Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 118, No. 5
MATHEMATICS
B. M. Budak and A. D. Gorbunov
THE METHOD OF LINES FOR SOLVING A NONLINEAR BOUNDARY-VALUE PROBLEM IN A DOMAIN WITH A CURVILINEAR BOUNDARY
(Presented by Academician A. A. Dorodnitsyn on July 17, 1957)
Let it be required to find a continuously differentiable solution of the equation
\[ u_{xy}=f(x,y,u,u_x,u_y), \tag{1} \]
the right-hand side of which is defined and continuous in the aggregate of all arguments and satisfies a Lipschitz condition with constant \(L_1\) in \(u_x\) and \(u_y\) in the closed bounded domain
\[ \overline G:\quad 0\le x\le l_x,\quad g(x)\le y\le l_y,\quad |u|\le l_u,\quad |u_x|\le l_{u_x},\quad |u_y|\le l_{u_y}, \]
where \(g(x)\ge 0\) for \(0\le x\le l_x\), and \(g'(x)\ge 0\) and is continuous; let this solution be required to satisfy the boundary conditions
\[ u(x,g(x))=\varphi(x),\quad 0\le x\le l_x;\qquad u(0,y)=\psi(y),\quad 0\le y\le l_y, \tag{2} \]
where \(\varphi'(x)\) and \(\psi'(y)\) are continuous, and
\(M_\varphi+2M_\psi<l_u,\quad M_{\varphi'}+M_{\psi'}M_{g'}<l_{u_x},\quad
M_\psi<l_u,\quad M_{\psi'}<l_{u_y}\)*.
The method of lines was previously applied to the solution of an analogous problem for the case in which \(f\) does not depend on \(u_x\) and \(u_y\).
In §1, under the stated conditions, the existence of at least one solution of problem (1), (2) is established. In §§2 and 3, under additional conditions, the uniqueness of the solution of this problem and its continuous dependence on the boundary conditions are asserted, and an estimate is given for the error incurred when the exact solution is replaced by an approximate one.
§1. Extend \(f\) by setting \(f(x,y,u,u_x,u_y)=f(x,l_y,u,u_x,u_y)\) for \(y\ge l_y\), and \(f(x,y,u,u_x,u_y)=f(x,g(x),u,u_x,u_y)\) for \(y\le g(x)\). On each line \(x=x_k=kh\) \((k=0,1,2,\ldots;\ h>0)\), replace the sought solution approximately by a function \(u_k(y)\), where \(u_k(y)\) is determined step by step from the system
\[ \frac{\Delta u'_k}{h} = f\left(x_k,y,u_k,\frac{\Delta u_k}{h},u'_k\right), \]
\[ \Delta u_k=u_{k+1}(y)-u_k(y),\qquad \Delta u'_k=u'_{k+1}(y)-u'_k(y),\qquad k=0,1,2,\ldots \tag{3} \]
with the boundary conditions
\[ u_0(y)=\psi(y),\qquad u'_0(y)=\psi'(y),\qquad u(g(x_k))=\varphi(x_k),\qquad k=0,1,\ldots \tag{4} \]
\[ \text{* By } M_\varphi,\ M_{\varphi'}\text{, etc., are denoted the upper bounds of the moduli of } \varphi,\ \varphi'\text{, etc., in the domain in which they are specified.} \]
Rewriting equation (3) in the form
\[ u_{k+1}'(y)=u_k'(y)+hf\left(x_k,y,u_k(y),\frac{u_{k+1}(y)-u_k(y)}{h},u_k'(y)\right), \tag{3′} \]
we note that the right-hand side of this differential equation with the unknown function \(u_{k+1}(y)\) satisfies the Lipschitz condition with constant \(L_1\) with respect to \(u_{k+1}\) and is continuous in the aggregate of the arguments \(y,u_{k+1}\) for all \(k\) and \(y\) for which the inequalities
\[ 0\leq x_k\leq l_x,\qquad |u_k(y)|\leq l_u,\qquad \left|\frac{u_{k+1}(y)-u_k(y)}{h}\right|\leq l_{u_x},\qquad |u_k'(y)|\leq l_{u_y} \]
are satisfied; therefore (3′) has a unique solution \(u_{k+1}'(y)\) with continuous derivative.
Define the domain
\[ \bar G_{xy}^{*}:\ 0\leq x\leq l_x^{*},\qquad g(x)-l_y^{*}\leq y\leq g(x)+l_y^{*}, \]
in which solutions of the system (3), (4) exist.
Replacing \(k\) by \(i\) in (3′) and summing over \(i\) from \(0\) to \(k-1\), we obtain
\[ u_k'(y)=\psi'(y)+h\sum_{i=0}^{k-1} f_i(y), \tag{5} \]
where
\[ f_i(y)=f\left(x_i,y,u_i(y),\frac{\Delta u_i(y)}{h},u_i'(y)\right). \]
Integrating (5) with respect to \(y\) from \(g(x_k)\) to \(y\), we shall have
\[ u_k(y)=\varphi(x_k)+\psi(y)+\psi(g(x_k)) +h\int_{g(x_k)}^{y}\sum_{i=0}^{k-1} f_i(\eta)\,d\eta . \tag{6} \]
Replacing \(k\) by \(k+1\) in (6), subtracting (6) from the equality obtained and dividing the result by \(h\), we find
\[ \frac{\Delta u_k(x)}{h} = \frac{\Delta\varphi(x_k)}{h} - \frac{\Delta\psi(g(x_k))}{h} + \int_{g(x_{k+1})}^{y} f_k(\eta)\,d\eta - \int_{g(x_k)}^{g(x_{k+1})}\sum_{i=0}^{k-1} f_i(\eta)\,d\eta . \tag{7} \]
From (6), (7), and (5) we obtain restrictions on \(l_x^{*}, l_y^{*}\):
\[ |u_k(\eta)|\leq M_\varphi+2M_\psi+M_f l_x^{*}l_y^{*}\leq l_u; \tag{8} \]
\[ \left|\frac{\Delta u_k(y)}{h}\right| \leq M_{\varphi'}+M_{\psi'}M_{g'}+M_f l_y^{*}+M_f M_g l_x^{*}\leq l_{u_x}; \tag{9} \]
\[ |u_k'(y)|\leq M_{\psi'}+M_f l_x^{*}\leq l_{u_y}. \tag{10} \]
If \(l_x^{*}\) and \(l_y^{*}\) satisfy conditions (8), (9), (10), and \(l_x^{*}\leq l_x\), \(l_y^{*}\leq l_y\), then the system (3), (4) will be solvable in the domain \(\bar G_{xy}^{*}\).
Next, form the domain
\[ \bar G_{xy}^{**}:\ 0\leq x\leq l_x^{**},\qquad g(x)\leq y\leq l_y^{*}, \]
choosing \(l_x^{**}\) so that for \(0\leq x\leq l_x^{**}\) the inequality \(g(x)\leq l_y^{*}\) is satisfied.
Put
\[ \tilde u(x,y)=u_k(y)+\frac{\Delta u_k(y)}{h}(x-x_k),\qquad x_k\leq x\leq x_{k+1}; \]
\[ g(x_{k+1})-l_y^{*}\leq y\leq g(x_k)+l_y^{*} \quad \text{if } g(x_{k+1})\geq g(x_k); \tag{11} \]
\[ g(x_k)-l_y^{*}\leq y\leq g(x_{k+1})+l_y^{*} \quad \text{if } g(x_{k+1})<g(x_k). \]
Then
\[ \tilde u_x(x,y)=\frac{\Delta u_k(y)}{h}; \tag{12} \]
\[ \tilde u_y(x,y)=u'_k(y)+\frac{\Delta u'_k(y)}{h}(x-x_k). \tag{13} \]
The functions \(\tilde u(x,y)\) and \(\tilde u_y(x,y)\) are continuous everywhere in their domain of definition, while \(\tilde u_x(x,y)\) is continuous with respect to \(y\) and undergoes a finite jump when passing through the lines \(x=x_k,\ k=1,2,\ldots\).
From (11) it follows that there exists a number \(h_0\) such that every function \(\tilde u(x,y)\), \(0<h\le h_0\), is defined at least in the domain \(\bar G_{xy}^{**}\). Differentiating (12) with respect to \(y\), we obtain
\[ \tilde u_{xy}(x,y)=\frac{\Delta u'_k(y)}{h} = f\left(x_k,y,u_k(y),\frac{\Delta u_k(y)}{h},u'_k(y)\right), \qquad x_k\le x<x_{k+1}. \tag{14} \]
Therefore one may write
\[ \tilde u_{xy}(x,y) = f\bigl(x,y,\tilde u(x,y),\tilde u_x(x,y),\tilde u_y(x,y)\bigr) +\tilde\Theta(x,y), \tag{15} \]
where
\[ \tilde\Theta = f\left(x_k,y,u_k,\frac{\Delta u_k}{h},u'_k\right) - f(x,y,\tilde u,\tilde u_x,\tilde u_y). \tag{16} \]
By virtue of the continuity of \(f\) in the aggregate of its arguments and by virtue of relations (11), (12), (13), we have \(\tilde\Theta\to 0\) as \(h\to 0\). Integrating (15) with respect to \(x\) from \(0\) to \(x\) and with respect to \(y\) from \(g(x)\) to \(y\), we obtain
\[ \tilde u(x,y)=\tilde u(x,g(x))+\psi(y)-\psi(g(x))+ \]
\[ + \int_{g(x)}^{y} d\eta \int_{0}^{x} \bigl[ f(\xi,\eta,\tilde u(\xi,\eta),\tilde u_x(\xi,\eta),\tilde u_y(\xi,\eta)) +\tilde\Theta \bigr]\,d\xi; \tag{17} \]
\[ \tilde u_x(x,y)=\tilde u_x(x,g(x))+ \]
\[ + \int_{g(x)}^{y} \bigl[ f(x,\eta,\tilde u(x,\eta),\tilde u_x(x,\eta),\tilde u_y(x,\eta)) +\tilde\Theta(x,\eta) \bigr]\,d\eta; \tag{18} \]
\[ \tilde u_y(x,y)=\psi'(y)+ \int_{0}^{x} \bigl[ f(\xi,y,\tilde u(\xi,y),\tilde u_x(\xi,y),\tilde u_y(\xi,y)) +\tilde\Theta(\xi,y) \bigr]\,d\xi. \tag{19} \]
Theorem 1. The families of functions \(\tilde u(x,y)\), \(\tilde u_x(x,y)\), \(\tilde u_y(x,y)\), depending on the parameter \(h\), \(0<h\le h_0\), are uniformly bounded and equicontinuous\({}^{(2)}\) in the domain \(\bar G_{xy}^{**}\) as \(h\to 0\).
Theorem 2. Under the conditions formulated earlier, there exists a sequence \(h_\nu\to 0\) as \(\nu\to 0\) such that \(\tilde u^{(\nu)}(x,y)\), \(\tilde u_x^{(\nu)}(x,y)\), \(\tilde u_y^{(\nu)}(x,y)\) converge uniformly, respectively, to the continuous functions \(u(x,y)\), \(v(x,y)\), \(w(x,y)\), with \(u_x(x,y)=v(x,y)\), \(u_y(x,y)=w(x,y)\); \(u(x,y)\) satisfies equation (1) and the boundary conditions (2) in the domain \(\bar G_{xy}^{**}\).
\(2^\circ\). Analogously to how this is done in \({}^{(2)}\) or \({}^{(3)}\), the following theorems can be proved:
Theorem 3. If the right-hand side of equation (1) is continuous in the aggregate of all its arguments and satisfies a Lipschitz condition in the last three arguments, then the boundary-value problem (1), (2) in the domain \(\bar G_{xy}^{**}\) has a unique continuously differentiable solution \(u(x,y)\), and \(\tilde u(x,y)\), \(\tilde u_x(x,y)\), \(\tilde u_y(x,y)\) converge uniformly, respectively, to \(u(x,y)\), \(u_x(x,y)\), \(u_y(x,y)\) for any manner in which \(h\) tends to zero.
Theorem 4. Under the conditions of Theorem 3, the solution of the boundary-value problem (1), (2) and its first derivatives depend continuously on the boundary conditions and on their first derivatives.
3°. Theorem 5. If \(f\) satisfies a Lipschitz condition with constant \(L_f\) in all its arguments, and \(\varphi'\), \(\psi'\), \(g'\) satisfy Lipschitz conditions with constants \(L_{\varphi'}\), \(L_{\psi'}\), \(L_{g'}\), respectively, then in \(\overline{G}^{**}\) the estimate
\[ |\tilde u-u|+|\tilde u_x-u_x|+|\tilde u_y-u_y| \leq \varepsilon(h)e^{L^*M(x+y)}, \tag{20} \]
where
\[ \begin{aligned} \varepsilon(h)=h\{&l_{u_x}(1+M_g)+l_{u_y}(1+L_{g'}) +M_fM_{g'}+{}\\ &+M_{\varphi'}+L_{\varphi'}+L_f(1+l_{u_x}+M_f)(2l_x^*+l_y^*+l_x^*l_y^*)+{}\\ &+2M_g^2e^{l_x^*L_1} +2M_{g'}^2[1+l_{u_y}+(1+h)M_f](e^{L_1l_x^*}-1)\frac{L_f}{L_1}\}^{*}. \end{aligned} \tag{21} \]
Moscow State University
named after M. V. Lomonosov
Received
8 VII 1957
CITED LITERATURE
\({}^{1}\) B. M. Budak, Vestn. MGU, No. 1 (1956). \({}^{2}\) A. D. Gorbunov, B. M. Budak, Vestn. MGU, No. 4 (1957). \({}^{3}\) B. M. Budak, A. D. Gorbunov, DAN, 117, No. 4 (1957).
* We recall that \(L_1\) denotes the Lipschitz constant of the function \(f\) with respect to the arguments \(u_x\) and \(u_y\); \(L_f\) is the Lipschitz constant of \(f\) with respect to all arguments jointly.