New Fifth-Kind Chebyshev Collocation Scheme for First-Order Hyperbolic and Second-Order Convection-Diffusion Partial Differential Equations

The fifth type of Chebyshev polynomials was used in tandem with the spectral tau method to achieve a semianalytical solution for the partial differential equation of the hyperbolic first order. For this purpose, the problem was diminished to the solution of a set of algebraic equations in unspecified expansion coefficients. The convergence and error analysis of the proposed expansion were studied in-depth. Numerical trials have confirmed the applicability and the accuracy.


Introduction
e rst-order partial di erential equations (PDEs) modelled various real-life and physical problems. Hyperbolic PDEs characterize the time-dependent physical systems and may be worn to model many phenomena including wave and advection transportation of the material. Advection equations form a special category of conservative hyperbolic rst-order PDEs which deliver a given property at a xed rate through a method. In advection equations, the space and time derivatives of the conserved quantity u(x, t) are proportional to each other, and the interested readers are referred to [1][2][3][4], for further implementations on hyperbolic PDE.
Spectral methods play a very important role in numerical analysis, especially in the eld of numerical solution of ordinary, partial, and di erential equations.
Chebyshev polynomials are very important in many mathematical divisions, especially in numerical analysis.
e key idea of Chebyshev polynomials is that they form the foundation for the extension of the di erential and integral equation solution. Four well-known Chebyshev polynomial groups are used in the literature. e author in [14] presented the extended Sturm-Liouville di erential problem in the fascinating PhD thesis of Masjed-Jamei, and he introduced a basic category of orthogonal-symmetric polynomials.
at class has four criteria. Some basic properties are also included, such as a seminal di erential equation of order two and a generic relationship containing a three-term recursive relation. For more essential formulae about these polynomials, the reader is referred to the work of [14]. e advantage of this class is that two new types of Chebyshev polynomials, fth and sixth, are inherited. Such polynomials were used only once in literature by seminal work in the numerical solution of fractional di erential equations, and Abd-Elhameed and Youssri [15][16][17][18][19][20] rst used the fth-type Chebyshev polynomials to handle ODEs and PDEs.

Mathematical Preliminaries.
is section presents the properties of the basic class of symmetric orthogonal polynomials (BCSOP) that formed in [14]. e key concept to develop this class of polynomial is focused on the use of an extended Sturm-Liouville differential problem. More precisely, in [14], the author assumed that y � ϕ i (z) is a sequence of symmetric functions that satisfies the following differential equation of second order: where A i (z), 1 ≤ i ≤ 5 are independent functions, and μ i are constants. In [14], it has been shown that A i (z), i � 1, 3, 4, 5 is even, and A 2 (z) is odd. You may obtain the desired symmetric category of orthogonal polynomials if A i (z), 1 ≤ i ≤ 5, and μ i are chosen as follows: where the m, n, r, s parameters are real numbers. By using the above relations, we have the following differential equation: e solution of (3) is the generalized polynomials G m,n,r,s i (z) which have the explicit form: where Moreover, the author introduced orthogonal symmetric polynomials in [14], denoted by G m,n,r,s i with the initials: And A i,m,n,r,s � rsi 2 + (m − 2r)s − (− 1) i rn i Many properties of G m,n,r,s i (t) may be found in [14]. ere are many specific categories of important orthogonal polynomials of G i,j,m,n,r,s (z).
e four different types of Chebyshev polynomials could be formed through the expressions: and T i (z), U i (z), V i (z), W i (z) are the first, second, third, and fourth kinds of Chebyshev polynomials. All these polynomials can be obtained as specific special cases of G m,n,r,s i (z). e two types of orthogonal polynomials in [14], 2 Mathematical Problems in Engineering especially, Chebyshev's fifth-and sixth-kind polynomials, may also be defined, respectively, as We focus our study on the Chebyshev fifth kind and their shifted polynomials. e property of orthogonality of where A i,m,n,r,s is defined in (9). Alternatively, the orthogonality formula above is written as And It is more reasonable to normalize fifth-type Chebyshev polynomials. For this specific purpose, we set Accordingly, X i (z) are orthonormal on I � [− 1, 1]:

Fifth Type of Shifted Orthonormal Chebyshev Polynomials
Also, h i is defined in (14). From (16), it is easy to note that C i (z), i ≥ 0 are orthonormal on I * . Directly, we have And e following results are needed in the sequel.

Theorem 1.
e polynomials C i (z) in (9) are connected with T * i (z) by the following formula: where And Proof. See [15] □ Theorem 2. e polynomials C i (z) in (9) are connected with T * i (z) by two formulae as follows: where δ r is defined in (12).
(23) e next corollary shows the above-intended purpose.

Corollary 1.
Chebyshev polynomials of the fifth type have the following trigonometric representations: And Mathematical Problems in Engineering □ e following connection theorem is needed in the sequel. e following two theorems are important.

Theorem 3.
e analytical form Ci(z) is specifically given as where

Theorem 4. e reflection relation (17) of the analytical relation may be stated as
where Proof. See [15] □ e derivative formula is also needed.

Corollary 2.
ese two identities hold for all nonnegative integer q: And

Mathematical Problems in Engineering
where (j) q is the pochhammer symbol.

Implementation of the Method
e aim of this section is to obtain two numerical algorithms for the solution of the first-order hyperbolic differential equation and the second-order convection-diffusion equation.

First-Order Hyperbolic Equation.
Consider the following first-order hyperbolic partial differential equation subject to the initial condition And the boundary condition

Mathematical Problems in Engineering
where ξ 1 , ξ 2 are two constants. We expand the exact solution v, the derivatives D t v and D x v by the fifth-type Chebyshev expansion as We apply the typical tau method and make use of the boundary conditions to get a system of (N + 1) × (N + 1) algebraic equations in the required double-shifted fifthkind Chebyshev coefficients a ij , i, j � 0, 1, . . . , N that can be solved using any standard iteration technique, such as the iteration method of Newton. It is therefore possible to evaluate the semianalytic solution v N (x, t).

Second-Order Convection-Diffusion Equation.
Consider the following second-order convection-diffusion equation: subject to the initial condition And the boundary conditions subject to the initial condition And the boundary conditions where α, β are two constants. We expand the exact solution in terms of the shifted fifth-kind Chebyshev polynomials and derivatives of u based on the above-mentioned derivatives theorems as (47) Applying the inner product and using the orthogonality relation, we get (48) e obtained system of algebraic equations is solved by the use of Gaussian elimination to get the unknown expansion coefficients and, hence, the numerical solution.

Discussion of Convergence and
Error Analysis e following lemma is needed.

(t)|⩽L, and if its expansion is
(50) The series in (50) uniformly converges to f(t). Also, we have Theorem 8. Let u(t) satisfies the conditions of eorem 7, u N+1 (t), u N (t) are two approximate solutions of u(t), and we define e N (t) � u N+1 (t) − u N (t), and then, we get the following estimation of the error: where ‖e N (t)‖ 2,w * means the L 2 − norm of e N (t).
Theorem 9. Let f(t) ascertain conditions of Lemma 1, let e N (t) � ∞ ℓ�N+1 a ℓ C ℓ (t) be the truncation error, and then, e N (t) satisfy the following estimate Theorem 10. Assume that v(x, t) is separable C 3 function with bounded third-order derivatives, and then, the coefficients in (31) satisfy: where Ω is a generic positive constant. Proof.
e proof is a direct consequence of eorem 6. □ Theorem 11. If v(x, t) satisfies the conditions of eorem 10 and v N (x, t) is the approximation of v(x, t), we then get the following estimation of the error: where ‖.‖ 2,w * denotes the L 2 -norm. Proof.
e proof is a direct consequence of eorem 6. □ Using the orthogonality of C i (x)C j (t) and with the aid of eorem 9 and applying Parseval's identity, we get the desired result.

Numerical Results
We offer some numerical tests in this section to illustrate the precision, efficacy, and the wide applicability of the proposed system. We compare our method with Laguerre-Gauss-Radau scheme [21] which shows that our method is very where y N,M (x, t) is the numerical, and , y(x, t) is the exact solution at the node (x, t). e point-wise errors are evaluated by Example 1 (see [21]). Let us start with the following hyperbolic PDE: with the initials where e smooth solution is given by In Tables 1-3, we compare between our method for the case N � 16, t � 0.1, 0.5, 1. ese are obtained by the generalized collocation method Laguerre-Gauss-Radau [21]. In   Figure 1, we depict the exact solution of Example 1, and for N � 16, in Figure 2, we depict the maximum absolute error when N � 16 (Table 4).
Example 2 (see [21]).And, then the following hyperbolic PDE: with the initials, e exact smooth solution is y(x, t) � e − t− x sin(t).
(64) Table 5 presents the results developed by our method when N � 16and the method in [21]; in Figure 3, we illustrate the solution y(x, t), while in Figure 4, we depict the error when N � 16.
In Table 6, we list the maximum pointwise error for different values of N � 16. In Figure 5, we depict the maximum absolute error when N � 16.

Conclusions
A precise numerical technique for solving the hyperbolic partial differential equations is being constructed and applied in the present work. e fifth-type approach of Chebyshev spectral tau was used to simplify the solution of hyperbolic partial differential equations to a set of algebraic equations, which can be more conveniently solved. e numerical findings showed our method to be extremely effective and reliable.

Data Availability
No data were associated with this research.

Conflicts of Interest
e authors declare that they have no conflicts of interest.