MATHEMATICS

B. M. BUDAK and A. D. GORBUNOV

ON A DIFFERENCE METHOD FOR SOLVING A NONLINEAR GOURSAT PROBLEM

(Presented by Academician S. L. Sobolev on 31 V 1957)

Difference methods have been applied to the solution of the Goursat problem in the case of two independent variables in papers (¹–³). In (¹, ²) the right-hand side of equation (1) was assumed not to depend on (u_x, u_y), while in (³) it depended linearly on (u_x, u_y).

In the present note we consider the solution by the method of finite differences of the Goursat problem

[
u_{xy}=f(x,y,u,u_x,u_y),
\tag{1}
]

[
u(x,0)=\varphi(x),\quad 0\leq x\leq l_x;\qquad
u(0,y)=\psi(y),\quad 0\leq y\leq l_y;\qquad
\varphi(0)=\psi(0)
\tag{2}
]

for an arbitrary right-hand side (f(x,y,u,u_x,u_y)), continuous in the totality of all its arguments in the domain of definition

[
D:\qquad 0\leq x\leq l_x,\quad 0\leq y\leq l_y;
]

[
|u-u^0|\leq l_u,\qquad |u_x-u_x^0|\leq l_{u_x},\qquad |u_y-u_y^0|\leq l_{u_y}
\tag{3}
]

and sufficiently smooth with respect to (u, u_x, u_y), for arbitrary continuously differentiable boundary functions (\varphi(x), \psi(y)). In Sec. 1, on the basis of a certain general convergence criterion, the existence of a continuously differentiable solution is established in the case where (f) satisfies the Lipschitz condition with respect to (u_x) and (u_y). In Sec. 2 the existence and uniqueness of the solution are established in the case where the Lipschitz condition is also satisfied with respect to (u).* In Sec. 3 an a priori error estimate is given in terms of the steps in (x) and (y) and the maxima of the moduli of the given functions, their first derivatives, and the Lipschitz constants, under the assumption that (f, \varphi'(x), \psi'(y)) satisfy the Lipschitz condition with respect to all their arguments. In Sec. 4 the results are extended to a system of equations in (m)-dimensional space.

1°. Consider a function of two points (f(P,Q)), where (P) ranges over a closed bounded domain (G) of Euclidean space (R), and (Q) over an arbitrary domain (G^) of Euclidean space (R^). We shall regard the function (f(P,Q)) as a family (\mathfrak{M}) of functions defined in (G) and depending on the parameter (Q).

A sequence of functions (f(P,Q_n)), where (Q_n\to Q) as (n\to+\infty), will be called uniformly bounded if there exists a constant (C) such that (|f(P,Q_n)|<C) for all (n) and at once for all (P\in G).

A sequence of functions (f(P,Q_n)), where (Q_n\to Q) as (n\to+\infty), will be called equally smoothing if

* Under these conditions the existence is proved more simply.

for every $\varepsilon > 0$ there exist such $n_0 = n_0(\varepsilon)$ and $\delta = \delta(\varepsilon)$ that the inequality

[
|f(P', Q_n) - f(P'', Q_n)| < \varepsilon
]

will hold for all $n > n_0 = n_0(\varepsilon)$ and simultaneously for all $P'$ and $P''$ from $G$ for which the distance $\rho(P', P'') < \delta(\varepsilon)$.

Obviously, a uniformly bounded and equicontinuously smoothing sequence may consist of discontinuous functions.

Theorem 1. Let the point $Q_0$ belong to the domain $G^$ or to its boundary. We shall consider all possible sequences of points ${Q_n}$ from the domain $G^$ converging to the point $Q_0$, and the corresponding sequences of functions ${f(P, Q_n)}$ belonging to the family $\mathfrak M$. In order that from each sequence ${f(P, Q_n)}$ one can extract a subsequence converging uniformly in the domain $G$ to a continuous function, it is necessary and sufficient that from each such sequence one can extract a subsequence that is uniformly bounded and equicontinuously smoothing.

Remark. If $\mathfrak M \subset C$, where $C$ is the space of continuous functions defined in the usual way, then this theorem becomes a criterion of compactness of $\mathfrak M$ in $C$. However, in the present note we shall not use this.

Let us proceed to the construction of a solution of the Goursat boundary-value problem (1), (2), (3) by the finite-difference method. In order not to pass to the consideration of subdomains, we shall assume that

[
u^0 = u(0,0) = \varphi(0) = \psi(0), \qquad
u_x^0 = u_x(0,0) = \varphi'(0), \qquad
u_y^0 = u_y(0,0) = \psi'(0).
]

We prescribe along the $x$-axis a step $h$ and along the $y$-axis a step $k$, putting $x_i = ih$, $i = 0,1,2,\ldots$; $y_j = jk$, $j = 0,1,2,\ldots$; $0 \le x_i < l_x$, $0 \le y_j < l_y$. We replace equation (1) by the difference equation

[
\Delta_{hk}^{2} u_{ij} = hk f(x_i, y_j, u_{ij}, u_{xij}, u_{yij}), \tag{1$_\Delta$}
]

where the indices $i,j$ indicate the values of the quantities referred to the node with coordinates $x_i, y_j$, and the differences $\Delta_h u_{ij}$, $\Delta_k u_{ij}$, $\Delta_{hk}^{2} u_{ij}$ are defined in the usual manner:

[
\Delta_h u_{ij} = u_{i+1,j} - u_{ij}, \qquad
\Delta_k u_{ij} = u_{i,j+1} - u_{ij},
\tag{4}
]

[
\Delta_{hk}^{2} u_{ij} = \Delta_k(\Delta_h u_{ij})
= u_{i+1,j+1} - u_{i+1,j} - u_{i,j+1} + u_{ij}.
]

Here $u_{xij}$ and $u_{yij}$ denote the difference quotients

[
u_{xij} = \frac{\Delta_h u_{ij}}{h}, \qquad
u_{yij} = \frac{\Delta_k u_{ij}}{k}. \tag{5}
]

We replace the boundary conditions (2) by the boundary conditions

[
u_{0i} = \varphi(x_i), \qquad i = 0,1,2,\ldots; \qquad
u_{0j} = \psi(y_j), \qquad j = 0,1,2,\ldots. \tag{2$_\Delta$}
]

The solution of the difference equation (1$\Delta$) under the boundary conditions (2$\Delta$) is feasible in the domain defined by the inequalities

[
0 \le x_i \le l_x, \qquad 0 \le y_j \le l_y,
]

[
\max_{0 \le x \le l_x} |\varphi(x) - \varphi(0)|
+
\max_{0 \le y \le l_y} |\psi(y) - \psi(0)|
+
x_i y_j f_{\max} \le l_u,
]

[
|\varphi'(0)| + \varphi'{\max} + y_j f, \qquad} \le l_{u_x
|\psi'(0)| + \psi'{\max} + x_i f.} \le l_{u_y
\tag{6}
]

* Recall that the domain $G$ is assumed to be closed and bounded.

where by (f_{\max},\ \varphi'{\max},\ \psi') here and below is denoted the maximum of the modulus of the quantities (f(x,y,u,u_x,u_y)), (\varphi'(x)), and (\psi'(y)) in the domain of their definition. Let us construct an “approximate solution” (\tilde u(x,y;h,k)) of the problem (1), (2), (3), by setting

[
\tilde u(x,y;h,k)=u_{ij}+\frac{\Delta_h u_{ij}}{h}(x-x_i)+\frac{\Delta_k u_{ij}}{k}(y-y_j)+
]

[
+\frac{\Delta^2_{hk}u_{ij}}{hk}(x-x_i)(y-y_j)
\tag{7}
]

in each cell

[
\Pi_{ij}:\quad x_i\le x<x_{i+1},\ y_j\le y<y_{j+1}.
\tag{8}
]

It is continuous in the domain (6), and its partial derivatives

[
\tilde u_x(x,y;h,k),\qquad \tilde u_y(x,y;h,k)
\tag{9}
]

are continuous in each cell (\Pi_{ij}) and piecewise continuous in the domain (6).

Theorem 2. If (f(x,y,u,u_x,u_y)) is continuous jointly in all arguments in the domain (3) and satisfies the Lipschitz condition with respect to (u_x,u_y), then the families of functions (7) and (9) are uniformly bounded and equicontinuous in the domain (6) as (h) and (k) tend to zero, and, consequently, by Theorem 1, one can find a sequence of pairs ((h_n,k_n)), (h_n\to0,\ k_n\to0) as (n\to+\infty), such that

[
\tilde u(x,y;h_n,k_n)\rightrightarrows u(x,y),
]

[
\tilde u_x(x,y;h_n,k_n)\rightrightarrows v(x,y),\qquad
\tilde u_y(x,y;h_n,k_n)\rightrightarrows w(x,y)
]

uniformly in the domain (6), where (u,v,w) are continuous in this domain.

Theorem 3. In the domain (6), under the assumptions of Theorem 2, the function (u(x,y)) has continuous first partial derivatives (u_x,\ u_y), equal respectively to (v,w), has a continuous partial derivative (u_{xy}), and satisfies equation (1) and the boundary conditions (2).

(2^\circ.) Theorem 4. If (f(x,y,u,u_x,u_y)) is continuous jointly in all arguments and satisfies the Lipschitz condition with respect to the last three arguments

[
\left|f(x,y,\bar u,\bar u_x,\bar u_y)-f(x,y,\overline{\overline{u}},\overline{\overline{u}}_x,\overline{\overline{u}}_y)\right|\le
]

[
\le L{|\bar u-\overline{\overline{u}}|+|\bar u_x-\overline{\overline{u}}_x|+|\bar u_y-\overline{\overline{u}}_y|},
\tag{10}
]

then for the families (7) and (9), as ((h^2+k^2)\to0), the Cauchy criterion for uniform convergence is fulfilled. For their limits (u(x,y), v(x,y), w(x,y)), Theorem 3 holds. The boundary-value problem (1), (2), (3) cannot have two different continuously differentiable solutions in the domain (6).

(3^\circ.) Theorem 5. If the function (f(x,y,u,u_x,u_y)) satisfies the Lipschitz condition with constant (L) with respect to all arguments, and (\varphi'(x),\psi'(y)) satisfy the Lipschitz condition with constants (L_{\varphi'}, L_{\psi'}), then the following a priori error estimate holds:

[
|\tilde u-u|+|\tilde u_x-u_x|+|\tilde u_y-u_y|\le
]

[
\le \varepsilon+\varepsilon L\left{\frac{2}{LM}\left[e^{LM(x+y)}-1\right]
+\frac{1}{L^2M^2}\left[e^{LM(x+y)}-1-LM(x+y)\right]\right}
\tag{11}
]

in the domain (6), where

[
M = 2 + l_x + l_y,
\tag{12}
]

[
\varepsilon = \varepsilon(h,k) =
4(\varphi'{\max}h+\psi'k)+}k)+2(L_{\varphi}h+L_{\psi
]
[
+\,2L{l_x+l_y+l_xl_y}{[1+\varphi'{\max}+(1+l_y)f]h+
]
[
+[1+\psi'{\max}+(1+l_x)f};}]k+hkf_{\max
\tag{13}
]

and (u) is the exact solution of problem (1), (2), (3); (\tilde u) is the approximate solution of the problem, defined in § 1.

4°. Consider the boundary-value problem

[
u^{(\nu)}{x_1x_2\ldots x_m}
=
f,}(x_1,x_2,\ldots,x_m; u^{(1)},\ldots,u^{(N)
u^{(1)}{x_1},\ldots,u^{(N)}),
\qquad 1\leq \nu \leq N;
\tag{14}
]

[
u^{(\nu)}\big|{x_i=0}
=
\varphi^{(\nu)}_i(x_1,\ldots,x,\ldots,x_m),},x_{i+1
\qquad 1\leq \nu \leq N,\quad 1\leq i\leq m,
\tag{15}
]

where (\varphi^{(\nu)}i) and (f\nu) are defined and continuous for
(0\leq x_j\leq l_{x_j}), (j=1,\ldots,m),

[
|u^{(\nu)}-u^{(\nu)0}|\leq l_{u^{(\nu)}},\ldots,
\left|u^{(\nu)}{x_1\ldots x_m}-u^{(\nu)0}\right|
\leq l_{u^{(\nu)}_{x_1\ldots x_m}},
\qquad 1\leq \nu \leq N,
]

and in (f_\nu) there enter only the mixed derivatives of the functions (u^{(\nu)}) up to order (\leq m-1), while the boundary functions (\varphi^{(\nu)}_i) have all possible mixed continuous derivatives up to order (m-1) inclusive and satisfy the natural compatibility conditions.

By a solution of problem (14), (15) we shall mean a system of functions (u^{(\nu)}) satisfying (14), (15) and continuous together with the mixed derivatives up to order (m) inclusive.

Theorems 2, 3, 4, and 5 extend to the boundary-value problem (14), (15). In particular, if (f_\nu) satisfies a Lipschitz condition in all arguments with constant (L), and the ((m-1))-st derivatives of (\varphi^{(\nu)}_i) also satisfy a Lipschitz condition in all arguments with constants, respectively, (L^{(\nu)}_i), then for the error incurred in solving problem (14), (15) by the finite-difference method according to the scheme described above, we obtain the estimate

[
\sum_{\nu=1}^{N}
\left{
|\tilde u^{(\nu)}-u^{(\nu)}|
+
\sum_{i_1=1}^{m}
|\tilde u^{(\nu)}{x}}-u^{(\nu){x|}
+\ldots
\right.
]
[
\left.
\ldots+
\sum_{i_1,\ldots,i_{m-1}=1}^{m}
|\tilde u^{(\nu)}{x}\ldots x_{i_{m-1}}
-u^{(\nu)}{x|}\ldots x_{i_{m-1}}
\right}
\leq
]
[
\leq
\varepsilon+\varepsilon LM
\left{
\frac{C_m^{m-1}}{LNM}[e^{\omega}-1]
+
\frac{C_m^{m-2}}{L^2N^2M^2}[e^{\omega}-1-\omega]
+\ldots
\right.
]
[
\left.
\ldots+
\frac{C_m^{m-m}}{L^mN^mM^m}
\left[
e^{\omega}-1-\omega-\ldots-\frac{\omega^{m-1}}{(m-1)!}
\right]
+
[(m-1)!-1]\frac{\omega^m}{m!}
\right},
]

where (\varepsilon=\varepsilon(h_1,h_2,\ldots,h_m)) is expressed analogously to (13) in terms of the steps (h_1,h_2,\ldots,h_m) with respect to (x_1,x_2,\ldots,x_m); (C_m^k) are binomial coefficients; the constant (M) is expressed in terms of the dimensions of the domain; (\omega=LNM(x_1+x_2+\ldots+x_m)).

Moscow State University
named after M. V. Lomonosov

Received
31 V 1957

CITED LITERATURE

1. B. M. Budak, DAN, 109, No. 1 (1956).
2. B. M. Budak, DAN, 112, No. 2 (1957).
3. I. M. Yarysheva, Finite-difference methods for solving the Goursat problem, Abstract of Candidate’s dissertation, Leningrad State University, 1956.