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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

38622192
58493
10.1038/s41598-024-58493-x
Article
Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems
Rida Saad. Z. 1
Arafa Anas. A. M. 23
Hussein Hussein. S. 1
Ameen Ismail G. 1
Mostafa Marwa. M. M. marwa.masoud@sci.svu.edu.eg

1
1 https://ror.org/00jxshx33 grid.412707.7 0000 0004 0621 7833 Department of Mathematics, Faculty of Science, South Valley University, Qena, 83523 Egypt
2 https://ror.org/01wsfe280 grid.412602.3 0000 0000 9421 8094 Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia
3 https://ror.org/01vx5yq44 grid.440879.6 0000 0004 0578 4430 Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, Egypt
15 4 2024
15 4 2024
2024
14 868320 11 2023
29 3 2024
© The Author(s) 2024
2024
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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.

Keywords

Shifted Chebyshev polynomials of the second kind
Residual power series algorithm
Fractional derivatives
Hyperbolic equation with time fractional
Time fractional pseudo hyperbolic equations
Numerical results.
Subject terms

Mathematics and computing
Pure mathematics
South Valley UniversityOpen access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Mathematical modeling of various nonlinear phenomena, which can be expressed using nonlinear differential equations (DEs) is more complex and difficult than modeling linear phenomena. Such phenomena have an important role in the study of many scientific fields and often described by ordinary differential equations (ODEs) and PDEs. Although solving PDEs is more difficult than solving ODEs, these equations are widely used in physics and mathematical problems. The fractional arrangement has been used to generalize these equations by researchers in recent decades, and these equations have been known as fractional partial differential equations (FPDEs). It is difficult to obtain accurate solutions to such equations in their nonlinear state. In the literature, several mathematical methods are presented for solving these equations as Adomian decomposition method (ADM)1,2, variational iteration method (VIM)3, Iterative Laplace transform method4, Sumudu transform method5, finite difference method6, Tau method7, homotopy perturbation method (HPM)8, wavelet methods9, homotopy analysis method10, variational homotopy perturbation iteration method (VHPIM)11, finite element method12, modified HPM13 and Jacobi collocation14.

RPSA is an efficient, powerful and simple technique to create a power series solution that can be handled without discretization, linearization, and perturbation for linear and nonlinear equations. RPSA does not need any changes while transforming from lower to higher order. Hence, the technique can be utilized directly for problem by choosing suitable preliminary guess approximation. Researchers have used RPSA for solving different types of models, such as fuzzy differential equations15, fractional Burger types equations16, fractional gas dynamic equations17, KdV-Burgers equation18, Whitham–Broer–Kaup equations19, fractional time Cahn–Hilliard, Gardner equations20,21, Swift–Hohenberg equation22, fractional diffusion equation23, Burgers–Huxley equations24, Navier–Stokes equations25 and Lane–Emden equations26.

Hyperbolic PDEs is a type of more significance nonlinear models in physics of mathematical. In the last few years, there exist analytical and numerical methods to solve these problems27,28. In Ref.8 Odibat and Momani obtained the analytic and approximate solution for hyperbolic PDEs by using VIM and ADM. Khalid et al.29 constructed an efficient schemes called Perturbation iteration algorithm (PIA) to get approximate solutions for hyperbolic PDEs. Das and Gupta30 employed HAM for obtaining the approximate solution for nonlinear hyperbolic PDEs of fractional order. Pseudo-hyperbolic equations is type of high order PDEs with combination of partial derivatives concerning space and time, which describes various phenomena of physical including, diffusion of reaction, vibrations of longitudinal and physics of plasma31,32. In recent years, researchers and scientists have presented the numerical and analytical methods to solve the pseudo hyperbolic equation33–35. 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 1

37–39 The CFD of order β for a function Θ(t)∈Cq, q≥-1 is defined as belows:

1 DβΘ(t)=Im-βDmΘ(t)=1Γ(m-β)∫0t(t-ν)m-β-1DmΘ(ν)dν,t>0,m-1<β<m.

Definition 2

37–39 The Caputo fractional partial derivative (CFPD) of order β for a function Θ(x,t)∈Cq, q≥-1 is given by:

2 DtβΘ(x,t)=1Γ(m-β)∫0t(t-ν)m-β-1∂mΘ(x,ν)∂νmdν,m-1<β<m,∂mΘ(x,t)∂tm,β=m∈N.

The CFD satisfies linear property similar to integer order differentiation:Dβλ1Θ1(t)+λ2Θ2(t)+⋯+λmΘm(t)=λ1DβΘ1(t)+λ2DβΘ2(t)+⋯+λmDβΘm(t),

where λ1,λ2,…,λm are constants.

The major properties for the Caputo derivative are:3 Dβk=0,kisconstant.

4 Dβtυ=Γ(υ+1)Γ(υ+1-β)tυ-β,forυ∈N0,υ≥β,0,forυ∈N0,υ<β,

where ⌈β⌉ denote to the smallest integer greater than or equal to β, where N0={0,1,2,…}.

Definition 3

40,41 The power series which has the formula

∑l=0∞ϑl(τ-τ0)lβ=ϑ0+ϑ1(τ-τ0)β+ϑ2(τ-τ0)2β+⋯,0≤l-1<β≤l,l∈N,andτ≥τ0,

is called FPS about τ0.

There exist the three possibilities for convergence of the FPS ∑l=0∞ϑl(τ-τ0)lβ, which are: The series converges only for τ=τ0, that is, the radius of convergence equal zero.

The series converges for all τ≥τ0, that is, the radius of convergence equal ∞.

The series converges for τ0≤τ<τ0+R, for some positive real number R and diverges for τ>τ0+R , where R is the radius of convergence for the FPS.

Definition 4

40 The multiple FPS at τ=τ0 is defined as:

∑r=0∞∑j=0l-1Prj(x)(τ-τ0)rβ+j,0≤l-1<β≤l,l∈Nandτ0≤τ≤τ0+R.

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 Tm(s) over the interval [-1,1] can be defined as:Tm(s)=sin(m+1)ξsin(ξ),

where s=cos(ξ), ξ∈[0,π].

The orthogonality formula of CPSK with respect to weight function ω(s)=1-s2 as:<Tm(s),Tj(s)>=∫-11ω(s)Tm(s)Tj(s)ds=0ifm≠j,π2ifm=j.

The recurrence form of polynomials Tm(s) can be written as:Tm(s)=2sTm-1(s)-Tm-2(s),m=2,3,4,…,

whereT0(s)=1,T1(s)=2s.

The explicit formula of Tm(s) as:5 Tm(s)=∑i=0⌈m2⌉(-1)i2m-2iΓ(m-i+1)Γ(m-2i+1)Γ(i+1)sm-2i,m>0,

where ⌈m2⌉ denotes the integral part of m2.

Definition 6

42 The SCPSK Tm∗(x) is defined on [0, 1] as:Tm∗(x)=Tm(2x-1).

The orthogonal property of SCPSK with respect to weight function ω∗(x)=x-x2 is given as below:<Tm∗(x),Tj∗(x)>=∫01x-x2Tm∗(x)Tj∗(x)dx=0ifm≠j,π8ifm=j,

The recurrence relation of SCPSK:Tm∗(x)=2(2x-1)Tm-1∗(x)-Tm-2∗(x),m=2,3,4,…

whereT0∗(x)=1,T1∗(x)=4x-2.

The analytical expressions of SCPSK Tm∗(x) of degree m can be given as:6 Tm∗(x)=∑i=0m(-1)i22m-2iΓ(2m-i+2)Γ(2m-2i+1)Γ(i+1)xm-i,m>0.

The function u(x)∈L2[0,1] can be defined by SCPSK Ti∗(x) as follows:7 u(x)=∑i=0∞ϑiTi∗(x),

where the coefficients ϑi are given by:8 ϑi=8π∫01x-x2u(x)Ti∗(x)dx,i=0,1,2,⋯

In practice, we truncate the infinite series up to (n+1) terms of SCPSK as follows:9 un(x)=∑i=0nϑiTi∗(x).

Theorem 1

Assume that un(x) be series approximation of spectral SCPSK defined by Eq. (9), then Dβun(x) is given as:

10 Dβun(x)=∑i=⌈β⌉n∑k=0i-⌈β⌉ϑiΩi,k(β)xi-k-β,

where Ωi,k(β) is defined as:Ωi,k(β)=(-1)k22i-2kΓ(2i-k+2)Γ(i-k+1)Γ(k+1)Γ(2i-2k+2)Γ(i-k-β+1).

Proof

(see Ref.42). □

Error analysis

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

Theorem 2

Suppose a function Φ(x)∈[0,1] which is continuous and differentiable up to (n+1) times. Let   un(x)=∑i=0nϑiTi∗(x) be the best square approximation function of Φ(x), then11 ‖Φ(x)-un(x)‖≤M(Kn+1)2Γ(n+2)π2,

where M=maxx∈[0,1]Φ(n+1)(x) and K=max{x0,x-x0}.

Proof

We approximate function Φ(x) by Taylor series as:12 Φ(x)=Φ(x0)+Φ′(x0)(x-x0)Γ(2)+Φ′′(x0)(x-x0)2Γ(3)+⋯+Φ(n)(x0)(x-x0)nΓ(n+1)+Φ(n+1)(ζ)(x-x0)n+1Γ(n+2),

where x0∈[0,1] and ζ∈[x0,x].

Let13 yn(x)=Φ(x0)+Φ′(x0)(x-x0)Γ(2)+Φ′′(x0)(x-x0)2Γ(3)+⋯+Φ(n)(x0)(x-x0)nΓ(n+1),

then14 Φ(x)-yn(x)=Φ(n+1)(ζ)(x-x0)n+1Γ(n+2).

Since un(x)=∑i=0nϑii(x)∗, is the best square approximation function of Φ(x), we haveΦ(x)-un(x)2≤Φ(x)-yn(x)2=∫01w∗(x)Φ(x)-yn(x)2dx=∫01w∗(x)Φ(n+1)(ζ)(x-x0)n+1Γ(n+2)2dx≤M2(Γ(n+2))2∫01x-x2(x-x0)n+12dx.

Hence K=max{x0,x-x0}, we get15 Φ(x)-un(x)2≤M2K(2n+2)(Γ(n+2))2∫01x-x2dx=M2K(2n+2)(Γ(n+2))2π8.

By taking square root of both sides for Eq. (15), we get16 ‖Φ(x)-un(x)‖≤M(Kn+1)2Γ(n+2)π2.

□

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. Time fractional hyperbolic PDEs29 17 DtβΘ(x,t)-μDx2Θ(x,t)-LΘ(x,t)=0,1<β≤2,x∈[0,X],t>0,

with subject to initial conditions (ICs) and boundary conditions (BCs): 18 Θ(x,0)=f1(x),DtΘ(x,0)=f2(x),Θ(0,t)=A1(t),Θ(X,t)=A2(t),

where μ∈R and L is non linear operator. Assume Θn(x,t) is approximated as: 19 Θn(x,t)=∑i=0nϑi(t)Ti∗(x).

Let us to utilize the approximation of Θn(x,t) which is defined in Eq. (19) as following steps:Step (I) By applying Theorem (1) and Eqs. (17) and (19), we have 20 ∑i=0nDtβϑi(t)Ti∗(x)-μ∑i=⌈β⌉n∑k=0i-⌈β⌉ϑi(t)Ωi,k(β)xi-k-β-L∑i=0nϑi(t)Ti∗(x)=0.

Step (II) Now we collocate Eq. (20) at xp, p=0,1,2,ldotsn-⌈β⌉ and the collocation point of SCPSK Tn+1-⌈β⌉∗(x), we have a system of fractional order differential equations (FODEs) as: 21 ∑i=0nDtβϑi(t)Ti∗(xp)-μ∑i=⌈β⌉n∑k=0i-⌈β⌉ϑi(t)Ωi,k(β)xpi-k-β-L(∑i=0nϑi(t)Ti∗(xp))=0.

Step (III) Substituting Eq. (19) into Eqs. (18), we can obtain (⌈β⌉+1) algebraic equations as: 22 ∑i=0nϑi(0)Ti∗(x)=f1(x),∑i=0nDtϑi(0)Ti∗(x)=f2(x),

where BCs 23 ∑i=0nϑi(t)Ti∗(0)=A1(t),∑i=0nϑi(t)Ti∗(X)=A2(t).

To obtain the unknown coefficients ϑ0(t),ϑ1(t),ϑ2(t),…,ϑn(t), combing Eqs. (21)–(23), we have system of FODEs, which can be solved by utilizing RPSA. To determine the unknown coefficients of ϑ0(t),ϑ1(t),ϑ2(t),…,ϑn(t), we take n=2 and n=3 in Eq. (21), respectively: 24 Dtβϑ0(t)-Dtβϑ2(t)-32μϑ2(t)-L(ϑ0(t)-ϑ2(t))=0.

25 Dtβϑ0(t)-Dtβϑ1(t)+Dtβϑ3(t)-μ(32ϑ2(t)-96ϑ3(t))-L(ϑ0(t)-ϑ1(t)+ϑ3(t))=0,Dtβϑ0(t)+Dtβϑ1(t)-Dtβϑ3(t)-μ(32ϑ2(t)+96ϑ3(t))-L(ϑ0(t)+ϑ1(t)-ϑ3(t))=0

By solving Eq. (25), we get 26 Dtβϑ0(t)-32μϑ2(t)-12[L(ϑ0(t)-ϑ1(t)+ϑ3(t))+L(ϑ0(t)+ϑ1(t)-ϑ3(t))]=0,Dtβϑ1(t)-Dtβϑ3(t)-96μϑ3(t)+12[L(ϑ0(t)-ϑ1(t)+ϑ3(t))-L(ϑ0(t)+ϑ1(t)-ϑ3(t))]=0.

By solving Eq. (23) at n=2 and n=3, respectively. Then we get 27 ϑ1(t)=14(A2(t)-A1(t)),ϑ2(t)=16(A1(t)+A2(t))-13ϑ0(t).

28 ϑ2(t)=16(A1(t)+A2(t))-13ϑ0(t),ϑ3(t)=18(A2(t)-A1(t))-12ϑ1(t).

By substituting Eqs. (27) and (28) into Eqs. (24) and (26), then 29 Dtβϑ0(t)-18Dtβ(A1(t)+A2(t))-4μ(A1(t)+A2(t))+8μϑ0(t)-34L43ϑ0(t)-16[A1(t)+A2(t)],

30 Dtβϑ0(t)-16μ3[A1(t)+A2(t)]+32μ3ϑ0(t)-12[L(ϑ0(t)-32ϑ1(t)+18[A2(t)-A1(t)])+L(ϑ0(t)+32ϑ1(t)-18[A2(t)-A1(t)])],

31 Dtβϑ1(t)-112Dtβ(A2(t)-A1(t))-8μ[A2(t)-A1(t)]+32μϑ1(t)+13[L(ϑ0(t)-32ϑ1(t)+18[A2(t)-A1(t)])-L(ϑ0(t)+32ϑ1(t)-18[A2(t)-A1(t)])]=0.

RPSA assumes the solution of Eq. (29) using FPS at t0=0 as: 32 ϑ0(t)=Υ0+Υ1t+∑r=1∞∑j=0lhrjtrβ+jΓ(rβ+j+1).

Next, let ϑ0(s,l)(t) denote the sth truncated series of ϑ0(t) which take the form: 33 ϑ0(s,l)(t)=Υ0+Υ1t+∑r=1s∑j=0lhrjtrβ+jΓ(rβ+j+1),∀s=1,2,...andl=0,1,

where Υ0 and Υ1 can be obtained by solving Eqs. (22) and (27). The RPSA assumes the solution of Eqs. (30) and (31) using FPS at t0=0 as: 34 ϑ0(t)=Ψ0+Ψ1t+∑r=1∞∑j=0lfrjtrβ+jΓ(rβ+j+1),ϑ1(t)=η0+η1t+∑r=1∞∑j=0ldrjtrβ+jΓ(rβ+j+1).

Let ϑ0(s,l)(t) and ϑ1(s,l)(t) denote the sth truncated series of ϑ0(t) and ϑ1(t) which take the form: 35 ϑ0(s,l)(t)=Ψ0+Ψ1t+∑r=1s∑j=0lfrjtrβ+jΓ(rβ+j+1),ϑ1(s,l)(t)=η0+η1t+∑r=1s∑j=0ldrjtrβ+jΓ(rβ+j+1),∀s=1,2,…andl=0,1,

where Ψ0, Ψ1, η0 and η1 can be obtained by solving Eqs. (28) and (22). We can write the residual functions of Eqs. (29)–(31) as: 36 ℜes(s,l)(t)=Dtβϑ0(s,l)(t)-18Dtβ(A1(t)+A2(t))-4μ(A1(t)+A2(t))+8μϑ0(s,l)(t)-34L(43ϑ0(s,l)-16[A1(t)+A2(t)]),

37 ℜes1(s,l)(t)=Dtβϑ0(s,l)(t)-16μ3[A1(t)+A2(t)]+32μ3ϑ0(s,l)(t)-12[L(ϑ0(s,l)(t)-32ϑ1(s,l)(t)+18[A2(t)-A1(t)])+L(ϑ0(s,l)(t)+32ϑ1(s,l)(t)-18[A2(t)-A1(t)])],

38 ℜes2(s,l)(t)=Dtβϑ1(s,l)(t)-112Dtβ(A2(t)-A1(t))-8μ[A2(t)-A1(t)]+32μϑ1(s,l)(t)+13[L(ϑ0(s,l)(t)-32ϑ1(s,l)(t)+18[A2(t)-A1(t)])-L(ϑ0(s,l)(t)+32ϑ1(s,l)(t)-18[A2(t)-A1(t)])],

and39 Dt(r-1)βDtjℜes(s,l)(t0)=0,Dt(r-1)βDtjℜes1(s,l)(t0)=0,Dt(r-1)βDtjℜes2(s,l)(t0)=0,∀r=1,2,…,sandj=0,1,…,l.

Time fractional pseudo hyperbolic PDEs with nonlocal conditions34 40 DtβΘ(x,t)-εDtDx2Θ(x,t)-Dx2Θ(x,t)-W(x,t)=0,1<β≤2,x∈[0,X],t∈[0,T],

subject to ICs and BCs: 41 Θ(x,0)=V1(x),DtΘ(x,0)=V2(x),x∈[0,X],Θ(0,t)=ρ1(t)+∫0XΘ(x,t)dx=B1(t),t∈[0,T],Θ(X,t)=ρ2(t)+∫0XΘ(x,t)dx=B2(t),t∈[0,T].

Let us utilize the approximation of Θn(x,t) which defined in Eq. (19) as following steps:Step (I) By substituting Theorem (1) and Eq. (19) into Eq. (40), we obtain 42 ∑i=0nDtβϑi(t)Ti∗(x)-ε∑i=⌈β⌉n∑k=0i-⌈β⌉Dtϑi(t)Ωi,k(β)xi-k-β-∑i=⌈β⌉n∑k=0i-⌈β⌉ϑi(t)Ωi,k(β)xi-k-β-W(x,t)=0.

Step (II) By collocating Eq. (42) at the roots xp, p=0,1,2,…n-⌈β⌉ and the collocation point of SCPSK n+1-⌈β⌉∗(x), we get a system of FODEs as: 43 ∑i=0nDtβϑi(t)Ti∗(xp)-ε∑i=⌈β⌉n∑k=0i-⌈β⌉Dtϑi(t)Ωi,k(β)xpi-k-β-∑i=⌈β⌉n∑k=0i-⌈β⌉ϑi(t)Ωi,k(β)xpi-k-β-W(xp,t)=0.

Step (III) By substituting Eq. (19) into Eq. (41), we can obtain (⌈β⌉+1) algebraic equations as: 44 ∑i=0nϑi(0)Ti∗(x)=V1(x),∑i=0nDtϑi(0)Ti∗(x)=V2(x),

where BCs 45 ∑i=0nϑi(t)Ti∗(0)=B1(t),∑i=0nϑi(t)Ti∗(X)=B2(t).

To obtain the unknown coefficients ϑ0(t),ϑ1(t),ϑ2(t),…,ϑn(t), combing Eqs. (43)–(45), we have system of FODEs which can be solved by utilizing RPSA. To determine the unknown coefficients of ϑ0(t),ϑ1(t),ϑ2(t),…,ϑn(t), we take n=3 in Eq. (43), we have 46 Dtβϑ0(t)-Dtβϑ1(t)+Dtβϑ3(t)-εDt(32ϑ2(t)-96ϑ3(t))-(32ϑ2(t)-96ϑ3(t))-W14,t=0,Dtβϑ0(t)+Dtβϑ1(t)-Dtβϑ3(t)-εDt(32ϑ2(t)+96ϑ3(t))-(32ϑ2(t)+96ϑ3(t))-W34,t=0.

By solving Eq. (45), we get 47 ϑ2(t)=16(B1(t)+B2(t))-13ϑ0(t),ϑ3(t)=18(B2(t)-B1(t))-12ϑ1(t).

By solving Eq. (46), we obtain 48 Dtβϑ0(t)-32εDtϑ2(t)-32εϑ2(t)-12(W(14,t)+W(34,t))=0,Dtβϑ1(t)-Dtβϑ3(t)-96εDtϑ3(t)-96ϑ3(t)+12(W(14,t)-W(34,t))=0.

By substituting Eq. (47) into Eq. (48), then 49 Dtβϑ0(t)-16ϵ3Dt(B1(t)+B2(t))+32ϵ3Dtϑ0(t)-163(B1(t)+B2(t))+323ϑ0(t)-12(W(14,t)+W(34,t))=0,

50 Dtβϑ1(t)-112Dtβ(B2(t)-B1(t))-8εDt(B2(t)-B1(t))+32εDtϑ1(t)-8(B2(t)-B1(t))+32ϑ1(t)+13(W(14,t)-W(34,t))=0.

Let ϑ0(s,l)(t) and ϑ1(s,l)(t) denote the sth truncated series of ϑ0(t) and ϑ1(t) which defined in Eq. (35), then the residual functions of Eqs. (49) and (50) take the form: 51 Res1(s,l)(t)=Dtβϑ0(s,l)(t)-16ϵ3Dt(B1(t)+B2(t))+32ϵ3Dtϑ0(s,l)(t)-163(B1(t)+B2(t))+323ϑ0(s,l)(t)-12(W(14,t)+W(34,t)),

52 Res2(s,l)(t)=Dtβϑ1(s,l)(t)-112Dtβ(B2(t)-B1(t))-8εDt(B2(t)-B1(t))+32εDtϑ1(s,l)(t)-8(B2(t)-B1(t))+32ϑ1(s,l)(t)+13(W(14,t)-W(34,t)),

and53 Dt(r-1)βDtjRes1(s,l)(t0)=0,Dt(r-1)βDtjRes2(s,l)(t0)=0,∀r=1,2,…,sandj=0,1,…,l.

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 PDEs29 which are described in Eq. (17), where μ=0 and L(Θ(x,t))=∂∂x(Θ(x,t)∂Θ(x,t)∂x), then54 DtβΘ(x,t)-∂∂x(Θ(x,t)∂Θ(x,t)∂x)=0,1<β≤2,x∈[0,1],t>0,

with ICs and BCs:55 Θ(x,0)=x2,DtΘ(x,0)=-2x2,Θ(0,t)=0,Θ(1,t)=1(t+1)2.

The exact solution at β=2 is Θ(x,t)=x2(t+1)2.

Table 1 presents the approximate solutions obtained by CTSCSK-RPSA with VIM, ADM8, VHPIM, HPM11 and PIA29. Table 2 present the CTSCSK-RPSA approximate solutions at various 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. Table 1 The Comparison between CTSCSK-RPSA and other available methods for Problem 1.

t	x	Exact	CTSCSK-RPSA	VIM8	ADM8	HPM11	VHPIM11	PIA29	
n=2	n=3	
0.2	0.25	0.043403	0.043403	0.043403	0.043400	0.043395	0.043400	0.04320	0.043400	
0.2	0.5	0.173611	0.173611	0.173611	0.173600	0.173580	0.173600	0.172820	0.173599	
0.2	0.75	0.390625	0.390625	0.390625	0.390600	0.390556	0.390600	0.388844	0.390599	
0.2	1	0.694444	0.694444	0.694444	0.694400	0.694321	0.694400	0.691278	0.694399	
0.4	0.25	0.031888	0.031887	0.031888	0.031779	0.031567	0.031779	0.029913	0.031779	
0.4	0.5	0.127551	0.127551	0.127551	0.127118	0.126268	0.127118	0.119650	0.127118	
0.4	0.75	0.286990	0.286988	0.286990	0.286015	0.284103	0.286015	0.269212	0.286015	
0.4	1	0.510204	0.510204	0.510204	0.508471	0.505072	0.508471	0.478600	0.508472	
0.6	0.25	0.024414	0.024648	0.024444	0.023665	0.022005	0.023665	0.018860	0.023665	
0.6	0.5	0.097656	0.097968	0.097460	0.094660	0.088018	0.094660	0.075442	0.094659	
0.6	0.75	0.219727	0.219960	0.219403	0.212984	0.198040	0.212984	0.169743	0.212984	
0.6	1	0.390625	0.390625	0.390625	0.378638	0.352071	0.378638	0.301766	0.378638	

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

t	x	β=1.5	β=1.75	
ADM8	VIM8	n=2	n=3	ADM8	VIM8	n=2	n=3	
0.2	0.25	0.0592832	0.047502	0.043791	0.043325	0.0497012	0.043403	0.044368	0.043210	
0.2	0.5	0.237133	0.190007	0.174129	0.174129	0.194805	0.184170	0.174898	0.174898	
0.2	0.75	0.533549	0.427517	0.391013	0.391480	0.438311	0.414383	0.391590	0.392748	
0.2	1	0.948532	0.760029	0.694444	0.694444	0.779220	0.736680	0.694444	0.694444	
0.4	0.25	0.0654119	0.041853	0.043745	0.023960	0.037742	0.037742	0.032128	0.031840	
0.4	0.5	0.261647	0.167412	0.180405	0.180405	0.174992	0.150968	0.127872	0.174898	
0.4	0.75	0.588707	0.376676	0.326630	0.374198	0.393732	0.339679	0.287230	0.287519	
0.4	1	1.04659	0.669647	0.510204	0.510204	0.699969	0.603873	0.510204	0.510204	
0.6	0.25	0.063177	0.037722	0.220672	0.117212	0.381836	0.031457	0.185133	0.100000	
0.6	0.5	0.252710	0.150888	0.359333	0.359333	0.152735	0.125829	0.311949	0.311949	
0.6	0.75	0.568598	0.339499	0.318117	0.651493	0.343653	0.283114	0.415984	0.554257	
0.6	1	1.01084	0.603553	0.390625	0.390625	0.610938	0.503314	0.390625	0.390625	

Figure 1 Exact and approximate solutions at β=2 of Problem 1.

Figure 2 Behavior of approximate at different values of β for Problem 1.

Figure 3 2D graphics of exact and approximate solutions at different fractional order of β for Problem 1.

Problem 2. Consider time fractional pseudo hyperbolic PDEs with nonlocal conditions3456 DtβΘ(x,t)-εDtDx2Θ(x,t)-Dx2Θ(x,t)-et(x3-18x)=0,1<β≤2,x∈[0,1],t∈[0,2],

with ICs and BCs:57 Θ(x,0)=x3,DtΘ(x,0)=x3,Θ(0,t)=∫01Θ(x,t)dx-14et,Θ(1,t)=∫01Θ(x,t)dx+34et.

The exact solution at β=2 is Θ(x,t)=x3et.

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 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. Table 3 Numerical results of pseudo hyperbolic PDE with nonlocal conditions at ε=2 and β=2 for Problem 2.

x	t	Exact	Approximate solutions	Absolute error	
CTSCSK-RPSA	RPSA34	CTSCSK-RPSA	RPSA34	
0	0	0	0	0	0	0	
0.1	0.2	0.001221	0.001221	0.001221	6.49719×10-14	1×10-12	
0.2	0.4	0.011935	0.011935	0.011935	1.73472×10-18	0	
0.3	0.6	0.049197	0.049197	0.049197	2.77556×10-17	3×10-11	
0.4	0.8	0.142435	0.142435	0.142435	8.32667×10-17	1×10-10	
0.5	1	0.339785	0.339785	0.339785	0	3×10-10	
0.6	1.2	0.717145	0.717145	0.717145	1.11022×10-16	0	
0.7	1.4	1.390934	1.390934	1.390934	2.22045×10-16	0	
0.8	1.6	3.097420	3.097420	3.097420	4.44089×10-16	0	
0.9	1.8	4.410193	4.410193	4.410193	0	3×10-9	
1	2	7.389056	7.389056	7.389056	2.66454×10-15	2.8×10-8	

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

x	t	CTSCSK-RPSA	
β=1.95	β=1.85	β=1.75	
0	0	0	0	0	
0.1	0.2	0.007560	0.001264	0.001298	
0.2	0.4	0.012122	0.012529	0.012985	
0.3	0.6	0.050111	0.052087	0.054292	
0.4	0.8	0.145352	0.151656	0.158673	
0.5	1	0.347200	0.363209	0.381013	
0.6	1.2	0.733481	0.768741	0.807946	
0.7	1.4	1.423587	1.494070	1.572439	
0.8	1.6	2.596795	2.728137	2.874197	
0.9	1.8	4.517702	4.749821	5.008006	
1	2	7.571329	7.964938	8.402844	

Figure 4 Exact and approximate solutions at β=2 of Problem 2.

Figure 5 Behavior of approximate at different values of β for Problem 2.

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

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.

Author contributions

Saad. Z. Rida jointly supervised this study. Anas. A. M. Arafa conceived the study. Hussein. S. Hussein and I. Ameen revised the manuscript. Marwa. M. M. Mostafaconducted the analyses and wrote the manuscript. All authors agree to submit the manuscript in its current form.

Funding

Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).

Data availability

Data used to support the findings of this study are included in the article.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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