coupled Riemann–Stieltjes integral boundary conditions. Their results’ proofs are founded on the Banach fixed-point theorem and the Leray–Schau alternative. The objective of [17] is to establish a discontinuous Galerkin method with a one-sided flux for a singularly perturbed regular Volterra integrodifferential equation. In [18], the authors proposed adequate requirements for the estimated controllability of a class of fractional differential systems in Banach space with a fixed delay. The existence of a moderate solution is examined using the fixed-point theorem. Using generalized Gronwall’s inequality, Cauchy sequence, and functional analysis fundamentals, controllability results are derived.

In [19], the authors established the existence and uniqueness of results for a system of coupled differential equations involving both left Caputo and right Riemann–Liouville fractional derivatives and mixed fractional integrals, supplemented with nonlocal coupled fractional integral boundary conditions using standard fixed-point theorems. The objective of [20] is to devise an approximate spectral method for the nonlinear time-fractional partial integrodifferential equation with a weakly singular kernel. This strategy is predicated on establishing a new Hilbert space that satisfies the initial and boundary conditions. The new spectral collocation method is applied to acquire a precise numerical approximation using new basis functions based on shifted first-kind Chebyshev polynomials. The paper [21] investigated an effective method for solving fractional integrodifferential equations with various scientific applications. The proposed method is founded on the orthonormal polynomial of Legendre and the least squares method. The authors of [22] presented, analyzed, and devised spectral collocation procedures for a particular class of nonlinear singular Lane-Emden equations with generalized Caputo derivatives that manifest in the study of astronomical objects. Under the supposition that the exact solution is an element in $L^2$, the proposed solution is approximated as a truncated series of normalized shifted Jacobi polynomials. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients.

The purpose of [23] is to develop a new operational matrix method based on Vieta–Fibonacci polynomials and validate the proposed method using it to find an approximation to the solution of the fractional-order nonlinear reaction-advection-diffusion equation of the form

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {}_0^c{D}_t^{\eta }\phi (x,t)+{\lambda }_1\frac{\partial \phi }{\partial x}={\nabla }^2\phi +{\lambda }_2(1-\phi )\phi +g(x,t),0<\eta \le 1,\end{array}\end{array}$$

(1)

subject to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \phi (x,0)& ={\rho }_1\left(x\right),\\ \hfill \phi (0,t)& ={\rho }_2\left(t\right),\\ \hfill \phi (1,t)& ={\rho }_3\left(t\right).\end{array}\end{array}$$

(2)

The study in question [24] examined the following specific category of fractional optimal control problem:

$$\begin{array}{c}\hfill \min \left(\max \right)J[\phi ,y]={\int }_0^{1}f(s,y\left(s\right),\phi \left(s\right))ds,\end{array}$$

(3)

subject to

$$\begin{array}{c}\hfill {}_0^c{D}_s^{\nu }y\left(s\right)=g(s,y\left(s\right))+b\left(s\right)\phi \left(s\right),m-1<\nu \le m,m=1,2,3\cdots ,m\in {\mathbf{Z}}^{+},\end{array}$$

(4)

$$\begin{array}{cc}& y\left(0\right)=y_0,\dot{y}\left(0\right)=y_1,\cdots ,y^{(\lceil \nu \rceil -1)}\left(0\right)=y_{(\lceil \nu \rceil -1)},\end{array}$$

(5)

which was addressed by the utilization of an approximate methodology relying on fractional shifted Vieta–Fibonacci functions. The operational matrix of fractional integral for this method is derived.

In [25], the numerical solution of the following fractional-order stochastic integrodifferential equations

$$\begin{array}{ll}{}_0^c{D}_{\eta }^{\alpha }u\left(\eta \right)=& g\left(\eta \right)+{\lambda }_1{\int }_0^{\eta }{\kappa }_1(\eta ,\zeta )u\left(\zeta \right)d\zeta +{\lambda }_2{\int }_0^{\eta }{\kappa }_2(\eta ,\zeta )u\left(\zeta \right)dB\left(\zeta \right),\\ & \eta \in [0,1],u\left(0\right)=u_0\end{array}$$

is obtained using an operational matrix approach based on shifted Vieta–Fibonacci polynomials. Here, B is the Brownian motion.

The authors of [26] used a Vieta–Fibonacci collocation method to find a numerical solution to fractional delay integrodifferential equations with weakly singular kernels using the collocation method and resultant operational matrices linked to the Vieta–Fibonacci polynomials. Krasnoselskii’s fixed-point theorem shows that the solution to this type of fractional delay singular integrodifferential equation exists and is unique.

Recently, many models, such as those in control interpreting, elasticity, signal analysis, and unusual propagation, have been formulated regarding fractional derivatives. These considerations lead us to propose a numerical method for solving a system of Caputo fractional integrodifferential equations. The reason for conducting our research on the Caputo derivative is because this kind of fractional operator, one of the most widely used fractional derivatives, in greater depth and takes into account the realistic initial conditions employed in physics. The following are some of the overarching goals of this work:

• Examining a system of fractional integrodifferential equations for the first time;
• The utilization of Vieta–Fibonacci polynomials was initially employed to numerically obtain the solution a system of fractional integrodifferential equations;
• Developing a projection method for addressing a system of fractional integrodifferential equations using Vieta–Fibonacci polynomials;
• Establishing a new error analysis in a Vieta–Fibonacci weighted space.

In this study, a new way of solving the system of fractional integrodifferential equations numerically is proposed. We use Vieta–Fibonacci polynomials as basis functions and derive for the first time the projection method for Caputo fractional order to obtain the approximation solution. The problem is reduced to a set of two separate equations by an excellent transformation. By solving two algebraic problems, we obtain the approximate solution to the problem. The suggested method is proven to work and be accurate. We show that there is a solution to the approximate problem. Numerical tests back up how the theory is interpreted.

2. Preliminaries

In this section, we begin by reviewing some of the basic terms and theoretical concepts used in fractional theory.

Let $\Gamma$ indicate the fundamental Euler Gamma function in the analysis of fractional differential equations.

Definition 1

([3,27]). The left-sided Riemann–Liouville fractional integral of order $\eta >0$ of an integrable function $\lambda :(0,\infty )\to \mathbb{R}$ is presented as follows:

$$\begin{array}{c}\hfill J_{0_{+}}^{\eta }\lambda \left(\kappa \right):=\frac{1}{\Gamma \left(\eta \right)}{\int }_0^{\kappa }\frac{\lambda \left(\rho \right)}{{(\kappa -\rho )}^{1-\eta }}d\rho ,\kappa >0.\end{array}$$

(6)

Remark 1.

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

$$\begin{array}{c}\hfill J_{0_{+}}^{\eta }\lambda \left(\kappa \right)=({\Psi }_{\eta }★\lambda )\left(\kappa \right),\end{array}$$

(7)

where

$$\begin{array}{c}{\Psi }_{\eta }\left(\kappa \right):=\left\{\begin{array}{ll}{\kappa }^{\eta -1}/\Gamma \left(\eta \right),& \kappa >0,\\ 0,& \kappa \le 0.\end{array}\right.\end{array}$$

(8)

Definition 2

([11,28]). The left-sided Riemann–Liouville fractional derivative of order $\eta >0$ of a continuous function $\lambda :(0,\infty )\to \mathbb{R}$ is defined by

$$\begin{array}{c}\hfill D_{0_{+}}^{\eta }\lambda \left(\kappa \right):=\frac{1}{\Gamma (1-\eta )}\frac{d}{d\kappa }{\int }_0^{\kappa }\frac{\lambda \left(\rho \right)}{{(\kappa -\rho )}^{\eta }}d\rho .\end{array}$$

(9)

Remark 2.

We have

$$\begin{array}{c}\begin{array}{c}{}^c{D}_{0_{+}}^{\eta }{\kappa }^{\alpha }=\left\{\begin{array}{l}0,for\alpha \in \mathbb{N}\mathit{and}\alpha <\left[\eta \right],\\ \frac{\Gamma (\alpha +1)}{\Gamma (\alpha +1-\eta )}{\kappa }^{\alpha -\eta },for\alpha \in \mathbb{N}and\alpha \ge \left[\eta \right]or\alpha \notin \mathbb{N}and\alpha >\left[\eta \right].\end{array}\right.\end{array}\end{array}$$

(10)

Remark 3.

For $\eta >0$, we have

$$\begin{array}{c}\hfill D_{0_{+}}^{\eta }\lambda \left(\kappa \right)=\frac{d}{d\kappa }{J}_{0_{+}}^{1-\eta }\lambda \left(\kappa \right).\end{array}$$

(11)

Also,

$$J_{0_{+}}^{\eta }{D}_{0_{+}}^{\eta }\left(\lambda \left(\kappa \right)-\lambda \left(0\right)\right)=\lambda \left(\kappa \right)-\lambda \left(0\right).$$

(12)

Definition 3

([3,28]). For an absolutely continuous function λ, the Caputo fractional derivative of order $\eta >0$ is defined by

$$\begin{array}{c}\hfill {}^c{D}_{0_{+}}^{\eta }\lambda \left(\kappa \right)=J_{0_{+}}^{1-\eta }\frac{d}{d\kappa }\lambda \left(\kappa \right)=\frac{1}{\Gamma (1-\eta )}{\int }_0^{\kappa }{(\kappa -\rho )}^{-\eta }{\lambda }^{\prime }\left(\rho \right)d\rho .\end{array}$$

(13)

Remark 4

([3,28]). For a continuous function λ, the relationship between the Caputo and Riemann–Liouville fractional derivatives is provided by

$${}^c{D}_{0_{+}}^{\eta }\lambda \left(\kappa \right)=D_{0_{+}}^{\eta }(\lambda \left(\kappa \right)-\lambda \left(0\right)),\eta >0.$$

(14)

In addition,

$$\begin{array}{c}\hfill {}^c{D}_{0_{+}}^{\eta }\lambda \left(\kappa \right)=D_{0_{+}}^{\eta }\left(\lambda \left(\kappa \right)-\sum _{k=0}^{r-1}\frac{{\kappa }^k}{k!}{\lambda }^{\left(k\right)}\left(0\right)\right),\kappa >0,r-1<\eta <r.\end{array}$$

(15)

3. System of Fractional Integrodifferential Equations

Let $\mathcal{H}:=L^2([0,1],\mathbb{R})$ be the space of real-valued Lebesgue square integrable functions on $[0,1]$.

The Vieta–Fibonacci polynomials in $\mathcal{H}$ are used in this paper to provide a projection method for solving a coupled system of fractional integrodifferential equations of the type:

$$\begin{array}{c}\left\{\begin{array}{l}{\displaystyle {}^c{D}_{0_{+}}^{\eta }\delta \left(\kappa \right)+\delta \left(\kappa \right)+{\int }_0^1\psi \left(\kappa ,\rho \right)\lambda \left(\rho \right)d\rho =\sigma \left(\kappa \right),0\le \kappa \le 1,}\\ {}^c{D}_{0_{+}}^{\eta }\lambda \left(\kappa \right)+\lambda \left(\kappa \right)+{\int }_0^1\psi \left(\kappa ,\rho \right)\delta \left(\rho \right)d\rho =\varsigma \left(\kappa \right),0\le \kappa \le 1,\\ {\displaystyle \sum _{k=0}^{r-1}}\frac{{\kappa }^k}{k!}{\delta }^{\left(k\right)}\left(0\right)=0,{\displaystyle \sum _{k=0}^{r-1}}\frac{{\kappa }^k}{k!}{\lambda }^{\left(k\right)}\left(0\right)=0,r-1<\eta <r,\end{array}\right.\hfill \end{array}$$

(16)

where $\delta$ and $\lambda$ are the unknown solutions and $\sigma$, $\varsigma$ and $\psi (.,.)$ are given real smooth functions. The function $\psi (.,.)$ is called the kernel.

We transform the above-coupled system into a system of two separable fractional integrodifferential equations, which is subsequently solved using the current method. To this end, as Mennouni suggests in [29,30], investigate the following transformation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \Delta :=\lambda -\delta ,\Sigma :=\varsigma +\sigma ,}\\ \Lambda :=\lambda +\delta ,\zeta :=\varsigma -\sigma .\end{array}\right.\end{array}$$

(17)

Lemma 1.

Problem (16) can be expressed in the following form:

$$\begin{array}{c}\left\{\begin{array}{l}{\displaystyle {}^c{D}_{0_{+}}^{\eta }\Lambda \left(\kappa \right)+\Lambda \left(\kappa \right)+{\int }_0^1\psi \left(\kappa ,\rho \right)\Lambda \left(\rho \right)d\rho =\Sigma \left(\kappa \right),0\le \kappa \le 1,}\\ {}^c{D}_{0_{+}}^{\eta }\Delta \left(\kappa \right)+\Delta \left(\kappa \right)-{\int }_0^1\psi \left(\kappa ,\rho \right)\Lambda \left(\rho \right)\Delta \left(\rho \right)d\rho =\zeta \left(\kappa \right),0\le \kappa \le 1,\\ {\displaystyle \sum _{k=0}^{r-1}}\frac{{\kappa }^k}{k!}{\Lambda }^{\left(k\right)}\left(0\right)=0,{\displaystyle \sum _{k=0}^{r-1}}\frac{{\kappa }^k}{k!}{\Delta }^{\left(k\right)}\left(0\right)=0,r-1<\eta <r.\end{array}\right.\end{array}$$

(18)

Proof. 

We have,

$$\begin{array}{c}\left\{\begin{array}{l}{\displaystyle \lambda =\frac{\Lambda +\Delta }{2},\sigma =\frac{\Sigma -\zeta }{2},}\\ \delta =\frac{\Lambda -\Delta }{2},\varsigma =\frac{\Sigma +\zeta }{2}.\end{array}\right.\end{array}$$

(19)

By putting them into (16), we obtain

$$\begin{array}{cc}\hfill {}^c{D}_{0_{+}}^{\eta }\left(\Lambda -\Delta \right)\left(\kappa \right)+\left(\Lambda -\Delta \right)\left(\kappa \right)+{\int }_0^1\psi \left(\kappa ,\rho \right)\left(\Lambda +\Delta \right)\left(\rho \right)d\rho & =\left(\Sigma -\zeta \right)\left(\kappa \right),\end{array}$$

(20)

$$\begin{array}{cc}\hfill {}^c{D}_{0_{+}}^{\eta }\left(\Lambda +\Delta \right)\left(\kappa \right)+\left(\Lambda +\Delta \right)\left(\kappa \right)+{\int }_0^1\psi \left(\kappa ,\rho \right)\left(\Lambda -\Delta \right)\left(\rho \right)d\rho & =\left(\Sigma +\zeta \right)\left(\kappa \right).\end{array}$$

(21)

Equations (20) and (21) are added together, and then (20) is subtracted from (21) to yield (18).  □

Denoting by $\mathcal{A}$ the integral operator, i.e.,

$$\begin{array}{c}\hfill \left(\mathcal{A}\delta \right)\left(\kappa \right):={\int }_0^1\psi \left(\kappa ,\rho \right)\delta \left(\rho \right)d\rho ,0\le \kappa \le 1,\end{array}$$

(22)

and setting

$$\mathcal{D}:=\left\{\lambda \in \mathcal{H}:{\lambda }^{\left(k\right)}\in \mathcal{H},\sum _{k=0}^{r-1}\frac{{\kappa }^k}{k!}{\lambda }^{\left(k\right)}\left(0\right)=0,r-1<\eta <r\right\}.$$

System (18) can be represented this way in operator form:

$$\left\{\begin{array}{l}{\displaystyle \Lambda \left(\kappa \right){+}^c{D}_{0_{+}}^{\eta }\Lambda \left(\kappa \right)+\mathcal{K}\Lambda \left(\kappa \right)=\Sigma \left(\kappa \right),}\\ \Delta \left(\kappa \right){+}^c{D}_{0_{+}}^{\eta }\Delta \left(\kappa \right)-\mathcal{K}\Delta \left(\kappa \right)=\zeta \left(\kappa \right),\end{array}\right.$$

(23)

Recall that the operator $\mathcal{A}$ is compact from $\mathcal{H}$ into itself.

Also,

$$J_{0_{+}}^{\eta }{}^c{D}_{0_{+}}^{\eta }\lambda \left(\kappa \right)=\lambda \left(\kappa \right)-\sum _{k=0}^{r-1}\frac{{\kappa }^k}{k!}{\lambda }^{\left(k\right)}\left(0\right),$$

(24)

and

$$J_{0_{+}}^{\eta }{}^c{D}_{0_{+}}^{\eta }\lambda \left(\kappa \right)=\lambda \left(\kappa \right),\mathrm{for} \mathrm{all}\lambda \in \mathcal{D}.$$

(25)

Moreover, $J_{0_{+}}^{\eta }:\mathcal{H}\to \mathcal{D}$ is compact.

4. Vieta–Fibonacci Polynomials

In this section, we look at a class of orthogonal polynomials. These polynomials can be used to construct a new family of orthogonal polynomials known as Vieta–Fibonacci polynomials using recurrence relations and an analytical formula.

Vieta–Fibonacci polynomials ${\mathcal{F}}_n$ of degree $n\in \mathbb{N}$ are defined as follows:

$${\mathcal{F}}_n\left(\kappa \right)=\frac{\sin \left((n+1)\theta \right)}{\sin \theta },\theta =\arccos \left(\frac{\kappa }{2}\right),\theta \in [0,\pi ]\mathrm{for} \mathrm{all}\left|\kappa \right|\le 2.$$

(26)

The following iterative formula can be used to generate polynomial ${\mathcal{F}}_n$:

$${\mathcal{F}}_n\left(\kappa \right)=\kappa {\mathcal{F}}_{n-1}\left(\kappa \right)-{\mathcal{F}}_{n-2}\left(\kappa \right),n=2,3,\dots ,{\mathcal{F}}_0\left(\kappa \right)=0,{\mathcal{F}}_1\left(\kappa \right)=1.$$

(27)

Also, the explicit power series formula shown below can be used to calculate ${\mathcal{F}}_n$:

$${\mathcal{F}}_n\left(\kappa \right)=\sum _{i=0}^{\lceil \frac{n-1}{2}\rceil }\frac{{(-1)}^i\Gamma (n-i)}{\Gamma (n-2i)\Gamma (i+1)}{\kappa }^{n-2i-1},n=\{2,3,\dots \}.$$

(28)

In addition, ${\mathcal{F}}_n$ are orthogonal polynomials with respect to the integral shown below:

$$⟨{\mathcal{F}}_m,{\mathcal{F}}_n⟩={\int }_{-2}^2\sqrt{4-{\kappa }^2}{\mathcal{F}}_m\left(\kappa \right){\mathcal{F}}_n\left(\kappa \right)d\kappa =\left\{\begin{array}{ll}0,& n\ne m,\\ 2\pi ,& m=n.\end{array}\right.$$

(29)

Let

$${\mathcal{F}}_k^{\mathcal{T}}\left(\kappa \right)={\mathcal{F}}_k(4\kappa -2)={\mathcal{F}}_{2k}\left(2\sqrt{\kappa }\right),k=0,1,2,3,\dots$$

(30)

denote the corresponding orthogonal polynomials on $[0,1]$. Additionally, ${\mathcal{F}}_k^{\mathcal{T}}$ are produced using the recurrence formula shown below:

$${\mathcal{F}}_{k+1}^{\mathcal{T}}\left(\kappa \right)=(4\kappa -2){\mathcal{F}}_{k-1}^{\mathcal{T}}\left(\kappa \right)-{\mathcal{F}}_{k-2}^{\mathcal{T}}\left(\kappa \right),k=1,2,\dots ,$$

(31)

with

$${\mathcal{F}}_0^{\mathcal{T}}\left(\kappa \right)=0.$$

(32)

We note that

$$\begin{array}{ccc}\hfill {\mathcal{F}}_p^{\mathcal{T}}\left(\kappa \right)& =& \sum _{j=0}^p{(-1)}^j\frac{2^{2p-2j-2}\Gamma (2p-2j)}{\Gamma (j+1)\Gamma (1-2j+2p)}{\kappa }^{p-j-1}\end{array}$$

(33)

$$\begin{array}{ccc}& =& \sum _{j=0}^p\frac{{(-1)}^{p-j-1}{2}^{2j}\Gamma (p+j+1)}{\Gamma (p-j)\Gamma (2j+2)}{\kappa }^j,p=1,2,3,\cdots \end{array}$$

(34)

Moreover,

$$\begin{array}{c}\hfill {⟨{\mathcal{F}}_k^{\mathcal{T}},{\mathcal{F}}_j^{\mathcal{T}}⟩}_{\omega }={\int }_0^1{\mathcal{F}}_k^{\mathcal{T}}\left(\kappa \right){\mathcal{F}}_j^{\mathcal{T}}\left(\kappa \right)\omega \left(\kappa \right)d\kappa =\left\{\begin{array}{ll}0,& k\ne j,\\ \frac{\pi }{8},& k=j,\end{array}\right.\end{array}$$

(35)

where

$$\omega \left(\kappa \right)=\sqrt{\kappa -{\kappa }^2}.$$

(36)

Letting

$${\mathcal{F}}_j^{\mathcal{N}}:=\frac{2\sqrt{2}{\mathcal{F}}_j^{\mathcal{T}}}{\sqrt{\pi }}.$$

(37)

Now, we introduce the first seven terms of ${\mathcal{F}}_j^{\mathcal{N}}$:

$$\begin{array}{cc}\hfill {\mathcal{F}}_0^{\mathcal{N}}\left(\kappa \right)& =0,\end{array}$$

(38)

$$\begin{array}{cc}\hfill {\mathcal{F}}_1^{\mathcal{N}}\left(\kappa \right)& =\frac{2\sqrt{2}}{\sqrt{\pi }},\end{array}$$

(39)

$$\begin{array}{cc}\hfill {\mathcal{F}}_2^{\mathcal{N}}\left(\kappa \right)& =\frac{4\sqrt{2}\left(2\kappa -1\right)}{\sqrt{\pi }},\end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mathcal{F}}_3^{\mathcal{N}}\left(\kappa \right)& =\frac{2\sqrt{2}\left(16{\kappa }^2-16\kappa +3\right)}{\sqrt{\pi }},\end{array}$$

(41)

$$\begin{array}{cc}\hfill {\mathcal{F}}_4^{\mathcal{N}}\left(\kappa \right)& =\frac{8\sqrt{2}\left(16{\kappa }^3-24{\kappa }^2+10\kappa -1\right)}{\sqrt{\pi }},\end{array}$$

(42)

$$\begin{array}{cc}\hfill {\mathcal{F}}_5^{\mathcal{N}}\left(\kappa \right)& =\frac{2\sqrt{2}\left(256{\kappa }^4-512{\kappa }^3+336{\kappa }^2-80\kappa +5\right)}{\sqrt{\pi }},\end{array}$$

(43)

$$\begin{array}{cc}\hfill {\mathcal{F}}_6^{\mathcal{N}}\left(\kappa \right)& =\frac{4\sqrt{2}\left(512{\kappa }^5-1280{\kappa }^4+1152{\kappa }^3-448{\kappa }^2+70\kappa -3\right)}{\sqrt{\pi }}.\end{array}$$

(44)

Let ${\pi }_n^{\mathcal{F}}$ be the chain of bounded finite rank orthogonal projections described by

$${\pi }_n^{\mathcal{F}}\psi :=\sum _{j=0}^{n-1}{⟨\psi ,{\mathcal{F}}_j^{\mathcal{N}}⟩}_{\omega }{\mathcal{F}}_j^{\mathcal{N}},\mathrm{where}{⟨\psi ,{\mathcal{F}}_j^{\mathcal{N}}⟩}_{\omega }:={\int }_0^1\omega \left(\sigma \right)\psi \left(\sigma \right){\mathcal{F}}_j^{\mathcal{N}}\left(\sigma \right)d\sigma .$$

(45)

Denote by $\parallel ·\parallel$ the corresponding norm on $\mathcal{H}$. Thus,

$$\underset{n\to \infty }{\lim }\parallel {\pi }_n^{\mathcal{F}}\vartheta -\vartheta \parallel =0,\mathrm{for} \mathrm{all}\vartheta \in \mathcal{H}.$$

(46)

Let ${\mathcal{H}}_n$ represent the space covered by the first n-orthonormal shifted Vieta–Fibonacci polynomials. Recall that

$$J_{0_{+}}^{\eta }{s}^k=\frac{\Gamma (k+1)}{\Gamma (k+\eta +1)}{s}^{k+\eta }.$$

(47)

So,

$$J_{0_{+}}^{\eta }\left({\mathcal{H}}_n\right)={\mathcal{H}}_{n+1}.$$

(48)

5. Development of the Method

Clearly that ${\Lambda }_n,{\Delta }_n\in \mathcal{D}\cap {\mathcal{H}}_{n+1}$. Therefore, the system

$$\left\{\begin{array}{l}{\displaystyle \Lambda +J_{0_{+}}^{\eta }\Lambda +J_{0_{+}}^{\eta }\mathcal{A}\Lambda =J_{0_{+}}^{\eta }\Sigma ,}\\ \Delta +J_{0_{+}}^{\eta }\Delta -J_{0_{+}}^{\eta }\mathcal{A}\Delta =J_{0_{+}}^{\eta }\zeta \end{array}\right.$$

(49)

is approached by

$$\left\{\begin{array}{l}{\displaystyle {\Lambda }_n+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}{\Lambda }_n+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}{\Lambda }_n=J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\Sigma ,}\\ {\Delta }_n+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}{\Delta }_n-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}{\Delta }_n=J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\zeta .\end{array}\right.$$

(50)

Alternatively,

$$\left\{\begin{array}{l}{\displaystyle {\Lambda }_n+J_{0_{+}}^{\eta }{\Lambda }_n+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}{\Lambda }_n=J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\Sigma ,}\\ {\Delta }_n+J_{0_{+}}^{\eta }{\Delta }_n-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}{\Delta }_n=J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\zeta .\end{array}\right.$$

(51)

We suppose that $-1$ is not eigenvalue of $J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }\mathcal{A}$ and $J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }\mathcal{A}$, respectively. Thus, both operators $\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }\mathcal{A}$ and $\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }\mathcal{A}$ are invertible.

We know that $J_{0_{+}}^{\eta }$ is compact and

$$\underset{n\to \infty }{\lim }\parallel \left(J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}-J_{0_{+}}^{\eta }\mathcal{A}\right)J_{0_{+}}^{\eta }\mathcal{A}\parallel =0,\underset{n\to \infty }{\lim }\parallel \left(J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}-J_{0_{+}}^{\eta }\mathcal{A}\right)J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{F}}\mathcal{A}\parallel =0.$$

(52)

Writing

$$\left\{\begin{array}{l}{\displaystyle {\Lambda }_n=\sum _{k=0}^n{\beta }_{n,k}{\mathcal{F}}_k^{\mathcal{N}},}\\ {\Delta }_n={\displaystyle \sum _{k=0}^n}{\gamma }_{n,k}{\mathcal{F}}_k^{\mathcal{N}}.\end{array}\right.$$

(53)

We obtain $2n+2$ unknowns ${\beta }_{n,k}$ and ${\gamma }_{n,k}$ by solving the following two separate linear systems,

$$\left\{\begin{array}{l}{\displaystyle \sum _{k=0}^n{\beta }_{n,k}\left[\mathcal{I}+D_{0_{+}}^{\eta }{\mathcal{F}}_k^{\mathcal{N}}+{\pi }_n^{\mathcal{F}}\mathcal{A}{\mathcal{F}}_k^{\mathcal{N}}\right]={\pi }_n^{\mathcal{F}}\Sigma ,\mathrm{with}\sum _{k=0}^n{\beta }_{n,k}{\mathcal{F}}_k^{\mathcal{N}}\left(0\right)=0,}\\ {\displaystyle \sum _{k=0}^n}{\gamma }_{n,k}\left[\mathcal{I}+D_{0_{+}}^{\eta }{\mathcal{F}}_k^{\mathcal{N}}-{\pi }_n^{\mathcal{F}}\mathcal{A}{\mathcal{F}}_k^{\mathcal{N}}\right]={\pi }_n^{\mathcal{F}}\zeta ,\mathrm{with}{\displaystyle \sum _{k=0}^n}{\gamma }_{n,k}{\mathcal{F}}_k^{\mathcal{N}}\left(0\right)=0.\end{array}\right.$$

(54)

As a result, two separate linear systems are produced,

$$\left\{\begin{array}{l}{\displaystyle {\beta }_{n,k}+\sum _{m=0}^n{P}_n(k,m){\beta }_{n,m}=T_{n,k},k=0,\cdots ,n,}\\ {\gamma }_{n,k}+{\displaystyle \sum _{m=0}^n}{\widehat{P}}_n(k,m){\gamma }_{n,m}={\widehat{T}}_{n,k},k=0,\cdots ,n,\end{array}\right.$$

(55)

where, for $k=0,\cdots ,n-1$ and $m=0,\cdots ,n$,

$$\begin{array}{c}\left\{\begin{array}{l}{\displaystyle P_n(k,m):=\left[{\int }_0^1{}^c{D}_{0_{+}}^{\eta }{\mathcal{F}}_m^{\mathcal{N}}\left(\kappa \right){\mathcal{F}}_k^{\mathcal{N}}\left(\kappa \right)\Lambda \left(\kappa \right)d\kappa +{\int }_0^1\left({\int }_0^1{\mathcal{F}}_m^{\mathcal{N}}\left(\rho \right)\psi \left(\kappa ,\rho \right)d\rho \right){\mathcal{F}}_k^{\mathcal{N}}\left(\kappa \right)\Lambda \left(\kappa \right)d\kappa \right],}\\ {\widehat{P}}_n(k,m):=\left[{\int }_0^1{}^c{D}_{0_{+}}^{\eta }{\mathcal{F}}_m^{\mathcal{N}}\left(\kappa \right){\mathcal{F}}_k^{\mathcal{N}}\left(\kappa \right)\Lambda \left(\kappa \right)d\kappa -{\int }_0^1\left({\int }_0^1{\mathcal{F}}_m^{\mathcal{N}}\left(\rho \right)\psi \left(\kappa ,\rho \right)d\rho \right){\mathcal{F}}_k^{\mathcal{N}}\left(\kappa \right)\Lambda \left(\kappa \right)d\kappa \right],\\ P_n(n,m):={\mathcal{F}}_m^{\mathcal{N}}\left(0\right),{\widehat{P}}_n(n,m):={\mathcal{F}}_m^{\mathcal{N}}\left(0\right),\\ T_n\left(k\right):={\int }_0^1\Sigma \left(\kappa \right){\mathcal{F}}_k^{\mathcal{N}}\left(\kappa \right)\Lambda \left(\kappa \right)d\kappa ,{\widehat{T}}_n\left(k\right):={\int }_0^1\zeta \left(\kappa \right){\mathcal{F}}_k^{\mathcal{N}}\left(\kappa \right)\Lambda \left(\kappa \right)d\kappa ,\\ T_n\left(n\right):=0,{\widehat{T}}_n\left(n\right):=0.\end{array}\right.\end{array}$$

(56)

6. Convergence Analysis

We now show how the current method converges. To that end, consider $L_{\varpi }^2([0,1],$$\mathbb{R})$ to be the weighted space and ${\parallel .\parallel }_{\varpi }$ to be its norm.

Denote with $\mathcal{I}$ the identity operator. We recall that there exists $B>0$ such that

$${∥(\mathcal{I}-{\pi }_n^{\mathcal{N}})\psi ∥}_{\varpi }\le \frac{B}{{(n+1)}^3},\mathrm{for} \mathrm{all}\psi \in L_{\varpi }^2([0,1],\mathbb{R}).$$

(57)

Since $J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }\mathcal{A}$ and $J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }\mathcal{A}$ are compact, the operators ${(\mathcal{I}+{\pi }_n^{\mathcal{N}}{J}_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A})}^{-1}$ and ${(\mathcal{I}+{\pi }_n^{\mathcal{N}}{J}_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A})}^{-1}$ exist for n large enough and are uniformly bounded with respect to n.

Theorem 1.

Assume that $\varsigma ,\sigma ,\mathcal{A}\lambda ,\mathcal{A}\delta \in L_{\varpi }^2([0,1],\mathbb{R})$. Then, there exist $B_1,B_2>0$ such that

$$\parallel {\Lambda }_n{-\Lambda \parallel }_{\varpi }\le \frac{B_1}{{(n+1)}^3},$$

(58)

and

$$\parallel {\Delta }_n{-\Delta \parallel }_{\varpi }\le \frac{B_2}{{(n+1)}^3}.$$

(59)

Proof. 

In fact,

$$\left\{\begin{array}{l}{\displaystyle {\Lambda }_n+J_{0_{+}}^{\eta }{\Lambda }_n+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}{\Lambda }_n=J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\Sigma ,}\\ {\Delta }_n+J_{0_{+}}^{\eta }{\Delta }_n-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}{\Delta }_n=J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\zeta .\end{array}\right.$$

(60)

Moreover,

$$\begin{array}{ll}\Lambda -{\Lambda }_n& =\left[{\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\Sigma -{\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\Sigma \right]\\ \hfill & +{\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\Sigma -{\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\Sigma \\ \hfill & ={\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\left[\left({\pi }_n^{\mathcal{N}}-\mathcal{I}\right)\mathcal{A}\Lambda +\left(\mathcal{I}-{\pi }_n^{\mathcal{N}}\right)\Sigma \right].\end{array}$$

(61)

In addition,

$$\begin{array}{ll}\Delta -{\Delta }_n& =\left[{\left(\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\zeta -{\left(I+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\zeta \right]\\ \hfill & +{\left(\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\zeta -{\left(\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\zeta \\ \hfill & ={\left(\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\left[\left({\pi }_n^{\mathcal{N}}-\mathcal{I}\right)\mathcal{A}\Delta +\left(\mathcal{I}-{\pi }_n^{\mathcal{N}}\right)\zeta \right].\end{array}$$

(62)

Hence,

$$\begin{array}{cc}\hfill \Lambda -{\Lambda }_n& ={\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\left[\left({\pi }_n^{\mathcal{N}}-\mathcal{I}\right)\mathcal{A}(\lambda +\delta )+\left(\mathcal{I}-{\pi }_n^{\mathcal{N}}\right)(\varsigma +\sigma )\right].\end{array}$$

(63)

Also,

$$\begin{array}{cc}\hfill \Delta -{\Delta }_n& ={\left(\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}{J}_{0_{+}}^{\eta }\left[\left({\pi }_n^{\mathcal{N}}-\mathcal{I}\right)\mathcal{A}(\lambda -\delta )+\left(\mathcal{I}-{\pi }_n^{\mathcal{N}}\right)(\varsigma -\sigma )\right].\end{array}$$

(64)

Letting

$$\begin{array}{cc}\hfill \mathsf{{\rm Y}}& :=∥{\left(\mathcal{I}+J_{0_{+}}^{\eta }+J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}∥∥J_{0_{+}}^{\eta }∥,\end{array}$$

(65)

$$\begin{array}{cc}\hfill \Xi & :=∥{\left(\mathcal{I}+J_{0_{+}}^{\eta }-J_{0_{+}}^{\eta }{\pi }_n^{\mathcal{N}}\mathcal{A}\right)}^{-1}∥∥J_{0_{+}}^{\eta }∥,\end{array}$$

(66)

we obtain

$$\begin{array}{cc}\hfill {∥\Lambda -{\Lambda }_n∥}_{\varpi }& \le \mathsf{{\rm Y}}\left[{∥\left(I-{\pi }_n^{\mathcal{N}}\right)\mathcal{A}(\lambda +\delta )∥}_{\varpi }+{∥\left(I-{\pi }_n^{\mathcal{N}}\right)(\varsigma +h)∥}_{\varpi }\right]\end{array}$$

(67)

$$\begin{array}{cc}\hfill & \le \mathsf{{\rm Y}}\left[\frac{B_3}{{(n+1)}^3}+\frac{B_4}{{(n+1)}^3}\right],\mathrm{for} \mathrm{some} \mathrm{constants}{B}_3,B_4>0.\end{array}$$

(68)

Moreover,

$$\begin{array}{cc}\hfill {∥\Delta -{\Delta }_n∥}_{\varpi }& \le \Xi \left[{∥\left(I-{\pi }_n^{\mathcal{N}}\right)\mathcal{A}(\lambda -\delta )∥}_{\varpi }+{∥\left(I-{\pi }_n^{\mathcal{N}}\right)(\varsigma -\sigma )∥}_{\varpi }\right]\end{array}$$

(69)

$$\begin{array}{cc}\hfill & \le \Xi \left[\frac{B_5}{{(n+1)}^3}+\frac{B_6}{{(n+1)}^3}\right],\mathrm{for} \mathrm{some} \mathrm{constants}{B}_5,B_6>0.\end{array}$$

(70)

Letting

$$B_1:=\mathsf{{\rm Y}}\max \left\{B_3,B_4\right\}\mathrm{and}{B}_2:=\Xi \max \left\{B_5,B_6\right\},$$

we obtain the desired results. □

7. Numerical Example

This section establishes numerical experiments to illustrate the results stated in the preceding one. The Maple programming language was used in these numerical evaluations.

Example 1.

We study the fractional integrodifferential system (16) in this example, which has the exact solution as follows:

$$\begin{array}{c}\hfill \delta \left(\kappa \right)=\frac{1}{2}\left({\kappa }^3-{\kappa }^4\right),\lambda \left(\kappa \right)=\frac{1}{2}\left({\kappa }^3+{\kappa }^4\right),\eta =\frac{3}{4},\end{array}$$

(71)

and

$$\psi \left(\kappa ,\rho \right)=\rho -\kappa .$$

So,

$$\begin{array}{c}\hfill \Lambda \left(\kappa \right)={\kappa }^3,\Delta \left(\kappa \right)={\kappa }^4,\end{array}$$

(72)

and

$$\begin{array}{ccc}\hfill \Sigma \left(\kappa \right)& =& \frac{1}{60\pi }\left[256{\kappa }^{3/4}\sqrt{2}\Gamma \left(\frac{3}{4}\right)+60{\kappa }^3\pi -60\kappa \pi +12\pi \right].\end{array}$$

(73)

Also,

$$\begin{array}{c}\hfill \zeta \left(\kappa \right)=\frac{1}{390\pi }\left[2048{\kappa }^{13/4}\sqrt{2}\Gamma \left(\frac{3}{4}\right)+390{\kappa }^4\pi +78\kappa \pi -65\pi \right].\end{array}$$

(74)

We perform some numerical tests to demonstrate the effectiveness of this example. For example, for $n=5$, unknowns ${\beta }_{5,0}\cdots {\beta }_{5,5}$ are described as follows:

$$\begin{array}{l}{\beta }_{5,0}=0.,{\beta }_{5,1}=0.10284,\\ {\beta }_{5,2}=0.11752,{\beta }_{5,3}=0.66104\times {10}^{-1},\\ {\beta }_{5,4}=0.19586\times {10}^{-1},{\beta }_{5,5}=0.24481\times {10}^{-2}.\end{array}$$

The approximate solution ${\Lambda }_5$ is offered by

$$\begin{array}{ll}{\Lambda }_5\left(\kappa \right)& =\frac{0.79785}{\sqrt{\kappa \left(1-\kappa \right)}}[0.13711\sin \left(2\arcsin \sqrt{\kappa }\right)-0.13710\sin \left(4\arcsin \sqrt{\kappa }\right)\\ & +0.58759\times {10}^{-1}\sin \left(6\arcsin \sqrt{\kappa }\right)-0.97930\times {10}^{-2}\sin \left(8\arcsin \sqrt{\kappa }\right)\\ & -1.1824\times {10}^{-7}\sin \left(10\arcsin \sqrt{\kappa }\right)].\end{array}$$

(75)

Moreover,

$$\begin{array}{l}{\gamma }_{5,0}=-0.,{\gamma }_{5,1}=0.10284,\\ {\gamma }_{5,2}=0.11752,{\gamma }_{5,3}=0.66104\times {10}^{-1},\\ {\gamma }_{5,4}=0.19586\times {10}^{-1},{\gamma }_{5,5}=0.24481\times {10}^{-2}.\end{array}$$

The approximate solution ${\Delta }_5$ is presented by

$$\begin{array}{ll}{\Delta }_5\left(\kappa \right)& =\frac{0.79785}{\sqrt{\kappa \left(1-\kappa \right)}}[0.10284\sin \left(2\arcsin \sqrt{\kappa }\right)-0.11752\sin \left(4\arcsin \sqrt{\kappa }\right)\\ & +0.66104\times {10}^{-1}\sin \left(6\arcsin \sqrt{\kappa }\right)-0.19586\times {10}^{-1}\sin \left(8\arcsin \sqrt{\kappa }\right)\\ & +0.24481\times {10}^{-2}\sin \left(10\arcsin \sqrt{\kappa }\right)].\end{array}$$

(76)

Using our proposed method, Table 1 displays the numerical results obtained for Example 1.

Table 1. Numerical results for Example 1.

n	${∥\mathbf{\Lambda }-{\mathbf{\Lambda }}_{\mathit{n}}∥}_{\mathit{\varpi }}$	${∥\Delta -{\Delta }_{\mathit{n}}∥}_{\mathit{\varpi }}$
4	3.6466 × 10${}^{-5}$	7.0747 × 10${}^{-5}$
6	5.5219 × 10${}^{-6}$	2.2673 × 10${}^{-6}$
8	5.0263 × 10${}^{-8}$	8.2457 × 10${}^{-8}$
18	6.2351 × 10${}^{-16}$	8.2354 × 10${}^{-12}$
23	9.2548 × 10${}^{-18}$	8.1254 × 10${}^{-14}$

The achieved approximate solutions and exacts one for Example 1 are shown in Figure 1.

Figure 1. Comparison of exact solutions $\Lambda$ and $\Delta$ and approximate ones ${\Omega }_n$ and ${\Delta }_n$, respectively, for $n=5$.

Example 2.

In this example, we investigate the fractional red integrodifferential system (16), which has the following exact solution:

$$\begin{array}{c}\hfill \delta \left(\kappa \right)={\kappa }^5,\lambda \left(\kappa \right)={\kappa }^6,\eta =\frac{2}{5},\end{array}$$

(77)

and

$$\psi \left(\kappa ,\rho \right)={\rho }^2-{\kappa }^2.$$

So,

$$\begin{array}{c}\hfill \Lambda \left(\kappa \right)={\kappa }^5-{\kappa }^6,\Delta \left(\kappa \right)={\kappa }^5+{\kappa }^6,\end{array}$$

(78)

and

$$\begin{array}{ll}\Sigma \left(\kappa \right)& =\frac{-1}{150,696\Gamma \left(\frac{3}{5}\right)}[468,750{\kappa }^{28/5}-437,500{\kappa }^{23/5}+150,696{\kappa }^6\Gamma \left(\frac{3}{5}\right)-150,696{\kappa }^5\Gamma \left(\frac{3}{5}\right)\\ & +3588{\kappa }^2\Gamma \left(\frac{3}{5}\right)-2093\Gamma \left(\frac{3}{5}\right)].\end{array}$$

(79)

Also,

$$\begin{array}{ll}\zeta \left(\kappa \right)& =\frac{1}{150,696\Gamma \left(\frac{3}{5}\right)}[468,750{\kappa }^{28/5}+437,500{\kappa }^{23/5}+150,696{\kappa }^6\Gamma \left(\frac{3}{5}\right)+150,696{\kappa }^5\Gamma \left(\frac{3}{5}\right)\\ & +46,644{\kappa }^2\Gamma \left(\frac{3}{5}\right)-35,581\Gamma \left(\frac{3}{5}\right)].\end{array}$$

(80)

The numerical outcomes for the second case using the current approximation are shown in Figure 2.

Figure 2. Comparison of exact solutions $\Lambda$ and $\Delta$ and approximate ones ${\Omega }_n$ and ${\Delta }_n$, respectively, for $n=10$.

8. Conclusions

In this research, we generalized a projection method to solve a system fractional integrodifferential via the shifted Vieta–Fibonacci polynomials. The current method reduces the provided problem into two systems of algebraic equations. Approximate solutions to the given problem are obtained by solving the acquired systems. This integrodifferential system has obvious relevance to problems in mathematical research, especially those involving interactions in physics. The fractional operator has been found to affect the growth of numerical results significantly. This method can be used to study and find solutions to various fractional integrodifferential and integral problems. This method has the potential to be employed as a future project for solving the system of fractional partial differential equations outlined below under some conditions:

$$\begin{array}{ccc}\hfill {}^c{D}_{0_{+}}^{\eta }\delta (\kappa ,t)+\delta (\kappa ,t)+{\int }_0^1\psi \left(\kappa ,\rho \right)\lambda (\kappa ,\rho )d\rho & =& \sigma (\kappa ,t),0\le \kappa ,t\le 1,\end{array}$$

(81)

$$\begin{array}{ccc}\hfill {}^c{D}_{0_{+}}^{\eta }\lambda (\kappa ,t)+\lambda (\kappa ,t)+{\int }_0^1\psi \left(\kappa ,\rho \right)\delta (\kappa ,\rho )d\rho & =& \varsigma (\kappa ,t),0\le \kappa ,t\le 1.\end{array}$$

(82)