Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems

In this paper, two problems involving nonlinear time fractional hyperbolic partial differential equations (PDEs) and time fractional pseudo hyperbolic PDEs with nonlocal conditions are presented. Collocation technique for shifted Chebyshev of the second kind with residual power series algorithm (CTSCSK-RPSA) is the main method for solving these problems. Moreover, error analysis theory is provided in detail. Numerical solutions provided using CTSCSK-RPSA are compared with existing techniques in literature. CTSCSK-RPSA is accurate, simple and convenient method for obtaining solutions of linear and nonlinear physical and engineering problems.

derivatives concerning space and time, which describes various phenomena of physical including, diffusion of reaction, vibrations of longitudinal and physics of plasma 31,32 .In recent years, researchers and scientists have presented the numerical and analytical methods to solve the pseudo hyperbolic equation [33][34][35] .In Refs. 32,36, the authors studied uniqueness, existence and stability analysis of numerical solutions for pseudo hyperbolic PDES.
The fundamental target of this study is to employ an approximate solution for time fractional hyperbolic PDEs and time fractional pseudo hyperbolic PDEs with nonlocal conditions.The method of solution is to apply properties of shifted Chebyshev polynomials of second kind (SCPSK) to reduce space hyperbolic PDEs and pseudo hyperbolic PDEs with nonlocal conditions into system of fractional ODEs, these FODEs system have been solved by employing RPSA.
The outline work is prepared as: The main definitions of Caputo fractional derivative (CFD) and fractional power series (FPS) are given in Section "Preliminaries".Some characteristics for Chebyshev polynomials of the second kind (CPSK) are presented in Section "General characteristics of spectral Chebyshev polynomials".The theorem utilized to discuss the method's error analysis is presented in Section "Error analysis".The methodology has been applied to two applications in Section "Applications of methodology".Numerical solutions and simulations to show CTSCSK-RPSA efficiency are presented in Section "Numerical simulation".In Section "Conclusion", a final conclusion is drawn.

Preliminaries
In this section, we give some essential definitions of CFD and FPS.
Definition 3 40,41 The power series which has the formula is called FPS about τ 0 .There exist the three possibilities for convergence of the FPS ∞ l=0 ϑ l (τ − τ 0 ) lβ , which are: • The series converges only for τ = τ 0 , that is, the radius of convergence equal zero.
• The series converges for all τ ≥ τ 0 , that is, the radius of convergence equal ∞.
• The series converges for τ 0 ≤ τ < τ 0 + R , for some positive real number R and diverges for τ > τ 0 + R , where R is the radius of convergence for the FPS.

General characteristics of spectral Chebyshev polynomials
We recall some main expressions of spectral SCPSK that are utilized in this paper.
Definition 5 42 The spectral CPSK T m (s) over the interval [−1, 1] can be defined as: The orthogonality formula of CPSK with respect to weight function ω(s) = √ 1 − s 2 as: The recurrence form of polynomials T m (s) can be written as: where The explicit formula of T m (s) as: where ⌈ m 2 ⌉ denotes the integral part of m 2 .
Definition 6 42 The SCPSK T * m (x) is defined on [0, 1] as: The orthogonal property of SCPSK with respect to weight function ω * (x) = √ x − x 2 is given as below: The recurrence relation of SCPSK: where The analytical expressions of SCPSK T * m (x) of degree m can be given as: The function u(x) ∈ L 2 [0, 1] can be defined by SCPSK T * i (x) as follows: where the coefficients ϑ i are given by: In practice, we truncate the infinite series up to (n + 1) terms of SCPSK as follows: Theorem 1 Assume that u n (x) be series approximation of spectral SCPSK defined by Eq. (9), then D β u n (x) is given as: ( i,k is defined as: Proof (see Ref. 42 ).

Error analysis
In this section, the following theorem proves an error analysis of the method.

Proof
We approximate function Φ(x) by Taylor series as: where , is the best square approximation function of Φ(x) , we have Hence K = max{x 0 , x − x 0 } , we get By taking square root of both sides for Eq. ( 15), we get (10) . Vol.:(0123456789)

Applications of methodology
The principal objective of this section is to obtain an approximate solution for time fractional hyperbolic PDEs and time fractional pseudo hyperbolic PDEs with nonlocal conditions.

Numerical simulation
Two problems are established in this section to demonstrate the effectiveness and applicability of the CTSCSK-RPSA.Problem 1. Suppose the following nonlinear time fractional hyperbolic PDEs 29 which are described in Eq. (17), where µ = 0 and L Θ(x, t) = ∂ ∂x Θ(x, t) ∂Θ(x, t) ∂x , then with ICs and BCs: The exact solution at Table 1 presents the approximate solutions obtained by CTSCSK-RPSA with VIM, ADM 8 , VHPIM, HPM 11 and PIA 29 .Table 2 present the CTSCSK-RPSA solutions at values of β .Figure 1 represents comparison between exact and approximate solutions at β = 2 .Figure 2 shows the 3D graph of approximate solution at β = {1.9,1.8, 1.7} .Figure 3 displays the behavior of approximate solution for fractional order β and t = 0.1 in two dimensional graphs.
Problem 2. Consider time fractional pseudo hyperbolic PDEs with nonlocal conditions 34 with ICs and BCs: The exact solution at β = 2 is Θ(x, t) = x 3 e t .Table 3 shows the numerical solution obtained by CTSCSK-RPSA and RPSA with absolute error.Table 4 present the CTSCSK-RPSA approximate solutions at various values of β .Figure 4 represents comparison between exact and approximate solutions at β = 2 .Figure 5 shows the 3D graph of approximate solution at (49) www.nature.com/scientificreports/β = {1.9,1.8, 1.7} .Figure 6 displays the behavior of approximate solution for fractional order β and t = 1 in two dimensional graphs.

Conclusion
In this study, the CTSCSK-RPSA is successfully applied to solve nonlinear time fractional hyperbolic PDEs and time fractional pseudo hyperbolic PDEs with nonlocal conditions.Error analysis of the proposed problems was studied.It is clear that the numerical and simulation results obtained by CTSCSK-RPSA at β = 2 are close to the exact solutions and they are more accurate than previous methods in the literature.All results were done with MATLAB R2017b (9.3.0.713579).Finally, we point out that CTSCSK-RPSA is a convenient and efficient solutions for for various types of fractional linear and nonlinear problems that arise in engineering and applied physics.

Figure 6 .
Figure 6.2D graphics of exact and approximate solutions at different fractional order of β for Problem 2. ).

Table 1 .
The Comparison between CTSCSK-RPSA and other available methods for Problem 1.

Table 2 .
Numerical results of CTSCSK-RPSA at different values of β for Problem 1.

Table 4 .
Approximate solution for different values of β for Problem 2.