# An Approximation of Ultra-Parabolic Equations

Authors: Allaberen Ashyralyev; Serhat Yılmaz
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The first and second order of accuracy difference schemes for the approximate solution of the initial boundary value problem for ultra-parabolic equations are presented. Stability of these difference schemes is established. Theoretical results are supported by the result of numerical examples.


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## Body

## 1. Introduction

Mathematical models that are formulated in terms of ultraparabolic equations are of great importance in various problems for instance in age-dependent population model, in the mathematical model of Brownian motion, in the theory of boundary layers, and so forth, see [1–5]. We refer also to [6–9] and the references therein for existence and uniqueness results and other properties of these models. On the other hand, Akrivis et al. [10] and Ashyralyev and Yılmaz [11, 12] developed numerical methods for ultraparabolic equations. In this paper, our interest is studying the stability of first- and second-order difference schemes for the approximate solution of the initial boundary value problem for ultraparabolic equations(1.1)∂u(t,s)∂t+∂u(t,s)∂s+Au(t,s)=f(t,s),0<t,s<T,u(0,s)=ψ(s),0≤s≤T,u(t,0)=φ(t),0≤t≤T,
in an arbitrary Banach space E with a strongly positive operator A. For approximately solving problem (1.1), the first-order of accuracy difference scheme(1.2)uk,m-uk-1,mτ+uk-1,m-uk-1,m-1τ+Auk,m=fk,m,fk,m=f(tk,sm),tk=kτ,sm=mτ,1≤k,m≤N,Nτ=1,u0,m=ψm,0≤m≤N,uk,0=φk,0≤k≤N
and second-order of accuracy difference scheme(1.3)uk,m-uk-1,mτ+uk-1,m-uk-1,m-1τ+12A(uk,m+uk-1,m-1)=fk,m,fk,m=f(tk-τ2,sm-τ2),tk=kτ,sm=mτ,1≤k,m≤N,Nτ=1,u0,m=ψm,0≤m≤N,uk,0=φk,0≤k≤N
are presented. The stability estimates for the solution of difference schemes (1.2) and (1.3) are established. In applications, the stability in maximum norm of difference shemes for multidimensional ultraparabolic equations with Dirichlet condition is established. Applying the difference schemes, the numerical methods are proposed for solving one-dimensional ultraparabolic equations.Theorem 1.1.
For the solution of (1.2), we have the following stability inequality:
(1.4)max⁡1≤k≤Nmax⁡1≤m≤N‖uk,m‖E≤C(max⁡0≤m≤N‖ψm‖E+max⁡0≤k≤N‖φk‖E+max⁡1≤k≤Nmax⁡1≤m≤N‖fk,m‖E),
where C is independent of τ, ψm, φk, and fk,m.Proof.
Using (1.2), we get
(1.5)uk,m-uk-1,m-1τ+Auk,m=fk,m.
From that it follows
(1.6)uk,m=Ruk-1,m-1+τRfk,m,
where R=(I+τA)-1. By the mathematical induction, we will prove that
(1.7)uk,m=Rnuk-n,m-n+∑j=1nτRn-j+1fk-n+j,m-n+j
is true for all positive integers n. It is obvious that for n=1,2 formula (1.7) is true. Assume that for n=r(1.8)uk,m=Rruk-r,m-r+∑j=1rτRr-j+1fk-r+j,m-r+j
is true. In formula (1.6), replacing k and m with k-r and m-r, respectively, we have
(1.9)uk-r,m-r=Ruk-r-1,m-r-1+τRfk-r,m-r.
Then, using (1.8) and (1.9), we get
(1.10)uk,m=Rr+1uk-r-1,m-r-1+τRr+1fk-r,m-r+∑j=1rτRr-j+1fk-r+j,m-r+j.
From that it follows
(1.11)uk,m=Rr+1uk-r-1,m-r-1+∑j=1r+1τRr-j+2fk-r-1+j,m-r-1+j
is true for n=r+1. So, formula (1.7) is proved. For m>k, replacing n with k in formula (1.7), we obtain that
(1.12)uk,m=Rkψm-k+∑j=1kτRk-j+1fj,m-k+j.
Using estimate (see [13])
(1.13)‖Rk‖E→E≤M
and triangle inequality, we get
(1.14)‖uk,m‖E≤‖Rk‖E→E‖ψm-k‖E+∑j=1kτ‖Rk-j+1‖E→E‖fj,m-k+j‖E≤M[max⁡0≤k,m≤N‖ψm-k‖+max⁡1≤j≤Nmax⁡1≤m≤N‖fj,m‖]
for any k and m. For k>m, replacing n with m in formula (1.7), we get
(1.15)uk,m=Rmφk-m+∑j=1mτRm-j+1fk-m+j,j.
From estimate (1.13) and triangle inequality, it follows that
(1.16)‖uk,m‖E≤‖Rm‖E→E‖φk-m‖E+∑j=1mτ‖Rm-j+1‖E→E‖fk-m+j,j‖E≤M[max⁡0≤k,m≤N‖φk-m‖+max⁡1≤j≤Nmax⁡1≤m≤N‖fj,m‖]
for any k and m. Thus, Theorem 1.1 is proved.Theorem 1.2.
For the solution of (1.3), we have the following stability inequality:
(1.17)max⁡1≤k≤Nmax⁡1≤m≤N‖uk,m+uk-1,m2‖E+max⁡1≤k≤Nmax⁡1≤m≤N‖uk,m+uk,m-12‖E≤C(max⁡0≤m≤N‖ψm‖E+max⁡0≤k≤N‖φk‖E+max⁡1≤k≤Nmax⁡1≤m≤N‖fk,m‖E),
where C is independent of τ, ψm, φk, and fk,m.The proof of Theorem1.2 is based on the following formulas:(1.18)uk,m=Buk-1,m-1+τBCfk,m,uk,m=Bkψm-k+∑j=1kτBk-jCfj,m-k+j,m>k,uk,m=Bmφk-m+∑j=1mτBm-jCfk-m+j,j,k>m
for the solution of difference scheme (1.3) and the following estimate [14]:(1.19)‖BkC‖E→E≤M,
where B=(I-τA/2)(I+τA/2)-1 and C=(I+τA/2)-1.


## 2. Application

LetΩ be the unit open cube in the n-dimensional Euclidean space ℝn(0<xk<1,  1≤k≤n) with boundary S,Ω¯=Ω∪S. In [0,1]×[0,1]×Ω¯, we consider the boundary-value problem for the multidimensional parabolic equation(2.1)∂u(t,s,x)∂t+∂u(t,s,x)∂s-∑r=1nαr(x)∂2u(t,s,x)∂xr2+δu(t,s,x)=f(t,s,x),x=(x1,…,xn)∈Ω,0<t,s<1,u(0,s,x)=ψ(s,x),s∈[0,1],u(t,0,x)=φ(t,x),t∈[0,1],x∈Ω,¯u(t,s,x)=0,t,s∈[0,1],x∈S,
where αr(x)>a>0(x∈Ω) and f(t,s,x)(t,s∈(0,1),x∈Ω) are given smooth functions and δ>0 is a sufficiently large number.We introduce the Banach spacesC01β(Ω¯)(β=(β1,…,βn), 0<xk<1, k=1,…,n) of all continuous functions satisfying a Hölder condition with the indicator β=(β1,…,βn), βk∈(0,1), 1≤k≤n, and with weight xkβk(1-xk-hk)βk, 0≤xk<xk+hk≤1, 1≤k≤n, which is equipped with the norm(2.2)‖f‖C01β(Ω¯)=‖f‖C(Ω¯)+sup⁡0≤xk<xk+hk≤11≤k≤n|f(x1,…,xn)-f(x1+h1,…,xn+hn)|×∏k=1nhk-βkxkβk(1-xk-hk)βk,
where C(Ω¯) stands for the Banach space of all continuous functions defined on Ω¯, equipped with the norm(2.3)‖f‖C(Ω¯)=max⁡x∈Ω¯|f(x)|.It is known that the differential expression(2.4)Av=-∑r=1nαr(x)∂2v(t,s,x)∂x2+δv(t,s,x)
defines a positive operator A acting on C01β(Ω¯) with domain D(Ax)⊂C012+β(Ω¯) and satisfying the condition v=0 on S.The discretization of problem (2.1) is carried out in two steps. In the first step, let us define the grid sets(2.5)Ω̃h={x=xm=(h1m1,…,hnmn),m=(m1,…,mn),0≤mr≤Nr,hrNr=L,r=1,…,n},Ωh=Ω̃h∩Ω,Sh=Ω̃h∩S.We introduce the Banach spacesCh=Ch(Ω̃h), Chβ=C01β(Ω̃h) of grid functions φh(x)={φ(h1m1,…,hnmn)} defined on Ω̃h, equipped with the norms(2.6)‖φh‖C(Ω¯h)=max⁡x∈Ω¯h|φh(x)|,‖φh‖C01β(Ω¯h)=‖φh‖C(Ωh¯)+sup⁡0≤xk<xk+hk≤11≤k≤n|φh(x1,…,xn)-φh(x1+h1,…,xn+hn)|×∏k=1nhk-βkxkβk(1-xk-hk)βk.To the differential operatorA generated by problem (2.1), we assign the difference operator Ahx by the formula(2.7)Ahxuxh=-∑r=1nar(x)(u-xr̅h)xr,jr
acting in the space of grid functions uh(x), satisfying the condition uh(x)=0 for all x∈Sh.With the help ofAhx, we arrive at the initial boundary-value problem(2.8)∂uh(t,s,x)∂t+∂uh(t,s,x)∂s+Ahxuh(t,s,x)=fh(t,s,x),0<t,s<1,x∈Ωh,uh(0,s,x)=ψh(s,x),0≤s≤1,uh(t,0,x)=φh(t,x),0≤t≤1,x∈Ω̃h
for an infinite system of ordinary differential equations.In the second step, we replace problem (2.8) by difference scheme(1.2)(2.9)uk,mh-uk-1,mhτ+uk-1,mh-uk-1,m-1hτ+Ahxuk,mh=fk,mh(x),x∈Ωh,fk,mh(x)=fh(tk,sm,x),tk=kτ,sm=mτ,1≤k,m≤N,x∈Ω̃h,u0,mh=ψmh,0≤m≤N,uk,0h=φkh,0≤k≤N
and by difference scheme(1.3)(2.10)uk,mh-uk-1,mhτ+uk-1,mh-uk-1,m-1hτ+12Ahx(uk,m+uk-1,m-1)=fk,mh(x),x∈Ωh,fk,mh(x)=fh(tk-τ2,sm-τ2,x),tk=kτ,sm=mτ,1≤k,m≤N,x∈Ω̃h,u0,mh=ψmh,0≤m≤N,uk,0h=φkh,0≤k≤N.It is known thatAhx is a positive operator in C(Ω̃h) and C01β(Ω¯h). Let us give a number of corollaries of Theorems 1.1 and 1.2.Theorem 2.1.
For the solution of difference scheme (2.9), we have the following stability inequality:
(2.11)max⁡1≤k≤Nmax⁡1≤m≤N‖uk,mh‖C(Ω̃h)≤C1(max⁡0≤m≤N‖ψmh‖C(Ω̃h)+max⁡0≤k≤N‖φkh‖C(Ω̃h)+max⁡1≤k≤Nmax⁡1≤m≤N‖fk,mh‖C(Ω̃h)),
where C1 is independent of τ, ψmh, φkh, and fk,mh.Theorem 2.2.
For the solution of difference scheme (2.10), we have the following stability inequality:
(2.12)max⁡1≤k≤Nmax⁡1≤m≤N‖uk,mh+uk-1,m-1h2‖C(Ω̃h)≤C1(max⁡0≤m≤N‖ψmh‖C(Ω̃h)+max⁡0≤k≤N‖φkh‖C(Ω̃h)+max⁡1≤k≤Nmax⁡1≤m≤N‖fk,mh‖C(Ω̃h)),
where C1 does not depend on τ, ψmh, φkh, and fk,mh.


## 3. Numerical Analysis

In this section, the initial boundary value problem(3.1)∂u(s,t,x)∂t+∂u(s,t,x)∂s-∂2u(s,t,x)∂x2+2u(t,s,x)=f(t,s,x),f(t,s,x)=e-(t+s)sin⁡⁡πx,0<s,t<1,0<x<1,u(0,t,x)=e-tsin⁡⁡πx,0<t<1,0<x<1,u(s,0,x)=e-ssin⁡⁡πx,0<s<1,0<x<1,u(s,t,0)=u(s,t,π)=0,0<s,t<1
for one-dimensional ultraparabolic equations is considered.The exact solution of problem (3.1) is(3.2)u(t,s,x)=e-(t+s)sin⁡⁡πx.Using the first order of accuracy int and s implicit difference scheme (2.9), we obtain the difference scheme first order of accuracy in t and s and second-order of accuracy in x(3.3)unk,m-unk-1,mτ-unk-1,m-unk-1,m-1τ-un+1k,m-2unk,m+un-1k,mh2+2unk,m=fk,mh,fk,mh=f(tk,sm,xn)=e-(tk+sm)sin⁡xn,1≤k,m≤N,1≤n≤M-1,un0,m=e-smsin⁡xn,0≤m≤N,0≤n≤M,unk,0=e-tksin⁡xn,0≤k≤N,0≤n≤M,u0k,m=uMk,m=0,0≤k,m≤N,tk=kτ,sm=mτ,1≤k,m≤N,Nτ=1,xn=nh,1≤n≤M,Mh=π
for approximate solutions of initial boundary value problem (3.3). It can be written in the matrix form(3.4)Aun+1+Bun+Cun-1=φn,1≤n≤M-1,u0=0⃗,uM=0⃗.Here(3.5)A=[0⋯⋯⋯⋯0000⋯⋯⋯⋯0⋮⋱⋱⋮⋮⋮⋮0⋯0a0⋯000⋯0a0⋯⋮⋱⋱⋱⋱⋱⋮0⋯⋯⋯⋯⋯a](N+1)2×(N+1)2,B=[10⋯⋯⋯⋯000010⋯⋯⋯⋯000010⋯⋯⋯⋯0⋮⋮⋱⋱⋱⋱⋱⋱⋮0b0⋯cd0⋯000b0⋯cd0⋯⋮⋱⋱⋱⋱⋱⋱⋱⋮00⋯⋯b0⋯cd](N+1)2×(N+1)2,C=[0⋯⋯⋯⋯0000⋯⋯⋯⋯0⋮⋱⋱⋮⋮⋮⋮0⋯0a0⋯000⋯0a0⋯⋮⋱⋱⋱⋱⋱⋮0⋯⋯⋯⋯⋯a](N+1)2×(N+1)2,
where
(3.6)a=-1h2,b=-1τ,c=-1τ,d=1τ+1τ+2h2+2,φn=[φn0,0φn1,0⋮φnN,0φn0,1φn1,1⋮φnN,1⋮φn0,Nφn1,N⋮φnN,N](N+1)2×1,φnk,m=f(tk,sm,xn)=e-(tk+sm)sin⁡⁡xn,un=[un0,0un1,0⋮unN,0un0,1un1,1⋮unN,1⋮un0,Nun1,N⋮unN,N](N+1)2×1.
This type system was used by Samarskii and Nikolaev [15] for difference equations. For the solution of matrix equation (3.4), we will use the modified Gauss elimination method. We seek a solution of the matrix equation by the following form:(3.7)un=αn+1un+1+βn+1,n=M-1,…,2,1,
where uM=0⃗, αj(j=1,…,M-1) are (N+1)2×(N+1)2 square matrices, βj(j=1,…,M-1) are (N+1)2×1coloumn matrices, α1, β1 are zero matrices, and(3.8)αn+1=-(B+Cαn)-1A,βn+1=(B+Cαn)-1(φn-Cβn),n=1,2,3,…,(M-1).Using the second-order of accuracy int and s implicit difference scheme (2.10), we obtain the difference scheme second-order of accuracy in t and s and second-order of accuracy in x(3.9)unk,m-unk-1,mτ-unk-1,m-unk-1,m-1τ-12[un+1k,m-2unk,m+un-1k,mh2+2unk,m+un+1k-1,m-1-2unk-1,m-1+un-1k-1,m-1h2+2unk-1,m-1]=fk,mh,fk,mh=f(tk,sm,xn)=e-(tk+sm-τ)sin⁡⁡xn,1≤k,m≤N,1≤n≤M-1,un0,m=e-smsin⁡xn,0≤m≤N,0≤n≤M,unk,0=e-tksin⁡xn,0≤k≤N,0≤n≤M,u0k,m=uMk,m=0,0≤k,m≤N,tk=kτ,sm=mτ,1≤k,m≤N,Nτ=1,xn=nh,1≤n≤M,Mh=π
for approximate solutions of initial boundary value problem (3.9). The matrix form (3.4) can be written. Here,
(3.10)A=[0⋯⋯⋯⋯0000⋯⋯⋯⋯0⋮⋱⋱⋮⋮⋮⋮a0⋯a0⋯00a0⋯a0⋯⋮⋱⋱⋱⋱⋱⋮0⋯⋯a0⋯a](N+1)2×(N+1)2(N+1)2×(N+1)2,B=[10⋯⋯⋯⋯000010⋯⋯⋯⋯000010⋯⋯⋯⋯0⋮⋮⋱⋱⋱⋱⋱⋱⋮b00⋯1c0⋯00b00⋯1c0⋯⋮⋱⋱⋱⋱⋱⋱⋱⋮00⋯b00⋯1c](N+1)2×(N+1)2,C=[0⋯⋯⋯⋯0000⋯⋯⋯⋯0⋮⋱⋱⋮⋮⋮⋮a0⋯a0⋯00a0⋯a0⋯⋮⋱⋱⋱⋱⋱⋮0⋯⋯a0⋯a](N+1)2×(N+1)2,
where
(3.11)a=-12h2,b=-1τ+1h2+1,c=1τ+1h2+1,φn=[φn0,0φn1,0⋮φnN,0φn0,1φn1,1⋮φnN,1⋮φn0,Nφn1,N⋮φnN,N](N+1)2×1,φnk,m=f(tk-τ2,sm-τ2,xn)=e-(tk+sm)sin⁡⁡xn,un=[un0,0un1,0⋮unN,0un0,1un1,1⋮unN,1⋮un0,Nun1,N⋮unN,N](N+1)2×1.We seek a solution of the matrix equation by the same algorithm (3.7) and (3.8).


## 4. Error Analysis

The errors are computed by(4.1)ENK,M=max⁡1≤k,m≤N,1≤n≤M-1|u(tk,sm,xn)-unk,m|
of the numerical solutions, where u(tk,sm,xn) represents the exact solution and unk,m represents the numerical solution at (tk,sm,xn), and the results are given in Table 1.Table 1Difference schemesN=M=10N=M=20N=M=40N=M=80Difference scheme (3.3)0.0281000.0144000.0068110.003225Difference scheme (3.9)0.0005110.0001210.0000280.000006It may be noted from Table1 that as N, M increase, the value of the errors associated with difference scheme (3.3) decreases by a factor of approximately 1/2 and the errors associated with difference scheme (3.9) decrease by a factor of approximately 1/4. This confirms that difference scheme (3.3) is first order and difference scheme (3.9) is second-order as stated in Section 1. Moreover, the results show that the second-order of accuracy difference scheme (3.9) are more accurate comparing with the first order of accuracy difference scheme (3.3).



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