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Reports of the Academy of Sciences of the USSR
1958, Volume 119, No. 4
MATHEMATICS
E. A. GRIGOR’EVA
THE METHOD OF LINES IN MIXED PROBLEMS FOR PARABOLIC SYSTEMS
(Presented by Academician S. L. Sobolev on 1 VII 1957)
In the present note we consider the question of solving the system of equations
\[ \frac{\partial u_i}{\partial t}=a_i\frac{\partial^2 u_i}{\partial x^2} \quad (i=1,2,\ldots,m) \tag{1} \]
under homogeneous initial conditions
\[ u_i(x,0)=0 \tag{2} \]
and under nonhomogeneous boundary conditions of general form
\[ \sum_{j=1}^{m} a_{i,j}u_j(0,t)+\frac{\partial u_i(0,t)}{\partial x}=f_i(t), \]
\[ i=1,2,\ldots,m, \tag{3} \]
\[ \sum_{j=1}^{m} a_{m+i,j}u_j(l,t)+\frac{\partial u_i(l,t)}{\partial x}=f_{m+i}(t), \]
where \(a_i,a_{k,j}\) are constants \((a_i>0)\); \(f_k(t)\) are continuous functions.
To this problem one can apply the method of integral equations. We choose as unknowns the values of the functions \(u_i(0,t)\), \(u_i(l,t)\), \(i=1,\ldots,m\). To determine these unknowns, from equations (1) and conditions (3) a system of Volterra-type integral equations is derived. After, for each unknown function \(u_i(x,t)\), the values of \(u_i\) and \(\partial u_i/\partial x\) at \(x=0\) and at \(x=l\) are known, the original problem is solved by applying Green’s formula.
Another possible method for the practical solution of our problem—the method of lines—consists in replacing the derivatives with respect to \(t\) by difference quotients
\[ \frac{u(x,t)-u(x,t-h)}{h} \]
and then solving a system of ordinary differential equations in \(x\). In this case, at the boundary of the domain we obtain a system of algebraic equations with a matrix that decomposes into triangular ones.
Establishing the convergence of the method of lines reduces to establishing the possibility of passing to the limit in this system of linear algebraic equations in order to obtain Volterra-type integral equations satisfied by \(u_i(0,t)\), \(u_i(l,t)\), \(i=1,\ldots,m\). It is difficult to establish directly the uniform convergence of the difference operator to the integral operator here. However, we show a proper approximation of this operator to the operator that is integral in the sense of S. L. Sobolev \((^1)\). From this convergence there follows the validity of the limiting passage, and at the same time the justification of the method of lines.
We shall proceed from the assumption that there exists a solution of problem (1) under conditions (2), (3), and that for this solution Green’s formula can be written.
\[ \int_0^t a_i\left( v_i\frac{\partial u_i}{\partial \xi}-u_i\frac{\partial v_i}{\partial \xi}\right)_{\xi=l}d\tau - \int_0^t a_i\left( v_i\frac{\partial u_i}{\partial \xi}-u_i\frac{\partial v_i}{\partial \xi}\right)_{\xi=0}d\tau = u_i(x,t), \tag{4} \]
where
\[ v_i(x-\xi,t-\tau) = \frac{1}{2\sqrt{\pi}\sqrt{a_i(t-\tau)}}e^{-(x-\xi)^2/4a_i(t-\tau)} = \]
\[ = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ip(x-\xi)}e^{-a_i(t-\tau)p^2}\,dp. \tag{5} \]
Suppose that the solution of the system of equations
\[ u_{ih}(x,t+h)-u_{ih}(x,t)=a_i\frac{\partial^2 u_{ih}(x,t+h)}{\partial x^2}, \qquad i=1,\ldots,m, \]
under conditions (2) and (3) also exists for sufficiently small \(h>0\). Here \(t\) takes the values \(nh\), \(n=0,\ldots,[T/h]\). The functions \(u_{ih}(x,nh)\) are determined successively for all \(n\) as solutions of a system of ordinary differential equations with boundary conditions (3), written for \(u_{ih}(x,nh)\). Put \(u_{ih}(x,t)=u_{ih}(x,nh)\) for \(nh\leq t\leq (n+1)h\).
Theorem 1. The functions \(u_{ih}(x,t)\) converge uniformly as \(h\to 0\) to \(u_i(x,t)\) on the interval \(0\leq t\leq T\).
Denote \(u_{ih}(x,nh)=u_{ih,n}\). It can be shown that for \(u_{ih,n}\) the following formula, analogous to (4), is valid:
\[ a_i h\sum_{k=0}^{n-1} \left( v_{ih,n-k}\frac{\partial u_{ih,k+1}}{\partial \xi} - \frac{\partial v_{ih,n-k}}{\partial \xi}u_{ih,k+1} \right)_{\xi=l} \]
\[ {}- a_i h\sum_{k=0}^{n-1} \left( v_{ih,n-k}\frac{\partial u_{ih,k+1}}{\partial \xi} - \frac{\partial v_{ih,n-k}}{\partial \xi}u_{ih,k+1} \right)_{\xi=0} = u_{ih,n}, \tag{6} \]
where
\[ v_{ih,n-k}=\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{e^{ip(x-\xi)}}{(a_i h p^2+1)^{\,n-k}}\,dp. \]
Let us pass in formula (4) to the limit as \(x\to 0\), and then as \(x\to l\); we obtain:
\[ \int_0^t a_i\left[ v_i(-l,t-\tau)\left.\frac{\partial u_i}{\partial \xi}\right|_{\xi=l} - \left.\frac{\partial v_i}{\partial \xi}(-l,t-\tau)u_i\right|_{\xi=l} \right]d\tau - \]
\[ - \int_0^t \left. a_i v_i(0,t-\tau)\frac{\partial u_i}{\partial \xi} \right|_{\xi=0} d\tau = \frac12 u_i(0,t); \tag{7} \]
\[ \int_0^t \left. a_i v_i(0,t-\tau)\frac{\partial u_i}{\partial \xi} \right|_{\xi=l} d\tau - \int_0^t a_i\left[ \left. v_i(l,t-\tau)\frac{\partial u_i}{\partial \xi} \right|_{\xi=0} - \left.\frac{\partial v_i}{\partial \xi}(l,t-\tau)u_i\right|_{\xi=0} \right]d\tau = \frac12 u_i(l,t). \]
Similarly, one can pass to the limit as \(x\to 0\), and then as \(x\to l\), in formula (6). We obtain:
\[ a_i h\sum_{k=0}^{n-1} \left[ \left. v_{ih,n-k}(-l)\frac{\partial u_{ih,k+1}}{\partial \xi} \right|_{\xi=l} - \left. \frac{\partial v_{ih,n-k}(-l)}{\partial \xi}u_{ih,k+1} \right|_{\xi=l} \right] - \]
\[ - a_i h\sum_{k=0}^{n-1} \left. v_{ih,n-k}(0)\frac{\partial u_{ih,k+1}}{\partial \xi} \right|_{\xi=0} = \frac12 u_{ih,n}(0); \tag{8} \]
\[ a_i h \sum_{k=0}^{n-1} v_{ih,n-k}(0) \left.\frac{\partial u_{ih,k+1}}{\partial \xi}\right|_{\xi=l} - a_i h \sum_{k=0}^{n-1} \left[ v_{ih,n-k}(l) \left.\frac{\partial u_{ih,k+1}}{\partial \xi}\right|_{\xi=0} - \left.\frac{\partial v_{ih,n-k}(l)}{\partial \xi}\,u_{ih,k+1}\right|_{\xi=0} \right] = \frac12 u_{ih,n}(l). \tag{8} \]
We do not give the proofs of the validity of formulas (6) and (8). In equations (7) and (8) we eliminate the boundary values of the derivatives by means of conditions (3). Then systems (7) and (8) can be written in the form
\[ \psi=f+A(\psi); \tag{9} \]
\[ \psi_h=f_h+A_h(\psi_h), \tag{10} \]
where
\[ \psi=[u_1(0,t)\ldots u_m(0,t),\; u_1(l,t),\ldots,u_m(l,t)]; \]
\[ \psi_h=[u_{1h}(0,nh),\ldots,u_{mh}(0,nh),\; u_{1h}(l,nh),\ldots,u_{mh}(l,nh)]. \]
We shall show that for every \(\varepsilon\) there exists such an \(h_0\) that \(|\psi_h-\psi|<\varepsilon\) for \(h<h_0\) and \(t=nh,\ 0\le t\le T\). For this purpose we shall use a theorem of S. L. Sobolev (1).
Theorem. Suppose there is an equation
\[ \psi=f+A\psi, \]
where \(A\) is a completely continuous operator and the operator \((E-A)^{-1}=E+\Gamma\) exists. Suppose there is an equation
\[ \psi_h=f+A_h\psi_h, \]
where \(A_h\) approximates the operator \(A\) regularly, i.e.: 1) \(A_h\) converges strongly to \(A\); 2) \(A_h\) are uniformly completely continuous, i.e. the set \(\sum_N A_h\varphi\) is compact when \(\varphi\) ranges over some bounded set \(\Phi\). Then all operators \(E-A_h\), starting with some one, have an inverse \((E-A_h)^{-1}=E-\Gamma_h\), and the operators \(\Gamma_h\) regularly approximate the operator \(\Gamma\).
We shall show that the operators \(A\) and \(A_h\) in (9) and (10) satisfy the conditions of the theorem. We shall assume that the operator \(A_h\) defines a function not only at those points \(t\) where \(t/h\) is an integer, but also at the remaining points, at which we set the values of \(t/h\) equal to the ordinates of the polygonal line joining the points \((t,A_h(\psi(t)))\), where \(t/h\) is an integer. Then \(A_h\) will be an operator on the space of continuous functions, mapping this space into itself. Similarly we define the function \(f_h\) in (10) at points where \(t/h\) is not an integer. At integral points \(t/h\) the relations (10) are not thereby disturbed.
It is not difficult to verify that \(A\) is a completely continuous operator. The operator \((E-A)^{-1}\) exists, since (9) is a system of Volterra equations of the second kind, where the kernels are continuous functions with a singularity of the type of a root in the denominator. To prove that \(A_h\) regularly approximates the operator \(A\), we establish the following lemmas.
Lemma 1. For any \(\delta>0\) there exists such an \(h_0\) that for \(h<h_0\) the inequalities
\[ \left|v_{ih,n-k}(x-\xi)-v_i(x-\xi,t-\tau)\right|<\delta; \tag{11} \]
\[ \left| \frac{\partial v_{ih,n-k}(x-\xi)}{\partial \xi} - \frac{\partial v_i(x-\xi,t-\tau)}{\partial \xi} \right|<\delta \tag{12} \]
hold for \(\varepsilon\le t-\tau\le T\), where \(\varepsilon>0\) is an arbitrary number and \(n-k=(t-\tau)/h\).
For the proof it is necessary to write the functions under consideration in the form of integrals, after which one passes to the limit as \(h \to 0\).
Lemma 2.
\[ \lim_{h\to 0}\sum_{k=\left[\frac{t-\varepsilon}{h}\right]+1}^{n-1} a_i h\frac{\partial v_{ih,n-k}(-l)}{\partial \xi} = \int_{t-\varepsilon}^{t} a_i \frac{\partial v_i(-l,t-\tau)}{\partial \xi}\,d\tau . \tag{13} \]
The proof of this assertion can be obtained if one considers Green’s formulas for the functions \(u(x,t)=1\), which is a solution of the heat equation, and \(u_h(x)=1\). Performing the subtraction and then using Lemma 1, we obtain the proof of relation (13). Let us write \(\partial v_{ih,n-k}(-l)/\partial \xi\) in the form of an integral and integrate it by parts. We obtain:
\[ \frac{\partial v_{ih,n-k}(-l)}{\partial \xi} = -\frac{l}{2a_i h(n-k-1)}v_{ih,n-k-1}(-l). \]
It is not hard to see that
\[ v_{ih,n-k-1}(-l) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{\cos pl}{(a_i h p^2+1)^{\,n-k+1}}\,dp>0. \]
Hence, and from Lemma 2, it follows that for any \(\delta>0\) there exists such an \(\varepsilon\) that
\[ \sum_{k=\left[\frac{t-\varepsilon}{h}\right]+1}^{n-1} a_i h \left| \frac{\partial v_{ih,n-k}(\pm l)}{\partial \xi} \right| <\delta \tag{14} \]
for \(h\) smaller than some \(h_0\).
We also give without proof the estimates
\[ \sum_{k=\left[\frac{t-\varepsilon}{h}\right]+1}^{n-1} a_i h|v_{ih,n-k}(\pm l)|<\delta, \qquad \sum_{k=\left[\frac{t-\varepsilon}{h}\right]+1}^{n-1} a_i h|v_{ih,n-k}(0)|<\delta \tag{15} \]
for sufficiently small \(\varepsilon\) and \(h\). Using Lemma 1 and estimates (14), (15), it is not hard to prove the strong convergence of the operators \(A_h\) to \(A\). In doing so, one must consider the strong convergence of the corresponding summands of the operators \(A_h\) and \(A\), for example
\[ a_i h\sum_{k=0}^{n-1} \frac{\partial v_{ih,n-k}(-l)}{\partial \xi}\,\varphi_{k+1} \Rightarrow \int_{0}^{t} a_i\frac{\partial v_i}{\partial \xi}(-l,t-\tau)\varphi\,d\tau, \]
where \(\varphi_k\) are the values of the function \(\varphi\), taken at the points \(kh\).
The uniform complete continuity of the operators \(A_h\) is also proved with the help of Lemma 1 and estimates (14) and (15).
It can likewise be shown that the free terms in equations (10) converge uniformly in the space \(C\) to the free terms \(f\) in equations (9). Hence, by the theorem of S. L. Sobolev, the uniform convergence of the boundary values \(u_{ih}(x)\) to the boundary values \(u_i(x,t)\) will follow. After this, Theorem 1 is proved without difficulty by means of Green’s formulas, Lemma 1, and estimates (14), (15).
The author takes this opportunity to express his deep gratitude to Academician S. L. Sobolev for his guidance in carrying out this work.
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
16 VI 1957
REFERENCES
- S. L. Sobolev, Izv. Akad. Nauk SSSR, Ser. Mat., 20, 413 (1956).