Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2021.014988

ARTICLE

Regarding on the Fractional Mathematical Model of Tumour Invasion and Metastasis

P. Veeresha1, Esin Ilhan2, D. G. Prakasha3, Haci Mehmet Baskonus4 and Wei Gao5,*

1Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, 560029, India
2Faculty of Engineering and Architecture, Kirsehir Ahi Evran University, Kirsehir, 40500, Turkey
3Department of Mathematics, Faculty of Science, Davangere University, Shivagangothri, Davangere, 577007, India
4Department of Mathematics and Science Education, Faculty of Education, Harran University, Sanliurfa, 63050, Turkey
5School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, China
*Corresponding Author: Wei Gao. Email: gaowei@ynnu.edu.cn
Received: 14 November 2020; Accepted: 09 February 2021

1  Introduction

The existence of cancer among the people around the globe has been developed swiftly, and globally it becomes the second foremost cause of demise after cardiovascular diseases [1]. The process of spreading and formation of secondary tumours is known as Metastasis and this nature of cancer cells is the main reason for the death in cancer patients. In addition, the prediction of size, stage, and evolution of a tumour is very critical for the treatment of cancer. Moreover, mathematics plays as an essential tool and aid us to analyze the behaviour of the tumour. The tumour growth has been mathematically modelled by the number of researchers and which are appeared in the literature [2–4]. The growth of the tumour invasion and metastasis are described by PDEs. Particularly, deterministic diffusion-reaction equations and these equations are employed to model the spatial spread of tumours at the initial development and later invasive moments [5].

The fractional-order derivatives are introduced by Leibnitz soon after the classical order derivatives. As compared to classical calculus, it was soon discovered that fractional calculus (FC) is more appropriate for describing real-world problems [6–10]. The calculus of arbitrary order turned out one of the most essential tools to describe biological phenomena. The human diseases which are modelled through derivative having fractional-order help us to incorporate the information about its present and past states [11–16]. Moreover, these models demonstrate the non-local distributed effects, hereditary properties and system memory. These properties are necessary to describe the biological models. In connection with this, recently many authors established the arbitrary-order model to analyze the diffusion equation and to forecast the effect of the tumour and they applied many powerful methods to find the solution for these models [17–32]. The pivotal aim of generalizing the integer to fractional order is to capture consequences related non-locality, long-range memory and time-based properties and also anomalous diffusion aspects. Many real-world problems exemplified with complex and nonlinear models are effectively, systematically and accurately illustrated and investigated by the aid of theory and fundamental concept of FC. Many pioneers nurtured and developed novel and distinct notions of fractional order for both differential and integral operators. Most familiarly hired operators to analyze many models are Riemann, Liouville, Caputo, Fabrizio and others. However, researchers pointed out some limitations while generalizing the system with these notions. The Riemann–Liouville derivative fails to elucidate the essence of initial conditions; the Caputo derivative has overcome this limitation and latter it has been widely applied to the numerous classes of mathematical models exemplifying real-world problems. But this fractional operator is unable to describe the singular kernel of the systems or problems. However, Caputo et al. [33] in 2015 overcome the foregoing obliges and then the number of authors employed CF derivative to investigate and study wide classes of complex and nonlinear problems. It has been proved by many researchers; CF fractional operator as great results compared to other fractional operators.

The study of mathematical models effectively exemplified diverse phenomena. However, as much as important of nurturing the phenomena with the system of equations finding the corresponding solution is also very vital and difficult. In this path, many authors examined diverse phenomena, for instance, the structured predator-prey model with prey refuge [34], COVID-19[35–39], Zika virus transmission [40], planar system-masses in an equilateral triangle [41], a harmonic oscillator with position-dependent mass [42], time fractional Burgers equation [43], fractional optimal control problems [44], Emden–Flower type equations [45], and many others [46–53]. These investigations help researchers to understand the importance of generalizing the classical concept into fractional operators, and efficiency and difference between diverse schemes.

Many physicists, engineers and mathematicians recently proposed and modified diverse solution procedure with a different approach with respect to increasing in accuracy and methodology, to reduces the complexity, many additional assumptions and consideration, huge time for evaluation and to save computer memory. Moreover, each method is suitable for some specific family of problems and they have their own limitations, including conversion of nonlinear to linear, partial to ordinary differential equations, splitting complex and nonlinear term to simple parts terms. In this connection, with the help of topological concept called homotopy, Liao Shijun who is a Chinese Mathematician proposed algorithm called homotopy analysis method (HAM) and illustrated to confirm it overcomes almost all the limitations raised while we solving nonlinear systems exists in sciences and other disciplines associated to mathematics [54]. The most familiar thing of employing this method by many authors is including it solves nonlinear problems without linearization and perturbation.

As science and technology-enhanced, mankind always expecting new tools or modifications in existing tools to improve the accuracy and reduces the time taken for finding needful. In this regard, some scholars pointed out similar things in HAM and suggested to union with existing and familiar transformation. Authors in [55] modified q-HAM with the help of Laplace transform (LT) and manifest new modified scheme is called q-HATM. This method is perceptible includes all merits which are achieved by HAM and also it attracted many researchers to analyze the diverse class of models and systems. For instance, the model exemplifying three Lakes pollution with the newly proposed fractional operator is investigated by authors in [56], fractional vibration equation is analyzed by authors in [57] with some interesting results, authors in [58] presented the efficiency of the projected scheme while analyzing Swift–Hohenberg equation having arbitrary order, the accuracy of the hired scheme in comparison with existing results is illustrated by authors in [59] with respect to the physical model, the convergence analysis of the considered method for Lienard’s equation is demonstrated in [60], many others analyzed various biological and physical phenomena by the assist of the projected scheme [61–62].

In the present study, we find the series solution for a system of nonlinear differential equations describing the model of tumour invasion and metastasis using q-HATM with the help of a novel fractional operator. By using the important results of fixed-point theory for the projected system the existence and uniqueness are demonstrated. The novelty of the projected scheme gives more freedom to choose the initial conditions and the novelty is it offers a simple solution procedure and associated with parameters to provide the swift convergence. Further, it contains the results achieved by other classical methods including ADM, HPM, q-HAM and some other methods [63–72]. In the present study, we analyzed the system describing the tumour invasion and metastasis with different time and space for different fractional-order using q-HATM within the frame of the novel fractional operator which can describe the singular kernel. This study can help us to analyze more complex and nonlinear mathematical models describing the deadly virus or diseases.

The rest of the manuscript is organized as follows: The basic and fundamentals are presented in the next section, the hired model is exemplified in Section 3, the basic procedure of the q-HATM is presented in Section 4, and its algorithm is illustrated for the considered model in Section 5. The existence and uniqueness for the archived results and error analysis are respectively presented in Sections 6 and 7. Moreover, with the aid of behaviour captured for the obtained result, the corresponding comments and conclusion are respectively exemplified in Sections 8 and9.

2  Preliminaries

The basic definitions are presented in this segment for the FC and Laplace transform. Specifically, we recall the notions related to Caputo-Fabrizio fractional operator [33,73].

Definition 1.The CF fractional derivative for f∈H1(a,b) is presented as [33]

Dtα(f(t))=M(α)1−α∫atf′(t)exp⁡[−αt−ϑ1−α]dϑ,b>a,α∈[0,1], (1)

where M(α) is a normalization function and admits M(0)=M(1)=1. Further, if f∉H1(a,b) then we have

Dtα(f(t))=αM(α)1−α∫at(f(t)−f(ϑ))exp⁡[−αt−ϑ1−α]dϑ. (2)

Definition 2. The CF fractional derivative for f∈H1(a,b) is presented as [73]

Itα(f(t))=2(1−α)(2−α)M(α)f(t)+2α(2−α)M(α)∫0tf(ϑ)dϑ,0<α<1,t≥0. (3)

Note: According to [73], the following must hold

2(1−α)(2−α)M(α)+2α(2−α)M(α)=1,0<α<1, (4)

which gives M(α)=22−α. By the assist of above equation researchers in [73] proposed a novel Caputo derivative as follows:

Dtα(f(t))=11−α∫0tf′(t)exp⁡[αt−ϑ1−α]dϑ,0<α<1. (5)

Definition 3. The LT for a CF derivative 0CFDtαf(t) is presented as [33] below

L[0CFCDt(α+n)f(t)]=sn+1L[f(t)]−snf(0)−sn−1f′(0)−…−f(n)(0)s+(1−s)α. (6)

3  Mathematical Model

On the basis of generic solid tumour growth and assuming it is in avascular stage, the mathematical model has been proposed [74]. In this stage, most of the tumours are asymptomatic and further there is a possibility of cells to migrate and escape to the lymph nodes. The considered system exemplifies the interfaces of the surrounding tissue with the tumour and it can be extended to incorporate tumour and the vasculature. Here, the projected system of equations illustrates the interactions of the matrix-degrading enzymes (MDE, signifies by E), extra cellular matrix (ECM, symbolised by C) and tumour cells (denoted by T). With respect to ECM, most of the macromolecules are essential for cell motility, spreading and adhesion. Further, the ECM associated with many macromolecules, for instances collagen, laminin and fibronectin. During the various stages of metastasis, invasion and turn our growth, MDEs play a vital role. The ECM locally degrades by MDEs which are produced by tumour cells. Further, the method wherein they interact with tumour cells, growth factors and inhibitors are highly intricate. The tumour cells in the considered system as haptotaxis and in order to integrate this concept in the model, the hypotactic flux is considered as [75,76]

Jhapto=χT∇C,

where χ>0 denotes haptotactic coefficient and which is constant. The random motion is another contribution to tumour cell motility and it helps to study ECM in isolation. Moreover, the flux is defined for the tumour cells with exemplified random motility is

Jrandom=−D(C,E)∇T,

where ∇T is the chemokinetic response, D(C,E) is the function of either the ECM or MDE concentration, or constant.

For the tumour cell density (T), the conservation equation is presented as

∂T∂t+∇⋅(Jhapto+Jrandom)=0,

and for the cell proliferation absence, the equation describing tumour cell motion is defined as

∂T∂t=∇⋅(D(C,M)∇T)−χ∇⋅(T∇C) (7)

For the notation, the random motility coefficient of tumour cell is considered as D(C,M)=DT and which is constant. Therefore, the degradation process is exemplified by the subsequent equation with positive constant δ

∂C∂t=−δEC. (8)

Active MDEs are formed by T, experience some form of decay and diffuse throughout the tissue. The equation modelling the evolution of MDE concentration is presented with MDE diffusion coefficient DE as

∂E∂t=DE∇2E+g(T,E)−h(T,E,C), (9)

where g=μT and h=λE, h and g are the functions respectively describing the MDE decay and the production of active MDEs. Moreover, in the surrounding tissues there is a linear relationship between the level of active MDEs and the density of tumour cells.

From the above description, the system is presented as [18,54,57]:

∂T∂t=DT∇2T⏟randommotility−χ∇⋅(T∇C)⏟haptotaxis,

∂C∂t=−δEC⏟,degradation

∂E∂t=DE∇2E⏟diffusion+μT⏟production−λE⏟.decay (10)

Here, with appropriate initial conditions, Eq. (10) is assumed to satisfy on a region of tissue or domain Ω. Moreover, the model is nurtured so that the MDEs and tumour cells remain inside the domain of tissue within deliberation and hence no-flux boundary conditions are executed on ∂Ω. The terms contained in the above system with ECM density (C0), tumour cell density (T0) and MDE concentration (E0) by setting

T~=TT0,C~=CC0,E~=EE0,x~=xl,t~=tτ, (11)

where l signifies scale length and τ is the time. Then we have a scaled system of equations by dropping the tildes for notational convenience [18,74,77]

∂T∂t=dT∇2T−γ∇⋅(T∇C),

∂C∂t=−ηEC,

∂E∂t=dE∇2E+αT−βE, (12)

where dT=DT/D, γ=χC0/D, η=τE0δ,dE=DE/D, α=τμT0/E0 and β=τλ.

The projected model can be protracted to integrate interactions between blood vessels and the tumour cells [74].

Now, we modify the time derivative by the CF derivative in Eq. (12) and given by

0CFDtαT(x,t)=dT∂2T∂x2−γ[∂T∂x∂C∂x+∂2C∂x2],

0CFDtαC(x,t)=−ηEC,

0CFDtαE(x,t)=dE∂2E∂x2+αT−βE, (13)

where α is fractional order. The associated initial conditions are

T(x,0)=e−x2ε,

C(x,0)=1−0.5e−x2ε,

E(x,0)=0.5e−x2ε. (14)

4  Fundamental Idea of the Considered Scheme

In this section, we hired the differential equation to present the basic procedure of the projected scheme with initial conditions

0CFDtαv(x,t)+Rv(x,t)+Nv(x,t)=f(x,t),n−1<α≤n, (15)

and

v(x,0)=𝓆(x). (16)

We obtained by applying LT on Eq. (15)

L[v(x,t)]−𝓆(x)s+s+(1−s)αs{L[Rv(x,t)]+L[Nv(x,t)]−L[f(x,t)]}=0. (17)

For φ(x,t;q), N is contracted as follows:

N[φ(x,t;q)]=L[φ(x,t;q)]−𝓆(x)s

+s+(1−s)αs{L[Rφ(x,t;q)]+L[Nφ(x,t;q)]−L[f(x,t)]}, (18)

where q∈[0,1𝓃]. Then, the homotopy is defined by results in [34]

(1−𝓃q)L[φ(x,t;q)−v0(x,t)]=ℏqN[φ(x,t;q)], (19)

where L is signifying LT. For q=0 and q=1n, the following conditions satisfies

φ(x,t;0)=v0(x,t),φ(x,t;1n)=v(x,t). (20)

By using Taylor theorem we get

φ(x,t;q)=v0(x,t)+∑m=1∞vm(x,t)qm, (21)

where

vm(x,t)=1m!∂mφ(x,t;q)∂qm|q=0. (22)

For the proper chaise of v0(x,t),n and ℏ the series (13) converges at q=1𝓃. Then

v(x,t)=v0(x,t)+∑m=1∞vm(x,t)(1n)m. (23)

After differentiating Eq. (19) m-times with q and multiplying by 1m! and substituting q=0, one can get

L[vm(x,t)−kmvm−1(x,t)]=ℏRm(v→m−1), (24)

where the vectors are defined as

v→m={v0(x,t),v1(x,t),…,vm(x,t)}. (25)

Eq. (24) reduces after employing inverse LT to

vm(x,t)=kmvm−1(x,t)+ℏL−1[Rm(v→m−1)], (26)

where

Rm(v→m−1)=L[vm−1(x,t)]−(1−kmn)(𝓆(x)s+s+(1−s)αsL[f(x,t)])

+s+(1−s)αsL[Rvm−1+Hm−1], (27)

and

km={0,m≤1,n,m>1. (28)

Here, Hm is homotopy polynomial and presented as

Hm=1m![∂mφ(x,t;q)∂qm]q=0and φ(x,t;q)=φ0+qφ1+q2φ2+…. (29)

By the help of Eqs. (26) and (27), we found

vm(x,t)=(km+ℏ)vm−1(x,t)−(1−kmn)L−1(𝓆(x)s+s+(1−s)αsL[f(x,t)])

+ℏL−1{s+(1−s)αsL[Rvm−1+Hm−1]}. (30)

By the help of q-HATM, the series solution is

v(x,t)=v0(x,t)+∑m=1∞vm(x,t)(1n)m. (31)

5  Implementation of the q-Homotopy Analysis Transform Method

Consider the system of equation cited in Eq. (13) describing the tumour invasion and metastasis in CF fractional derivative

0CFDtαT(x,t)−dT∂2T∂x2+γ[∂T∂x∂C∂x+∂2C∂x2]=0,

0CFDtαC(x,t)−ηEC=0,

0CFDtαE(x,t)−dE∂2E∂x2−αT+βE=0. (32)

Applying Laplace transform on Eq. (32) and then with the help of Eq. (13), we get

L[T(x,t)]−1s(e−x2ε)−s+(1−s)αsL{dT∂2T∂x2−γ[∂T∂x∂C∂x+∂2C∂x2]}=0,

L[C(x,t)]−1s(1−0.5e−x2ε)+s+(1−s)αsL{ηEC}=0,

L[E(x,t)]−1s(0.5e−x2ε)−s+(1−s)αsL{dE∂2E∂x2+αT−βE}=0. (33)

The non-linear operator N defined as

N1[ φ1(x,t;q),φ2(x,t;q),φ3(x,t;q) ]=L[ φ1(x,t;q) ]−1s(e−x2ε)−s+(1−s)αsL{ dT∂2φ1(x,t;q)∂x2                 −γ[ ∂φ1(x,t;q)∂x∂φ2(x,t;q)∂x+∂2φ2(x,t;q)∂x2 ] },N2[ φ1(x,t;q),φ2(x,t;q),φ3(x,t;q) ]=L[ φ2(x,t;q) ]−1s(1−0.5e−x2ε)                 +s+(1−s)αsL{ ηφ3(x,t;q)φ2(x,t;q) },

N3[ φ1(x,t;q),φ2(x,t;q),φ3(x,t;q) ]=L[ φ3(x,t;q) ]−1s(0.5e−x2ε)−s+(1−s)αs                 ×L{ dE∂2φ3(x,t;q)∂x2+αφ1(x,t;q)−βφ3(x,t;q) }. (34)

The m-th order deformation equation by the projected scheme at H(x,t)=1 is given by

L[Tm(x,t)−kmTm−1(x,t)]=ℏL−1{R1,m[T→m−1,C→m−1,E→m−1]},

L[Cm(x,t)−kmCm−1(x,t)]=ℏL−1{R2,m[T→m−1,C→m−1,E→m−1]},

L[Em(x,t)−kmEm−1(x,t)]=ℏL−1{R3,m[T→m−1,C→m−1,E→m−1]}, (35)

where

R1,m[T→m−1,C→m−1,E→m−1]=L[Tm−1(x,t)]−(1−kmn)1s(e−x2ε)−s+(1−s)αsL{dT∂2Tm−1∂x2−γ[∑i=0m−1∂Ti∂x∂Cm−1−i∂x+∂2Cm−1∂x2]},R2,m[T→m−1,C→m−1,E→m−1]=L[Cm−1(x,t)]−(1−kmn)1s(1−0.5e−x2ε)+s+(1−s)αsL{η∑i=0m−1EiCm−1−i},R3,m[T→m−1,C→m−1,E→m−1]=L[Em−1(x,t)]−(1−kmn)1s(0.5e−x2ε)−s+(1−s)αsL{dE∂2Em−1∂x2+αTm−1−βEm−1}. (36)

On employing inverse LT on Eq. (35), it simplifies to

Tm(x,t)=kmTm−1(x,t)+ℏL−1{R1,m[T→m−1,C→m−1,E→m−1]},

Cm(x,t)=kmCm−1(x,t)+ℏL−1{R2,m[T→m−1,C→m−1,E→m−1]},

Em(x,t)=kmEm−1(x,t)+ℏL−1{R3,m[T→m−1,C→m−1,E→m−1]}. (37)

By using T0(x,t),C0(x,t) and E0(x,t) and then solving the forgoing equations, we can obtain the terms of

T(x,t)=T0(x,t)+∑m=1∞Tm(x,t)(1n)m,

C(x,t)=C0(x,t)+∑m=1∞Cm(x,t)(1n)m,

E(x,t)=E0(x,t)+∑m=1∞Em(x,t)(1n)m. (38)

6  Existence and Uniqueness of Solutions

In this section, the existence and uniqueness are illustrated for the considered system with the assist of fixed-point theory. We consider the Eq. (32) as follows:

{0CFDtα[T(x,t)]=G1(x,t,T),0CFDtα[C(x,t)]=G2(x,t,C),0CFDtα[E(x,t)]=G3(x,t,E). (39)

Now, using Eq. (32) and results derived in [53], we obtained

{T(x,t)−T(x,0)=0CFItα{dT∂2T∂x2−γ(∂T∂x∂C∂x+∂2C∂x2)},C(x,t)−C(x,0)=0CFItα{−ηEC},E(x,t)−E(x,0)=0CFItα{dE∂2E∂x2+αT−βE}. (40)

Then we have from [73] as follows:

{T(x,t)−T(x,0)=2(1−α)M(α)G1(x,t,T)+2α(2−α)M(α)∫0tG1(x,ζ,T)dζ,C(x,t)−C(x,0)=2(1−α)M(α)G2(x,t,C)+2α(2−α)M(α)∫0tG2(x,ζ,C)dζ,E(x,t)−E(x,0)=2(1−α)M(α)G3(x,t,E)+2α(2−α)M(α)∫0tG3(x,ζ,E)dζ. (41)

Theorem 1. The kernel G1 admits the Lipschitz condition and contraction if 0≤(dTδ2−γ(λ1δ+λ2))<1 satisfies.

Proof. Let us consider the two functions u and u1 to prove the theorem, then

‖G1(x,t,T)−G1(x,t,T1)‖=‖dT(∂2T∂x2−∂2T1∂x2)−γ(∂C∂x(∂T∂x−∂T1∂x)+∂2C∂x2)‖

=‖(dTδ2−γ(∂C∂xδ+∂2C∂x2))[T(x,t)−T(x,t1)]‖

≤‖dTδ2−γ(∂C∂xδ+∂2C∂x2)‖‖T(x,t)−T(x,t1)‖

≤(dTδ2−γ(λ1δ+λ2))‖T(x,t)−T(x,t1)‖ (42)

where ‖∂C∂x‖≤λ2 and ‖∂2C∂2x‖≤λ3 be the bounded function. Putting η1=dTδ2−γ(λ1δ+λ2) in the above inequality, then we have

‖G1(x,t,T)−G1(x,t,T1)‖≤η1‖T(x,t)−T(x,t1)‖. (43)

Eq. (43) provides the Lipschitz condition for G1. Similarly, we can see that if 0≤dTδ2−γ(λ1δ+λ2)<1, then it implies the contraction. Similarly, we can prove

{‖G2(x,t,C)−G2(x,t,C1)‖≤η2‖C(x,t)−C(x,t1)‖,‖G3(x,t,E)−G3(x,t,E1)‖≤η3‖E(x,t)−E(x,t1)‖. (44)

By the assist of the above equations, Eq. (41) simplifies to

{T(x,t)=T(x,0)+2(1−α)(2−α)M(α)G1(x,t,T)+2α(2−α)M(α)∫0tG1(x,ζ,T)dζ,C(x,t)=C(x,0)+2(1−α)(2−α)M(α)G2(x,t,C)+2α(2−α)M(α)∫0tG2(x,ζ,C)dζ,E(x,t)=E(x,0)+2(1−α)(2−α)M(α)G3(x,t,E)+2α(2−α)M(α)∫0tG3(x,ζ,E)dζ. (45)

Then we get the recursive form as follows:

{Tn(x,t)=2(1−α)(2−α)M(α)G1(x,t,Tn−1)+2α(2−α)M(α)∫0tG1(x,ζ,Tn−1)dζ,Cn(x,t)=2(1−α)(2−α)M(α)G2(x,t,Cn−1)+2α(2−α)M(α)∫0tG2(x,ζ,Cn−1)dζ,En(x,t)=2(1−α)(2−α)M(α)G3(x,t,En−1)+2α(2−α)M(α)∫0tG3(x,ζ,En−1)dζ. (46)

The associated initial conditions are

T(x,0)=T0(x,t),C(x,0)=C0(x,t)andE(x,0)=E0(x,t). (47)

Now, between the terms, the successive difference is defined as

ϕ1n(x,t)=Tn(x,t)−Tn−1(x,t)

=2(1−α)(2−α)M(α)(G1(x,t,Tn−1)−G1(x,t,Tn−2))

+2α(2−α)M(α)∫0t(G1(x,t,Tn−1)−G1(x,t,Tn−2))dζ,

ϕ2n(x,t)=Cn(x,t)−Cn−1(x,t)

=2(1−α)(2−α)M(α)(G2(x,t,Cn−1)−G2(x,t,Cn−2))

+2α(2−α)M(α)∫0t(G2(x,t,Cn−1)−G1(x,t,Cn−2))dζ, (48)

ϕ3n(x,t)=En(x,t)−En−1(x,t)

=2(1−α)(2−α)M(α)(G3(x,t,Tn−1)−G3(x,t,Tn−2))

+2α(2−α)M(α)∫0t(G3(x,t,Tn−1)−G3(x,t,Tn−2))dζ.

Notice that

{Tn(x,t)=∑i=1nϕ1i(x,t),Cn(x,t)=∑i=1nϕ2i(x,t),En(x,t)=∑i=1nϕ3i(x,t). (49)

Then we have

‖ϕ1n(x,t)‖=‖Tn(x,t)−Tn−1(x,t)‖

=‖2(1−α)(2−α)M(α)(G1(x,t,Tn−1)−G1(x,t,Tn−2))

+2α(2−α)M(α)∫0t(G1(x,t,Tn−1)−G1(x,t,Tn−2))dζ‖, (50)

Application of the triangular inequality, Eq. (50) reduces to

‖ϕ1n(x,t)‖=‖Tn(x,t)−Tn−1(x,t)‖=2(1−α)(2−α)M(α)‖(G1(x,t,Tn−1)−G1(x,t,Tn−2))‖

+2α(2−α)M(α)‖∫0t(G1(x,t,Tn−1)−G1(x,t,Tn−2))dζ‖ (51)

The Lipschitz condition satisfied by the kernel T1, so, we have

‖ϕ1n(x,t)‖=‖Tn(x,t)−Tn−1(x,t)‖≤2(1−α)(2−α)M(α)η1‖ϕ1(n−1)(x,t)‖