Research article

Double composed metric-like spaces via some fixed point theorems

  • The manuscript introduces the concept of a double-composed metric-like space, which is an extension of the notion of a double-composed metric space. In this new space, the self-distance is not necessarily zero, but if the distance metric equals zero, it must be for identical points of distance. Furthermore, this paper presents several results related to this novel concept in the literature, with a particular focus on Hardy–Rogers type contractions. It establishes fixed point theorems accompanied by some illustrative examples that elucidate the findings. Finally, this research provides an application to nonlinear integral equation to substantiate our theorems.

    Citation: Anas A. Hijab, Laith K. Shaakir, Sarah Aljohani, Nabil Mlaiki. Double composed metric-like spaces via some fixed point theorems[J]. AIMS Mathematics, 2024, 9(10): 27205-27219. doi: 10.3934/math.20241322

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  • The manuscript introduces the concept of a double-composed metric-like space, which is an extension of the notion of a double-composed metric space. In this new space, the self-distance is not necessarily zero, but if the distance metric equals zero, it must be for identical points of distance. Furthermore, this paper presents several results related to this novel concept in the literature, with a particular focus on Hardy–Rogers type contractions. It establishes fixed point theorems accompanied by some illustrative examples that elucidate the findings. Finally, this research provides an application to nonlinear integral equation to substantiate our theorems.



    In this paper, the following one-sided tempered fractional diffusion equations are considered:

    {u(x,t)t=K(Dα,λa,xu(x,t)λαu(x,t)αλα1u(x,t)x)+f(x,t),(x,t)(a,b)×(0,T],u(x,0)=φ(x),x[a,b],u(a,t)=0,u(b,t)=ψr(t),t[0,T], (1.1)

    and

    {u(x,t)t=K(Dα,λx,bu(x,t)λαu(x,t)+αλα1u(x,t)x)+f(x,t),(x,t)(a,b)×(0,T],u(x,0)=φ(x),x[a,b],u(a,t)=ψl(t),u(b,t)=0,t[0,T], (1.2)

    where 1<α<2, λ0, the diffusion coefficient K is positive, and f(x,t) is the source term. Dα,λa,xu(x,t) and Dα,λx,bu(x,t) represent the left and right Riemann-Liouville tempered fractional derivatives, respectively, and are defined as

    {Dα,λa,xu(x,t)=eλxΓ(2α)2x2(xaeλτu(τ,t)(xτ)α1dτ),Dα,λx,bu(x,t)=eλxΓ(2α)2x2(bxeλτu(τ,t)(τx)α1dτ). (1.3)

    The concept of fractional derivatives appeared almost at the same time as integer derivatives, but the lack of application background of fractional derivatives at the beginning of their appearance made fractional models not widely developed until recent decades. Fractional models are widely used in physics [1,2,3,4], finance [5], biology [6], and hydrology [7,8,9]. Recently, the study of fractional diffusion equations has attracted a lot of attention, in which the integral derivatives of diffusion equations are replaced by the fractional derivatives to obtain fractional diffusion equations. Usually the time fractional derivatives describe anomalous sub-diffusion, and the space fractional derivatives describe anomalous super-diffusion [10].

    The analytical solutions of fractional differential equations usually cannot be obtained because of the nonlocality of fractional derivatives. Therefore, a lot of attention has been paid to the development of high-precision numerical methods, and a lot of results have been obtained [11,12,13,14,15]. For the space fractional advection-dispersion equations, Meerschaert and Tadjeran [11] point out that the standard Grünwald difference operator to approximate the Riemann-Liouville fractional derivative is unconditionally unstable regardless of the implicit and explicit Euler methods, so a modified Grünwald difference operator, called the shifted Grünwald difference operator, is proposed to solve this problem. Based on the shift Grünwald difference operator and its idea, more research has been done [16,17,18,19,20]. While developing numerical methods for fractional diffusion equations, some researchers have found that if the power law (waiting time or jump length) is tempered by an exponential factor, it has practical advantages [21,22,23]. Therefore, the time and space tempered fractional derivatives are obtained, and the time tempered fractional derivative yields the time tempered fractional diffusion equation [22]. Luo et al. [24] proposed an effective Lagrange-quadratic spline optimal collocation method to solve it. The space tempered fractional derivative yields the space tempered fractional diffusion equation [23]. It is very important to develop numerical methods for tempered fractional diffusion equations.

    A common way to solve the space tempered fractional diffusion equation is to develop a modified Grünwald difference operator based on the idea of the shifted Grünwald difference operator, called the tempered shifted Grünwald difference operator by Baeumer and Meerschaert [25], which is used to solve the tempered fractional advection-dispersion equation, but the space direction has only first-order precision. Based on the idea of the tempered shifted Grünwald difference operator, Li and Deng [26] obtained a class of second-order approximations of left and right Riemann-Liouville tempered fractional derivatives by using the weighted idea, called the tempered weighted and shifted Grünwald difference operators, which combined with the Crank-Nicolson method is used to solve the two-sided tempered fractional diffusion equations. See more studies on space-time tempered fractional differential equations [27,28], time tempered fractional differential equations [29,30], and space tempered fractional differential equations [31,32,33,34,35]. Similar to the quasi-compact scheme of fractional diffusion equations [36,37,38], it is natural to think of applying the quasi-compact technique to the numerical solutions of two-sided tempered fractional diffusion equations. However, due to the incompatibility of the left and right Riemann-Liouville tempered fractional quasi-compact operators, Yu et al. [39] considered the quasi-compact technique on the one-sided tempered fractional diffusion equations and obtained that the numerical schemes are stable and convergent with order O(τ+h3) and the schemes don't depend on time and space steps. The high-order quasi-compact scheme of the two-sided tempered fractional diffusion equation needs to be studied.

    The novelty of this paper is that the fourth-order numerical schemes of the one-sided tempered fractional diffusion equations are given, and the convergence accuracy is one order higher than that of the existing literature. In the process of proposing the numerical schemes, the left and right third-order Riemann-Liouville tempered derivatives will be generated, so it is necessary to study their effective second-order numerical approximations, and we obtain them through the definition of Riemann-Liouville tempered integer derivatives. Finally, the effective fourth-order numerical schemes for solving these two classes of problems are obtained. In addition, there is a simpler numerical scheme that will be available to solve the two-sided fractional advection-diffusion equations [38] by using the idea of this paper.

    The remaining sections of this paper are arranged as follows: Section 2 gives the fourth-order quasi-compact approximations of tempered fractional derivatives. In Section 3, the numerical scheme for solving problems (1.1) and (1.2) are derived. The stability and convergence of the numerical schemes are proved in Section 4. Section 5 shows by examples that the numerical schemes are effective. Finally, a brief summary of this work is given in Section 6.

    Define a fractional Sobolev space Sn+αλ(R),

    Sn+αλ(R)={ν|νL1(R),andR(|λ|+|w|)n+α|ˆν(w)|dw<},

    where ˆν(w)=Rν(x)eiwxdx is the Fourier transform of ν(x).

    Lemma 2.1. [25,26] Let 1<α<2, λ0. The shift number p is an integer, h is the step size, ν(x) is defined on the bounded interval [a,b], and belongs to Sn+αλ(R) after zero extension on the interval x(,a)(b,+). The tempered and shifted Grünwald type difference operators are defined as

    {Aα,λh,pν(x)=1hα[xah]+pk=0g(α)ke(kp)λhν(x(kp)h),ˆAα,λh,pν(x)=1hα[bxh]+pk=0g(α)ke(kp)λhν(x+(kp)h), (2.1)

    then

    {Aα,λh,pν(x)=Dα,λa,xν(x)+n1k=1cα,pkDk+α,λa,xν(x)hk+O(hn),ˆAα,λh,pν(x)=Dα,λx,bν(x)+n1k=1cα,pkDk+α,λx,bν(x)hk+O(hn), (2.2)

    where g(α)k=(1)k(αk)(k0) denotes the normalized Grünwald weights, that is,

    (1s)α=+k=0g(α)ksk, (2.3)

    cα,pk are the power series expansion coefficients of the function Wp(s)=eps(1es)αs=+k=0cα,pksk, and the first four coefficients are given as

    {cα,p0=1,cα,p1=pα2,cα,p2=12p212pα+α+3α224,cα,p3=8p312p2α+2p(α+3α2)α2α348.

    Remark 2.1. Let the equations be

    {γ1+γ0+γ1=1,γ1cα,11+γ0cα,01+γ1cα,11=0,γ1cα,13+γ0cα,03+γ1cα,13=0,

    then

    {γ1=112(α2+3α+2),γ0=16(α2+4),γ1=112(α23α+2),

    and there are the following approximations:

    {Bα,λhν(x)=1p=1γpAα,λh,pν(x)=1hα[xah]+1k=0w(α)kν(x(k1)h)=Dα,λa,xν(x)+α2+α+424D2+α,λa,xν(x)h2+O(h4)=(I+α2+α+424h2D2,λa,x)Dα,λa,xν(x)+O(h4),ˆBα,λhν(x)=1p=1γpˆAα,λh,pν(x)=1hα[bxh]+1k=0w(α)kν(x+(k1)h)=Dα,λx,bν(x)+α2+α+424D2+α,λx,bν(x)h2+O(h4)=(I+α2+α+424h2D2,λx,b)Dα,λx,bν(x)+O(h4), (2.4)

    where

    {w(α)k=(γ(α)1g(α)k+γ(α)0g(α)k1+γ(α)1g(α)k2)e(k1)λh(k0,g(α)2=g(α)1=0),w(α)0+w(α)2>0,w(α)1<0,w(α)k>0(k3),+k=0w(α)k=(γ(α)1eλh+γ(α)0+γ(α)1eλh)(1eλh)α, (2.5)

    in particular, when α=1, the following approximations are given as

    {(I+16h2D2,λa,x)D1,λa,xν(x)=1p=1γpA1,λh,pν(x)+O(h4)=12h(eλhν(x+h)eλhν(xh))+O(h4)=B1,λhν(x)+O(h4),(I+16h2D2,λx,b)D1,λx,bν(x)=1p=1γpˆA1,λh,pν(x)+O(h4)=12h(eλhν(xh)eλhν(x+h))+O(h4)=ˆB1,λhν(x)+O(h4). (2.6)

    For the first left and right Riemann-Liouville tempered derivatives D1,λa,xν(x) and D1,λx,bν(x), we know

    {D1,λa,xν(x)=eλxddx[eλxν(x)]=dν(x)dx+λν(x),D1,λb,xν(x)=eλxddx[eλxν(x)]=dν(x)dx+λν(x), (2.7)

    so the normalized left and right Riemann-Liouville tempered fractional derivatives can be rewritten as

    Dα,λa,xν(x)λαν(x)αλα1dν(x)dx=Dα,λa,xν(x)+(α1)λαν(x)αλα1[dν(x)dx+λν(x)]=Dα,λa,xν(x)+(α1)λαν(x)αλα1D1,λa,xν(x), (2.8)

    and

    Dα,λx,bν(x)λαν(x)+αλα1dν(x)dx=Dα,λx,bν(x)+(α1)λαν(x)αλα1[dν(x)dx+λν(x)]=Dα,λx,bν(x)+(α1)λαν(x)αλα1D1,λx,bν(x). (2.9)

    Let Λαx:=(I+ch2D2,λa,x) and Δαx:=(I+ch2D2,λx,b), where c=α2+α+424(112,16]. Now, we give the following theorem as the contribution of this paper.

    Theorem 2.1. Let ν(x)S4+αλ(R),1<α<2, λ0, and the continuous operators Λαx and Δαx are given to operate on the normalized left and right Riemann-Liouville tempered fractional derivatives, respectively. Then,

    ΛαxDα,λa,xν(x)+(α1)λαΛαxν(x)αλα1ΛαxD1,λa,xν(x)=ΛαxDα,λa,xν(x)+(α1)λαΛαxν(x)αλα1Λ1xD1,λa,xν(x)+αλα1(16c)h2D2,λa,x(D1,λa,xν(x)), (2.10)

    and

    ΔαxDα,λx,bν(x)+(α1)λαΔαxν(x)αλα1ΔαxD1,λx,bν(x)=ΔαxDα,λx,bν(x)+(α1)λαΔαxν(x)αλα1Δ1xD1,λx,bν(x)+αλα1(16c)h2D2,λx,b(D1,λx,bν(x)) (2.11)

    have a fourth-order approximation, respectively. The details are as follows:

    {(i).ΛαxDα,λa,xν(x)=Bα,λhν(x)+O(h4),(ii).Λ1xD1,λa,xν(x)=B1,λhν(x)+O(h4),(iii).Λαxν(x)=Cα,λhν(x)+O(h4),(iv).D2,λa,x(D1,λa,xν(x))=λ3ν(x)+3λ2δxν(x)+3λδ2xν(x)+δ3x,1ν(x)+O(h2), (2.12)

    and

    {(i).ΔαxDα,λx,bν(x)=ˆBα,λhν(x)+O(h4),(ii).Δ1xD1,λx,bν(x)=ˆB1,λhν(x)+O(h4),(iii).Δαxν(x)=ˆCα,λhν(x)+O(h4),(iv).D2,λx,b(D1,λx,bν(x))=λ3ν(x)3λ2δxν(x)+3λδ2xν(x)δ3x,2ν(x)+O(h2), (2.13)

    where

    {Cα,λhν(x)=ν(x)+ch2eλxδ2x[eλxν(x)],ˆCα,λhν(x)=ν(x)+ch2eλxδ2x[eλxν(x)],δxν(x)=12h[ν(x+h)ν(xh)],δ2xν(x)=1h2[ν(x+h)2ν(x)+ν(xh)],δ3x,1ν(x)=12h3[ν(x3h)6ν(x2h)+12ν(xh)10ν(x)+3ν(x+h)],δ3x,2ν(x)=12h3[ν(x+3h)+6ν(x+2h)12ν(x+h)+10ν(x)3ν(xh)]. (2.14)

    Proof. From Remark 2.1, the conclusion of (i) and (ii) in Eqs (2.12) and (2.13) can be obtained.

    For the conclusion (iii) in Eqs (2.12) and (2.13), it is easy to obtain the following:

    {Λαxν(x)=ν(x)+ch2eλxd2dx2[eλxν(x)]=ceλhν(x+h)+(12c)ν(x)+ceλhν(xh)+O(h4)=ν(x)+ch2eλxδ2x[eλxν(x)]+O(h4)=Cα,λhν(x)+O(h4),Δαxν(x)=ν(x)+ch2eλxd2dx2[eλxν(x)]=ceλhν(x+h)+(12c)ν(x)+ceλhν(xh)+O(h4)=ν(x)+ch2eλxδ2x[eλxν(x)]+O(h4)=ˆCα,λhν(x)+O(h4). (2.15)

    Finally, D2,λa,x(D1,λa,xν(x)) and D2,λx,b(D1,λx,bν(x)) can be represented as

    {D2,λa,x(D1,λa,xν(x))=eλxd2dx2{eλx[eλxddx(eλxν(x))]}=λ3ν(x)+3λ2dν(x)dx+3λd2ν(x)dx2+d3ν(x)dx3,D2,λx,b(D1,λx,bν(x))=eλxd2dx2{eλx[eλxddx(eλxν(x))]}=λ3ν(x)3λ2dν(x)dx+3λd2ν(x)dx2d3ν(x)dx3. (2.16)

    For derivatives of order 1 to 3, we adopt the following discretizations, respectively,

    {dν(x)dx=12h[ν(x+h)ν(xh)]+O(h2)=δxν(x)+O(h2),d2ν(x)dx2=1h2[ν(x+h)2ν(x)+ν(xh)]+O(h2)=δ2xν(x)+O(h2), (2.17)
    {d3ν(x)dx3=12h3[ν(x3h)6ν(x2h)+12ν(xh)10ν(x)+3ν(x+h)]+O(h2)=δ3x,1ν(x)+O(h2),d3ν(x)dx3=12h3[ν(x+3h)+6ν(x+2h)12ν(x+h)+10ν(x)3ν(xh)]+O(h2)=δ3x,2ν(x)+O(h2). (2.18)

    Thus, the whole conclusion of the theorem is proved.

    In this section, we consider the numerical scheme for solving problems (1.1) and (1.2). Here, we always assume that the function u(x,) in problem (1.1) and (1.2) belongs to S4+αλ(R) after zero extension.

    We make space grid {xi=a+ih}M0 and time grid {tn=nτ}N0, where h=baM and τ=TN represent the space step size and the time step size, respectively. Let uni=u(xi,tn), fni=f(xi,tn), and Uni represent the numerical solution at the point (xi,tn).

    For problem (1.1), the operator Λαx is applied to both the left and right ends of equation. We obtain

    Λαxu(x,t)t=K[ΛαxDα,λa,xu(x,t)+(α1)λαΛαxu(x,t)αλα1Λ1xD1,λa,xu(x,t)+αλα1h2(16c)D2,λa,x(D1,λa,xu(x,t))]+Λαxf(x,t), (3.1)

    then the backward Euler method is carried out at the point (xi,tn) to discrete the time partial derivative, and by Theorem 2.1, we get

    Cα,λh(uniun1iτ)=K[Bα,λhuni+(α1)λαCα,λhuniαλα1B1,λhuni+αλα1h2(16c)(λ3uni+3λ2δxuni+3λδ2xuni+δ3x,1uni)]+Cα,λhfni+Rni,1nN,1iM1, (3.2)

    where Rni=O(τ+h4) is the local truncation error.

    Removing the local truncation error in Eq (3.2) yields the numerical scheme as

    Cα,λh(UniUn1i)=τK[Bα,λhUni+(α1)λαCα,λhUniαλα1B1,λhUni+αλα1h2(16c)(λ3Uni+3λ2δxUni+3λδ2xUni+δ3x,1Uni)]+τCα,λhfni, (3.3)

    the matrix form of the numerical scheme (3.3) can be written as

    (C(α)τKPτKQ)Un=C(α)Un1+τC(α)fn+Fn, (3.4)

    where P=B(α)+(α1)λαC(α)αλα1B(1)+(16c)αλα+2h2E, Q=(16c)αλα1h2(3λ2D(1)+3λD(2)+D(3)), Un=(Un1,Un2,...,UnM2,UnM1)T, fn=(fn1,fn2,...,fnM2,fnM1)T, E is the identity matrix of order M1,

    B(α)=1hα(w(α)1w(α)0w(α)2w(α)1w(α)0w(α)M2w(α)M3w(α)M4w(α)1w(α)0w(α)M1w(α)M2w(α)M3w(α)2w(α)1), (3.5)
    C(α)=(12cceλhceλh12cceλhceλh12cceλhceλh12c), (3.6)
    B(1)=12h(0eλheλh0eλheλh0eλheλh0), (3.7)
    D(1)=12h(0110110110), (3.8)
    D(2)=1h2(2112112112), (3.9)
    D(3)=12h3(103121036121031612103161210), (3.10)

    and

    Fn=(Un10+τfn0Un0)(ceλh00)+(Un1M+τfnMUnM)(00ceλh)
    +τKUn0(wα2hα+c(α1)λαeλh+αλα1eλh2h+(16c)αλα1h2(3λ22h+3λh2+6h3)wα3hα+(16c)αλα1h2(3h3)wα4hα+(16c)αλα1h2(12h3)wα5hαwαMhα)
    +τKUnM(00wα0hα+c(α1)λαeλhαλα1eλh2h+(16c)αλα1h2(3λ22h+3λh2+32h3)). (3.11)

    The same idea is used to solve problem (1.2). The operator Δαx is applied to both the left and right ends of the equation, and we obtain

    Δαxu(x,t)t=K[ΔαxDα,λx,bu(x,t)+(α1)λαΔαxu(x,t)αλα1Δ1xD1,λx,bu(x,t) (3.12)
    +αλα1h2(16c)D2,λx,b(D1,λx,bu(x,t))]+Δαxf(x,t), (3.13)

    then the backward Euler method is carried out at the point (xi,tn) to discrete the time partial derivative. By Theorem 2.1, we get

    ˆCα,λh(uniun1iτ)=K[ˆBα,λhuni+(α1)λαˆCα,λhuniαλα1ˆB1,λhuni+αλα1h2(16c)(λ3uni+3λ2δxuni+3λδ2xuni+δ3x,1uni)]+ˆCα,λhfni+ˆRni,1nN,1iM1, (3.14)

    where ˆRni=O(τ+h4) is the local truncation error.

    Removing the local truncation error in Eq (3.14) yields the numerical scheme as

    ˆCα,λh(UniUn1i)=τK[ˆBα,λhUni+(α1)λαˆCα,λhUniαλα1ˆB1,λhUni+αλα1h2(16c)(λ3Uni3λ2δxUni+3λδ2xUniδ3x,2Uni)]+τˆCα,λhfni, (3.15)

    the matrix form of the numerical scheme (3.15) can be written as

    (ˆC(α)τKˆPτKˆQ)Un=ˆCUn1+τˆC(α)fn+ˆFn, (3.16)

    where ˆP=ˆB(α)+(α1)λαˆC(α)αλα1ˆB(1)+(16c)αλα+2h2E, ˆQ=(16c)αλα1h2(3λ2D(1)+3λD(2)ˆD(3)), ˆB(α)=(B(α))T, ˆB(1)=(B(1))T, ˆC(α)=(C(α))T, ˆP=PT, ˆD(3)=(D(3))T, ˆQ=QT,

    ˆFn=(Un10+τfn0Un0)(ceλh00)+(Un1M+τfnMUnM)(00ceλh)
    +τKUnM(wαMhαwα5hαwα4hα+(16c)αλα1h2(12h3)wα3hα+(16c)αλα1h2(3h3)wα2hα+c(α1)λαeλh+αλα1eλh2h+(16c)αλα1h2(3λ22h+3λh2+6h3))
    +τKUn0(wα0hα+c(α1)λαeλhαλα1eλh2h+(16c)αλα1h2(3λ22h+3λh2+32h3)00). (3.17)

    In this section, we use the energy method to prove in detail that numerical schemes (3.3) and (3.15) are valid. Before doing so, we first introduce some lemmas that will be used.

    Lemma 4.1. [40] A real matrix A of order M is positive definite if and only if D=A+AT2 is positive definite.

    Lemma 4.2. [41] Let T be a Toeplitz matrix with the generating function f(x) being a 2π-periodic continuous real-valued function. Denote λmin(T) and λmax(T) as the smallest and largest eigenvalues of T, respectively. Then, we have

    fmin(x)λmin(T)λmax(T)fmax(x),

    where fmin(x) and fmax(x) denote the minimum and maximum values of f(x), respectively. In particular, if f(x) is a nonpositive function and is not always zero, and fmin(x)fmax(x),

    fmin(x)<λ(T)<fmax(x).

    Lemma 4.3. [41] (Weyl's theorem). Let A,ECn×n be a Hermitian matrix and the eigenvalues λi(A),λi(E),λi(A+E) be arranged in an increasing order. Then, for each k = 1, 2, ..., n, we have

    λk(A)+λ1(E)λk(A+E)λk(A)+λn(E).

    Lemma 4.4. For 1<α<2,0<λh1, then the matrices P, Q, and C(α) in Eq (3.4) have the following properties:

    (i).{pi,i=wα1hα+(12c)λα(α1)+(16c)αλα+2h2<0,i=1,2,...,M1,pi,i+1=wα0hα+cλα(α1)eλh12hαλα1eλh,i=1,2,...,M2,pi+1,i=wα2hα+cλα(α1)eλh+12hαλα1eλh,i=1,2,...,M2,pi,i+1+pi+1,i>0,i=1,2,...,M2,pi+k,i=1hαwαk+1>0,i=1,2,...,Mk1,k=2,3,...,M2,pi,i+k=0,i=1,2,...,Mk1,k=2,3,...,M2,+k=0w(α)k+(λh)α(α1)[12c+c(eλh+eλh)]+(16c)α(λh)α+2+α(λh)α12(eλheλh)<0,

    and the matrix P is negative definite.

    (ii). The matrix Q is negative definite.

    (iii). For all given nonzero real column vectors ϵ, C(α) satisfies that

    112ϵTϵ<ϵTC(α)ϵ<43ϵTϵ.

    Proof. (i). By simple calculation, it is easy to obtain the properties of the elements of the matrix P; From the Gerschgorin disk theorem [42], we get that the matrix P is negative definite.

    (ii). Note that Q=(16c)αλα1h2(3λ2D(1)+3λD(2)+D(3)), so we just consider 3λ2D(1)+3λD(2)+D(3). First, it is easy to show that the matrix 3λ2D(1)+3λD(2) is negative definite; and then for the matrix D(3), let D=D(3)+(D(3))T2, which is a Toeplitz matrix, and the generating function [41] is

    f(x)=14h3(8cos3x24cos2x+24cosx8)=2h3(cosx1)30,x[π,π].

    From Lemma 4.2, we can see that the matrix D is negative definite. Further from Lemma 4.1, the matrix Q is negative definite.

    (iii). Let H=C(α)+(C(α))T2, and the generating function of the matrix H is

    f(x)=12c+c(eλh+eλh)cosx,x[π,π].

    From Lemma 4.2, we obtain 1c(eλh+eλh+2)λ(H)1+c(eλh+eλh2), and it is easy to check 112<λ(H)<43, which means 112ϵTϵ<ϵTHϵ<43ϵTϵ, that is, 112ϵTϵ<ϵTC(α)ϵ<43ϵTϵ.

    Thus, all the conclusions of the lemma are proved.

    Lemma 4.5. [43] Assume that {kn} and {pn} are nonnegative sequences, and the sequence {ϕn} satisfies

    ϕ0g0,ϕng0+n1l=0pl+n1l=0klϕl,n1,

    where g00. Then, the sequence {ϕn} satisfies

    ϕn(g0+n1l=0pl)exp(n1l=0kl),n1.

    To prove the stability and convergence of numerical schemes by the energy method, we define Uh={u|u={ui} as a grid function defined on {xi=a+ih}M1i=1}, for uUh, and the corresponding discrete L2-norm is defined as uL2=(hM1i=1u2i)1/2. Next, we present the theoretical analysis.

    Theorem 4.1. For α(1,2), let 0<λh1, then the numerical scheme (3.3) is stable for solving problem (1.1).

    Proof. Let Uni and Vni represent the solution obtained by solving problem (1.1) using scheme (3.3) from different initial values. Denoting εni=UniVni,εn=(εn1,εn2,...,εnM1)T, it can be seen from Eq (3.3) that

    (C(α)τKPτKQ)εn=C(α)εn1. (4.1)

    Multiplying the left side of both sides of Eq (4.1) by h(εn)T, we obtain

    h(εn)TC(α)εn=h(εn)T(τKP+τKQ)εn+h(εn)TC(α)εn1. (4.2)

    From Lemma 4.3, we know that matrix P+Q is negative definite, so

    h(εn)TC(α)εnh(εn)TC(α)εn112[h(εn)TC(α)εn+h(εn1)TC(α)εn1], (4.3)

    which implies that

    h(εn)TC(α)εnh(εn1)TC(α)εn1h(εn1)TC(α)εn1h(ε0)TC(α)ε0. (4.4)

    Finally, it follows from Lemma 4.4 that

    112εn2L2h(εn)TC(α)εnh(ε0)TC(α)ε043ε02L2, (4.5)

    that is,

    εn2L216ε02L2.

    At this point, we have proved that the numerical scheme (3.3) is stable.

    Theorem 4.2. For α(1,2), let 0<λh1, then the numerical scheme (3.3) is convergent for solving problem (1.1), that is, the following relationship is satisfied:

    en2L2C1(τ+h4),1nN,

    where en=(en1,en2,...,enM1)T, eni=uniUni, and C1 is a constant independent of n, τ, and h.

    Proof. Subtracting (3.3) from (3.2) and from scheme (3.4), we obtain

    (C(α)τKPτKQ)en=C(α)en1+τRn, (4.6)

    where Rn=(Rn1,Rn2,...,RnM1)T.

    Let's multiply both sides of (4.6) by h(en)T, which can be written as

    h(en)TC(α)en=τKh(en)T(P+Q)en+h(en)TC(α)en1+τh(en)TRn. (4.7)

    Denoting En=h(en)TC(α)en, similar to the proof of Theorem 4.1,

    112en2L2EnEn1+2τh(en)TRnE0+2τhnk=1(ek)TRk43e02L2+2τ(148τen2L2+12τRn2L2)+2τn1k=1(148ek2L2+12Rk2L2). (4.8)

    Noticing that e02L2=0, the Eq (4.8) can be rearranged as

    en2L2τn1k=1ek2L2+576τn1k=1Rk2L2+576τ2Rn2L2. (4.9)

    By the Lemma 4.5 (discrete Gronwall inequalities), we obtain

    en2L2eT(576τn1k=1Rk2L2+576τ2Rn2L2), (4.10)

    that is,

    enL2C1(τ+h4). (4.11)

    Thus, the theorem is proved.

    By the same idea, we can obtain the following theorem, whose proof we skip here.

    Theorem 4.3. For α(1,2), let 0<λh1, then the numerical scheme (3.15) is stable for solving problem (1.2).

    Theorem 4.4. For α(1,2), let 0<λh1, then the numerical scheme (3.15) is convergent for solving problem (1.2), that is, the following relationship is satisfied:

    en2L2C2(τ+h4),1nN,

    where en=(en1,en2,...,enM1)T, eni=uniUni, and C2 is a constant independent of n, τ, and h.

    In this section, we present some numerical experiments to verify the convergence accuracy of the numerical schemes. Let

    Order=logm(eL2,heL2,h/m),

    be the order of observation.

    Example 5.1. Consider the following left Riemann-Liouville tempered fractional diffusion equation with initial-boundary value problem

    {u(x,t)t=Dα,λ0,xu(x,t)λαu(x,t)αλα1u(x,t)x+f(x,t),(x,t)(0,1)×(0,1],u(0,t)=0,u(1,t)=etλ,t[0,1],u(x,0)=eλxx6,x(0,1), (5.1)

    where 1<α<2, and the linear source term is

    f(x,t)=etλx[(1+λααλα)x6+6αλα1x5Γ(7)Γ(7α)x6α],

    and the exact solution is u(x,t)=etλxx6.

    Set the relationship between time step size and space step size as τ=h4 for all numerical experiments. Choose different α, λ, and space step size h, and use numerical scheme (3.3) to solve Example 5.1. The error and observation order results obtained are shown in Table 1. From Table 1, it can be seen that the space convergence order reaches the fourth order, and the numerical results are in perfect agreement with the theoretical analysis.

    Table 1.  The numerical results obtained in Example 5.1 are computed by the numerical scheme (3.3) at t=1.
    λ=1/5 λ=1 λ=5
    α h eL2 Order eL2 Order eL2 Order Time(s)
    1/10 1.3130e-04 4.1820e-05 4.7268e-06 0.0975
    1.1 1/15 2.6460e-05 3.9506 8.3497e-06 3.9736 1.0110e-06 3.8038 0.7121
    1/20 8.4533e-06 3.9665 2.6544e-06 3.9836 3.3111e-07 3.8801 2.7296
    1/25 3.4822e-06 3.9745 1.0901e-06 3.9882 1.3823e-07 3.9147 8.2779
    1/10 2.8480e-04 1.3493e-04 2.7545e-05 0.1022
    1.5 1/15 5.7261e-05 3.9564 2.6583e-05 4.0065 5.8428e-06 3.8243 0.6929
    1/20 1.8275e-05 3.9700 8.3898e-06 4.0088 1.9067e-06 3.8926 2.9840
    1/25 7.5243e-06 3.9768 3.4300e-06 4.0084 7.9433e-07 3.9241 8.7314
    1/10 1.4830e-04 7.7791e-05 3.4075e-05 0.1108
    1.9 1/15 2.9868e-05 3.9521 1.4949e-05 4.0679 7.1197e-06 3.8615 0.7326
    1/20 9.5371e-06 3.9683 4.6496e-06 4.0596 2.3064e-06 3.9181 2.9485
    1/25 3.9271e-06 3.9763 1.8829e-06 4.0511 9.5663e-07 3.9438 8.6782

     | Show Table
    DownLoad: CSV

    Example 5.2. Consider the following right Riemann-Liouville tempered fractional diffusion equation with initial-boundary value problem

    {u(x,t)t=Dα,λx,1u(x,t)λαu(x,t)+αλα1u(x,t)x+f(x,t),(x,t)(0,1)×(0,1],u(0,t)=et,u(1,t)=0,t[0,1],u(x,0)=eλx(1x)8,x(0,1), (5.2)

    where 1<α<2, and the linear source term is

    f(x,t)=et+λx[(1+λααλα)(1x)8+8αλα1(1x)7Γ(9)Γ(9α)(1x)8α],

    and the exact solution is u(x,t)=et+λx(1x)8.

    The numerical scheme (3.15) is used to solve Example 5.2, and the obtained numerical results are presented in Table 2 for different α,λ, and space step size h. From Table 2, we find that the space convergence order is fourth order, and the numerical results are consistent with the theoretical results.

    Table 2.  The numerical results obtained in Example 5.2 are computed by the numerical scheme (3.15) at t=1.
    λ=1/5 λ=1 λ=5
    α h eL2 Order eL2 Order eL2 Order Time(s)
    1/10 9.1487e-04 7.0276e-04 1.1609e-03 0.1055
    1.1 1/15 1.8988e-04 3.8780 1.4515e-04 3.8900 2.5294e-04 3.7581 0.7646
    1/20 6.1547e-05 3.9161 4.6930e-05 3.9249 8.3946e-05 3.8340 2.9746
    1/25 2.5572e-05 3.9360 1.9468e-05 3.9431 3.5365e-05 3.8740 8.9580
    1/10 1.9184e-03 2.2558e-03 7.2422e-03 0.1153
    1.5 1/15 3.9926e-04 3.8712 4.7089e-04 3.8638 1.5730e-03 3.7659 0.7558
    1/20 1.2964e-04 3.9101 1.5301e-04 3.9075 5.2152e-04 3.8375 3.1384
    1/25 5.3930e-05 3.9305 6.3662e-05 3.9298 2.1958e-04 3.8766 9.1835
    1/10 1.1269e-03 1.5888e-03 6.4974e-03 0.1217
    1.9 1/15 2.3282e-04 3.8893 3.3031e-04 3.8738 1.3517e-03 3.8722 0.8132
    1/20 7.5358e-05 3.9210 1.0712e-04 3.9143 4.3885e-04 3.9104 3.1974
    1/25 3.1293e-05 3.9385 4.4519e-05 3.9348 1.8250e-04 3.9320 9.2827

     | Show Table
    DownLoad: CSV

    In this paper, we focus on fourth-order numerical algorithms for one-sided tempered fractional diffusion equations. By using the quasi-compact technique, the fourth-order quasi-compact approximations of the tempered fractional derivatives and the effective second-order approximations of the third-order tempered derivatives are given. It is proved that the numerical schemes are stable and convergent. Finally, some experiments are given to demonstrate the effectiveness of the numerical schemes.

    The author would like to express thanks to the editors and anonymous reviewers for their valuable comments and suggestions. This research is supported by the Xinjiang University of Political Science and Law.

    The author has no conflict of interest.



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