Full Text
UDC 517.948.32
ON A VARIANT OF THE AVERAGING METHOD OF FUNCTIONAL CORRECTIONS FOR SOLVING LINEAR INTEGRAL EQUATIONS OF MIXED TYPE
V. I. Tivonchuk
The method of averaging functional corrections was proposed in 1952 by Corresponding Member of the Academy of Sciences of the Ukrainian SSR Yu. D. Sokolov and was justified by him [1–4] for various classes of both linear and nonlinear differential and integral equations. Generalizations of this method to operator equations in various functional spaces belong to E. A. Chernysheko [5], A. Yu. Luchka [6], and N. S. Kurpel’ [7].
In [3] an exposition and justification are given of the basic variant of Yu. D. Sokolov’s method for solving integral equations of mixed type:
\[ u(P,t)=\varphi(P,t)+\int_{t_0}^{t}\int_G K(P,t;Q,\tau)u(Q,\tau)\,d\omega_Q\,d\tau, \tag{1} \]
where \(P,Q\) are points of some closed domain \(G\) of any number of dimensions, or of some surface or curve; \(d\omega_Q\) is an element of volume (area or arc length); the point \((t,\tau)\) belongs to the domain \(\Delta:\ t_0\leq \tau\leq t\leq t_0+T_0\).
This variant practically almost always leads to a more accurate result than the classical method of successive approximations. But along with this it has the drawback that the constructed algorithm does not converge, generally speaking, in the whole domain under consideration \(D_0=G\times(t_0,t_0+T_0)\).
In the present article a new variant of the method of averaging functional corrections is proposed, and an algorithm is constructed which, like the algorithm considered by us in [8–10], converges to the solution of equation (1) already in the whole domain \(D_0\).
§ 1. CONSTRUCTION OF THE ALGORITHM
In the first approximation we set
\[ u_1(P,t)=\varphi(P,t)+\int_{t_0}^{t}\int_G K(P,t;Q,\tau)a_1(\tau)\,d\omega_Q\,d\tau, \tag{2} \]
where
\[ a_1(t)=\frac{1}{B}\int_G u_1(P,t)\,d\omega_P, \tag{3} \]
\(B\) is the volume of the domain \(G\) (the area of the surface or the length of the arc).
Substituting (2) into (3), we obtain a linear Volterra integral equation for determining \(\alpha_1(t)\):
\[ \alpha_1(t)=\varphi_1(t)+\int_{t_0}^{t} S(t,\tau)\alpha_1(\tau)\,d\tau, \tag{4} \]
where
\[ \varphi_1(t)=\frac{1}{B}\int_G \varphi(P,t)\,d\omega_P, \]
\[ S(t,\tau)=\frac{1}{B}\int_G\int_G K(P,t;Q,\tau)\,d\omega_Q\,d\omega_P. \tag{5} \]
From equation (4) we find
\[ \alpha_1(t)=\varphi_1(t)+\int_{t_0}^{t} R(t,\tau)\varphi_1(\tau)\,d\tau, \tag{6} \]
where the function \(R(t,\tau)\) satisfies the equations
\[ R(t,\tau)-S(t,\tau)=\int_{\tau}^{t} S(t,\eta)R(\eta,\tau)\,d\eta =\int_{\tau}^{t} R(t,\eta)S(\eta,\tau)\,d\eta. \tag{7} \]
Substituting the value of \(\alpha_1(t)\), expressed by formula (6), into (2), we obtain the final expression for the first approximation.
In the \(n\)-th approximation we set
\[ u_n(P,t)=\varphi(P,t)+\int_{t_0}^{t}\int_G K(P,t;Q,\tau)\,[u_{n-1}(Q,\tau)+\alpha_n(\tau)]\,d\omega_Q\,d\tau, \tag{8} \]
where
\[ \alpha_n(t)=\frac{1}{B}\int_G \delta_n(P,t)\,d\omega_P \qquad (n=2,3,\ldots), \tag{9} \]
\[ \delta_n(P,t)=u_n(P,t)-u_{n-1}(P,t),\qquad \delta_1(P,t)=u_1(P,t). \tag{10} \]
On the basis of formulas (8)—(10) we have
\[ \delta_n(P,t)=\int_{t_0}^{t}\int_G H(P,t;Q,\tau)\delta_{n-1}(Q,\tau)\,d\omega_Q\,d\tau+ \]
\[ {}+B\int_{t_0}^{t} K_1(P,t;\tau)\alpha_n(\tau)\,d\tau \qquad (n=2,3,\ldots), \tag{11} \]
where
\[ K_1(P,t;\tau)=\frac{1}{B}\int_G K(P,t;Q,\tau)\,d\omega_Q, \]
\[ H(P,t;Q,\tau)=K(P,t;Q,\tau)-K_1(P,t;\tau). \tag{12} \]
Substituting (11) into (9), we obtain an equation for determining \(\alpha_n(t)\):
\[ \alpha_n(t)=\varphi_n(t)+\int_{t_0}^{t} S(t,\tau)\alpha_n(\tau)\,d\tau, \tag{13} \]
where
\[ \varphi_n(t)=\int_{t_0}^{t}\int_G H_1(t;Q,\tau)\delta_{n-1}(Q,\tau)\,d\omega_Q\,d\tau, \]
\[ H_1(t;Q,\tau)=\frac{1}{B}\int_G H(P,t;Q,\tau)\,d\omega_P. \tag{14} \]
If the solution of equation (13)
\[ \alpha_n(t)=\varphi_n(t)+\int_{t_0}^{t} R(t,\tau)\varphi_n(\tau)\,d\tau \tag{15} \]
is substituted into (8), we obtain the final expression for the \(n\)-th approximation.
Let us indicate two special cases.
1) If \(K(P,t;Q,\tau)=E(P,t;\tau)\), then from (2), (3), (8)—(10) we shall have
\[ \begin{aligned} u_n(P,t)-\varphi(P,t) &=\int_{t_0}^{t} E(P,t;\tau)\,d\tau\int_G u_{n-1}(Q,\tau)\,d\omega_Q \\ &\quad +B\int_{t_0}^{t}E(P,t;\tau)\alpha_n(\tau)\,d\tau =\int_{t_0}^{t}E(P,t;\tau)\,d\tau\int_G u_n(Q,\tau)\,d\omega_Q \\ &\equiv \int_{t_0}^{t}\int_G K(P,t;Q,\tau)u_n(Q,\tau)\,d\omega_Q\,d\tau \qquad (n=1,2,\ldots), \end{aligned} \]
i.e., in this case all successive approximations coincide with the exact solution.
2) If \(\int_G K(P,t;Q,\tau)\,d\omega_Q=0\), then by formula (2) \(u_1(P,t)=\varphi(P,t)\), and by formula (8)
\[ u_n(P,t)=\varphi(P,t)+\int_{t_0}^{t}\int_G K(P,t;Q,\tau)u_{n-1}(Q,\tau)\,d\omega_Q\,d\tau \qquad (n=2,3,\ldots), \]
i.e., in the case under consideration the algorithm described coincides with the usual iterative process.
§ 2. CONVERGENCE OF THE ALGORITHM AND ERROR ESTIMATE IN THE SPACE \(\widetilde C\)
- Passing to the proof of convergence of the algorithm described, we shall transform expression (11) for \(\delta_n(P,t)\) somewhat. To this end, instead of the function \(\varphi_n(t)\) in expression (15), we substitute its value from (14). We obtain
\[ \alpha_n(t)=\int_{t_0}^{t}\int_G R_1(t;Q,\tau)\delta_{n-1}(Q,\tau)\,d\omega_Q\,d\tau, \tag{16} \]
where
\[ R_1(t;Q,\tau)=H_1(t;Q,\tau)+\int_{\tau}^{t}R(t,\eta)H_1(\eta;Q,\tau)\,d\eta, \tag{17} \]
and the function \(R_1(t;Q,\tau)\) satisfies the equation
\[ R_1(t;Q,\tau)=H_1(t;Q,\tau)+\int_{\tau}^{t}S(t,\eta)R_1(\eta;Q,\tau)\,d\eta. \tag{18} \]
Substituting (16) into (11), we obtain the final expression for \(\delta_n(P,t)\):
\[ \delta_n(P,t)=\int_{t_0}^{t}\int_G M(P,t;Q,\tau)\delta_{n-1}(Q,\tau)\,d\omega_Q\,d\tau \qquad (n=2,3,\ldots), \tag{19} \]
where
\[ M(P,t;Q,\tau)=H(P,t;Q,\tau)+B\int_{\tau}^{t}K_1(P,t;\eta)R_1(\eta;Q,\tau)\,d\eta. \tag{20} \]
The proof of convergence of the sequence \(\{u_n(P,t)\}\), defined by formulas (2), (3), (8)—(10), is equivalent to the proof of convergence of the series
\[ \sum_{n=1}^{\infty}\delta_n(P,t), \]
whose \(n\)-th partial sum is equal to \(u_n(P,t)\).
- Suppose that the given function \(\varphi(P,t)\) belongs to the space \(\widetilde{C}_{D_0}\) of bounded measurable functions for \(T_0<\infty\), and that the kernel \(K(P,t;Q,\tau)\) satisfies one of the following conditions: \(A_1\). The function \(K(P,t;Q,\tau)\) is bounded and measurable with respect to \(P,t,\tau\) and summable with respect to \(Q\) in the domain \(D=G\times\Delta\). \(A_2\). The function \(K(P,t;Q,\tau)\) is a polar kernel of the form
\[ K(P,t;Q,\tau)=\frac{\overline{K}(P,t;Q,\tau)}{(t-\tau)^{1-\sigma}} \qquad (0<\sigma<1), \tag{21} \]
where the function \(\overline{K}(P,t;Q,\tau)\) satisfies condition \(A_1\).
Let, in the domain \(D\),
\[ \sup |u_1(P,t)|=\delta_1, \]
\[ \sup\left|\int_G K(P,t;Q,\tau)\,d\omega_Q\right|=K, \tag{22} \]
\[ \sup\int_G |H(P,t;Q,\tau)|\,d\omega_Q= \]
\[ =\sup\int_G |K(P,t;Q,\tau)-K_1(P,t;\tau)|\,d\omega_Q=H. \tag{23} \]
Then the following theorem is valid.
Theorem 1. If \(\varphi(P,t)\in\widetilde{C}_{D_0}\), and \(K(P,t;Q,\tau)\) satisfies condition \(A_1\) in the domain \(D\), then the sequence of functions \(\{u_n(P,t)\}\), defined by formulas (2), (3), (8)—(10), converges uniformly in the entire domain \(D_0\) under consideration to a solution \(u^*(P,t)\in\widetilde{C}_{D_0}\) of equation (1), and the error estimate holds
\[ |u^*(P,t)-u_n(P,t)| < \delta_1\frac{H}{K+H}e^{K(t-t_0)} \sum_{i=n}^{\infty} \frac{[H(t-t_0)]^{i-1}}{(i-1)!} \leq \]
\[ \leq \delta_1\frac{H}{K+H}e^{KT_0} \sum_{i=n}^{\infty} \frac{(HT_0)^{i-1}}{(i-1)!} \qquad (n=2,3,\ldots), \tag{24} \]
\[ |u^*(P,t)-u_1(P,t)| < \delta_1\frac{H}{K+H} \left[e^{(K+H)(t-t_0)}-1\right] \leq \]
\[ \leq \delta_1\frac{H}{K+H} \left[e^{(K+H)T_0}-1\right]. \tag{25} \]
Proof. On the basis of the notation (22) and (23), it follows from (5), (13), and (14) that in the domain \(D\)
\[ |S(t,\tau)|\leqslant K,\qquad \int_G |H_1(t;Q,\tau)|\,d\omega_Q\leqslant H,\qquad B|K_1(P,t;\tau)|\leqslant K, \tag{26} \]
whence, by virtue of relations (7), (17), and (20), we have
\[ \int_G |M(P,t;Q,\tau)|\,d\omega_Q\leqslant He^{K(t-\tau)}. \tag{27} \]
To simplify the subsequent calculations, introduce the notation
\[ \frac{[K(t-t_0)]^n}{n!}=T_n(t),\qquad \frac{H}{K}\bigl(e^{K(t-t_0)}-1\bigr)=N_0(t), \tag{28} \]
\[ \left(\frac{H}{K}\right)^{n+1} \left\{[T_n-T_{n-1}+\cdots+(-1)^{n-1}T_1]e^{K(t-t_0)} +(-1)^n\bigl[e^{K(t-t_0)}-1\bigr]\right\}=N_n(t) \quad (n=1,2,\ldots), \tag{29} \]
where the relations hold
\[ N_n(t)=\frac{H}{K}\frac{[H(t-t_0)]^n}{n!}e^{K(t-t_0)} -\frac{H}{K}N_{n-1}(t); \tag{30} \]
\[ H\int_{t_0}^{t}N_{n-1}(\tau)e^{K(t-\tau)}\,d\tau=N_n(t) \quad (n=1,2,\ldots). \tag{31} \]
Now from (19), on the basis of (22), (27), and (28), we find
\[ |\delta_2(P,t)|\leqslant \int_{t_0}^{t}\int_G |M(P,t;Q,\tau)|\,|u_1(Q,\tau)|\,d\omega_Q\,d\tau< \]
\[ <\delta_1H\int_{t_0}^{t}e^{K(t-\tau)}\,d\tau =\delta_1\frac{H}{K}\bigl(e^{K(t-t_0)}-1\bigr) =\delta_1N_0(t). \tag{32} \]
In general, if
\[ |\delta_n(P,t)|<\delta_1N_{n-2}(t), \tag{33} \]
then from (19), according to (27), (31), and (33), we obtain
\[ |\delta_{n+1}(P,t)|\leqslant \int_{t_0}^{t}\int_G |M(P,t;Q,\tau)|\,|\delta_n(Q,\tau)|\,d\omega_Q\,d\tau< \]
\[ <\delta_1\int_{t_0}^{t}N_{n-2}(\tau)\,d\tau \int_G |M(P,t;Q,\tau)|\,d\omega_Q\leqslant \]
\[ \leqslant \delta_1H\int_{t_0}^{t}N_{n-2}(\tau)e^{K(t-\tau)}\,d\tau =\delta_1N_{n-1}(t). \tag{34} \]
By virtue of relations (30) and (33), inequality (34) for \(n\geqslant 2\) may be represented in the form
\[ |\delta_{n+1}(P,t)|+\frac{H}{K}|\delta_n(P,t)|< \]
\[ <\delta_1\frac{H}{K}\frac{[H(t-t_0)]^{n-1}}{(n-1)!}e^{K(t-t_0)} \qquad (n=2,3,\ldots), \tag{35} \]
whence
\[ |\delta_n(P,t)|<\delta_1\frac{[H(t-t_0)]^{n-1}}{(n-1)!}e^{K(t-t_0)}\leq \]
\[ \leq \delta_1\frac{(HT_0)^{n-1}}{(n-1)!}e^{KT_0} \qquad (n=2,3,\ldots). \tag{36} \]
Since the numerical series \(\sum_{n=2}^{\infty}\frac{(HT_0)^{n-1}}{(n-1)!}\) converges, the series \(\sum_{n=1}^{\infty}\delta_n(P,t)\), by (22) and (36), converges absolutely and uniformly in the entire domain \(D_0\) under consideration, i.e., as \(n\to\infty\) the sequence of functions \(\{u_n(P,t)\}\), defined by formulas (2), (3), (8)—(10), converges uniformly in this domain to some function \(u^*(P,t)\in \widetilde C_{D_0}\), which, as is not difficult to verify, satisfies the original equation (1).
The required error estimates (24) and (25) follow from relations (28), (32), (35) and the inequality
\[ |u^*(P,t)-u_n(P,t)|\leq \sum_{i=n}^{\infty}|\delta_{i+1}(P,t)| \qquad (n=1,2,\ldots). \tag{37} \]
From the error estimate (24) it is clear that, in the case under consideration, the algorithm set forth in § 1 converges to the solution of equation (1) no more slowly than the classical method of simple iteration.
A cruder, but at the same time simpler than (24), error estimate has the form
\[ |u^*(P,t)-u_n(P,t)|<\delta_1\frac{H}{K+H}e^{K(t-t_0)} \frac{[H(t-t_0)]^{n-1}}{(n-1)!} \frac{1}{1-\dfrac{H(t-t_0)}{n}} \leq \]
\[ \leq \delta_1\frac{H}{K+H}e^{KT_0} \frac{(HT_0)^{n-1}}{(n-1)!} \frac{1}{1-\dfrac{HT_0}{n}}; \]
here it is assumed that \(n\) is so large that \(HT_0<n\).
- In the case when the kernel \(K(P,t;Q,\tau)\) has the form (21), it follows from relations (5), (12), (14), (18) and (20) that \(M(P,t;Q,\tau)\) has the form
\[ M(P,t;Q,\tau)=\frac{\overline M(P,t;Q,\tau)}{(t-\tau)^{1-\sigma}}, \tag{38} \]
where \(\overline M(P,t;Q,\tau)\) is a function satisfying condition \(A_1\) in the domain \(D\). Applying in this case to (19) arguments analogous to those given in [10], we arrive at the following theorem.
Theorem 2. If \(\varphi(P,t)\in \widetilde C_{D_0}\), and the kernel \(K(P,t;Q,\tau)\) satisfies condition \(A_2\) in the domain \(D\), then the sequence of functions defined by formulas (2), (3), (8)—(10) converges uniformly in the entire considered-
of the domain \(D_0\) to the solution \(u^*(P,t)\in \widetilde C_{D_0}\) of equation (1), and the error estimate holds
\[
\left|u^*(P,t)-u_n(P,t)\right|
<\delta_1\sum_{i=n}^{\infty}
\frac{\left[\Gamma(\sigma)\,\overline M\,(t-t_0)^\sigma\right]^i}{i\,\sigma\Gamma(i\sigma)}
\le
\]
\[
\le \delta_1\sum_{i=n}^{\infty}
\frac{\left[\Gamma(\sigma)\,\overline M T_0^\sigma\right]^i}{i\,\sigma\Gamma(i\sigma)}
\qquad (n=1,2,\ldots),
\tag{39}
\]
where the number \(\overline M\) in the domain \(D\) is subject to the condition
\[
\int_G |\overline M(P,t;Q,\tau)|\,d\omega_Q \le \overline M .
\]
Applying to (39) the well-known Stirling formula for gamma functions
\[
\Gamma(x)=x^{x-\frac12}e^{-x}\sqrt{2\pi}\,e^{\frac{\theta}{12x}}
\qquad (0<\theta<1),
\]
we obtain an error estimate more convenient for practical computations:
\[
\left|u^*(P,t)-u_n(P,t)\right|
<
\frac{\delta_1 L^n(t)}
{(\sigma^n n!)^\sigma\sqrt{\sigma(2\pi n)^{1-\sigma}}}
\,
\frac{1}
{1-\dfrac{L(t)}{[(n+1)\sigma]^\sigma}}
\le
\]
\[
\le
\frac{\delta_1 L^n}
{(\sigma^n n!)^\sigma\sqrt{\sigma(2\pi n)^{1-\sigma}}}
\,
\frac{1}
{1-\dfrac{L}{[(n+1)\sigma]^\sigma}},
\tag{40}
\]
where \(L(t)=\Gamma(\sigma)\overline M(t-t_0)^\sigma,\quad L=\Gamma(\sigma)\overline M T_0^\sigma\). Here \(n\) must satisfy the inequality \([(n+1)\sigma]^\sigma>L\).
§ 3. CONVERGENCE OF THE ALGORITHM AND ERROR ESTIMATE IN THE SPACE \(L^p\)
Suppose that 1) the given function \(\varphi(P,t)\) belongs to the space \(L^p_{D_0}\), \(p\ge 1\), i.e.
\[
\|\varphi\|_{L^p}
=
\left\{
\int_{t_0}^{t_0+T_0}\int_G |\varphi(P,t)|^p\,d\omega_P\,dt
\right\}^{\frac1p}
<\infty;
\tag{41}
\]
2) the given function \(K(P,t;Q,\tau)\) satisfies the condition
\[
\left\{
\int_{t_0}^{t_0+T_0}\int_G
\left[
\int_{t_0}^{t}\int_G |K(P,t;Q,\tau)|^q\,d\omega_Q\,d\tau
\right]^{\frac pq}
d\omega_P\,dt
\right\}^{\frac1p}
<\infty
\]
\[
\left(\frac1p+\frac1q=1\right),
\tag{42}
\]
which for \(p=1\) \((q=\infty)\) is replaced by the condition
\[
\int_{t_0}^{t_0+T_0}\int_G \operatorname*{vrai\,max}_{Q,\tau}|K(P,t;Q,\tau)|\,d\omega_P\,dt<\infty;
\]
3) the interval \((t_0,t_0+T_0)\) is finite or infinite.
Theorem 3. If
\[ \left\{ \int_{t_0}^{t_0+T_0}\int_G \left[ \int_{t_0}^{t}\int_G |M(P,t;Q,\tau)|^q\,d\omega_Q\,d\tau \right]^{\frac{p}{q}} d\omega_P\,dt \right\}^{\frac{1}{p}} = M < \infty, \tag{43} \]
then the algorithm set forth in § 1 converges in the mean with exponent \(p\) throughout the domain \(D_0\) under consideration to the solution \(u^*(P,t)\in L^p_{D_0}\) of equation (1), and the error estimate
\[ \|u^*-u_n\|_{L^p} < \|u_1\|_{L^p} \sum_{i=n}^{\infty}\frac{M^i}{\sqrt[p]{i!}} \qquad (n=1,2,\ldots), \tag{44} \]
holds, where
\[ \|u\|_{L^p} = \left\{ \int_{t_0}^{t_0+T_0}\int_G |u(P,t)|^p\,d\omega_P\,dt \right\}^{\frac{1}{p}}. \tag{45} \]
Proof. From assumptions (41) and (42) it follows first of all that all successive approximations \(u_n(P,t)\) belong to the space \(L^p_{D_0}\), as is easily verified by the method of complete mathematical induction. Further, from equation (19), on the basis of Hölder’s inequality, we obtain
\[ \int_{t_0}^{t}\int_G |\delta_n(P,\theta)|^p\,d\omega_P\,d\theta = \]
\[ = \int_{t_0}^{t}\int_G \left| \int_{t_0}^{\theta}\int_G M(P,\theta;Q,\tau)\delta_{n-1}(Q,\tau)\,d\omega_Q\,d\tau \right|^p d\omega_P\,d\theta \le \]
\[ \le \int_{t_0}^{t}\int_G \left[ \int_{t_0}^{\theta}\int_G |M(P,\theta;Q,\tau)|^q\,d\omega_Q\,d\tau \right]^{\frac{p}{q}} d\omega_P\,d\theta \int_{t_0}^{\theta}\int_G |\delta_{n-1}(Q,\tau)|^p\,d\omega_Q\,d\tau = \]
\[ = \int_{t_0}^{t}\int_G W(P,\theta)\,d\omega_P\,d\theta \int_{t_0}^{\theta}\int_G |\delta_{n-1}(Q,\tau)|^p\,d\omega_Q\,d\tau \qquad (n=2,3,\ldots), \tag{46} \]
where
\[ W(P,t)= \left[ \int_{t_0}^{t}\int_G |M(P,t;Q,\tau)|^q\,d\omega_Q\,d\tau \right]^{\frac{p}{q}}. \tag{47} \]
If we introduce the notation
\[ F_0(t)\equiv 1,\qquad F_n(t)= \int_{t_0}^{t}\int_G W(P,\theta)F_{n-1}(\theta)\,d\omega_P\,d\theta \qquad (n=1,2,\ldots), \tag{48} \]
then from (46), for \(n=2\), we obtain
\[ \int_{t_0}^{t}\int_G |\delta_2(P,\theta)|^p\,d\omega_P\,d\theta \le \]
\[ \le \int_{t_0}^{t}\int_G W(P,\theta)\,d\omega_P\,d\theta \int_{t_0}^{\theta}\int_G |u_1(Q,\tau)|^p\,d\omega_Q\,d\tau < \]
\[ < \|u_1\|_{L^p}^{p}\int_{t_0}^{t}\int_G W(P,\theta)\,d\omega_P\,d\theta = \|u_1\|_{L^p}^{p} F_1(t). \]
In general, if
\[ \int_{t_0}^{t}\int_G |\delta_n(P,\theta)|^p\,d\omega_P\,d\theta < \|u_1\|_{L^p}^{p} F_{n-1}(t), \]
then in an analogous way we establish that
\[ \begin{aligned} \int_{t_0}^{t}\int_G |\delta_{n+1}(P,\theta)|^p\,d\omega_P\,d\theta &\le \int_{t_0}^{t}\int_G W(P,\theta)\,d\omega_P\,d\theta \int_{t_0}^{\theta}\int_G |\delta_n(Q,\tau)|^p\,d\omega_Q\,d\tau \\ &< \|u_1\|_{L^p}^{p}\int_{t_0}^{t}\int_G W(P,\theta)F_{n-1}(\theta)\,d\omega_P\,d\theta = \|u_1\|_{L^p}^{p}F_n(t) \end{aligned} \tag{49} \]
\[ (n=1,2,\ldots). \]
By virtue of the easily verified identity
\[ F_n(t)=\frac{1}{n!}F_1^n(t) \qquad (n=1,2,\ldots) \]
inequality (49) can be rewritten in the form
\[ \int_{t_0}^{t}\int_G |\delta_{n+1}(P,\theta)|^p\,d\omega_P\,d\theta < \|u_1\|_{L^p}^{p}\frac{F_1^n(t)}{n!} \qquad (n=1,2,\ldots), \]
whence, according to (43), (47) and (48), we shall have
\[ \left\{ \int_{t_0}^{t_0+T_0}\int_G |\delta_{n+1}(P,t)|^p\,d\omega_P\,dt \right\}^{\frac1p} < \|u_1\|_{L^p}\, \frac{M^n}{\sqrt[p]{\,n!\,}} \qquad (n=1,2,\ldots). \tag{50} \]
Consequently, as \(n\to\infty\), \(\|u_{n+1}-u_n\|_{L^p}\to0\), i.e. the sequence of functions \(\{u_n(P,t)\}\) is Cauchy. And by virtue of the completeness of the space \(L^p\), this sequence converges in the mean with exponent \(p\) throughout the entire domain \(D_0\) under consideration to a function \(u^*(P,t)\in L^p_{D_0}\), satisfying, as is easy to verify, equation (1). Applying the generalized Minkowski inequality to (37) and taking into account inequality (50), we obtain the required error estimate (44).
A simpler error estimate has the form
\[ \|u^*-u_n\|_{L^p} < \|u_1\|_{L^p} \frac{M^n}{\sqrt[p]{\,n!\,}}\, \frac{1}{1-\dfrac{M}{\sqrt[p]{\,n+1\,}}}; \tag{51} \]
here the number \(n\) must satisfy the inequality \(\sqrt[p]{\,n+1\,}>M\).
Example. As an example, consider the equation
\[ u(x,t) = 1+\sqrt[3]{xt^2} +\frac{1}{10}(2+5x)t^{2/3} +\frac{1}{12}(1+2x)t^{4/3} - \frac{2}{9}\int_0^t\int_0^1 \frac{x+\xi}{\sqrt[3]{\xi\tau}}\, u(\xi,\tau)\,d\xi\,d\tau, \tag{52} \]
having as its solution the function \(u(x,t)=1+\sqrt[3]{xt^{2}}\). Applying to this equation the algorithm set forth in § 1, we obtain, in the first approximation,
\[ u_{1}(x,t)=1+\sqrt[3]{xt^{2}}+\frac{1}{10}(2+5x)t^{2/3} +\frac{1}{12}(1+2x)t^{4/3} - \]
\[ -\frac{1}{15}(2+5x)\int_{0}^{t}\frac{a_{1}(\tau)}{\sqrt[3]{\tau}}\,d\tau, \tag{53} \]
where \(a_{1}(t)\) is determined from the linear Volterra integral equation
\[ a_{1}(t)=1+\frac{6}{5}t^{2/3}+\frac{1}{6}t^{4/3} -\frac{3}{10}\int_{0}^{t}\frac{a_{1}(\tau)}{\sqrt[3]{\tau}}\,d\tau. \tag{54} \]
Substituting the solution of equation (54) into (53), we obtain the final expression for the first approximation
\[ u_{1}(x,t)=1+\sqrt[3]{xt^{2}} -\frac{1}{243}(1+2,5x)t^{2/3} +\frac{1}{27}(0,25-0,5x)t^{4/3} + \]
\[ +\frac{1}{2187}(20+50x)\left(1-e^{-0,45\sqrt[3]{t^{2}}}\right). \tag{55} \]
If in expression (53), instead of \(a_{1}(t)\), we restrict ourselves only to its first approximation, found by the method of simple iteration, then the corresponding formula for the first approximation will have the form
\[ u_{1}^{(1)}(x,t)=1+\sqrt[3]{xt^{2}} +\frac{1}{3}(0,025-0,0625x)t^{4/3} + \]
\[ +(0,02+0,05x)\left(\frac{31}{90}+0,0625t^{2/3}\right)t^{2}. \tag{56} \]
Applying to equation (52) the basic variant of Yu. D. Sokolov’s method [3] and the method of simple iteration, in the first approximation we obtain respectively
\[ \bar u_{1}(x,t)=1+\sqrt[3]{xt^{2}} -\frac{56+490x-(78x+69)t^{2/3}}{12(700+189t^{2/3})}\,t^{4/3}, \tag{57} \]
\[ U_{1}(x,t)=1+\sqrt[3]{xt^{2}}-(0,1x+0,05125)t^{4/3} -(0,025x+0,0125)t^{2}. \tag{58} \]
Carrying out computations by formulas (55)—(58), we obtain the results given in the table; here the sign “\(-\)” indicates an approximation from below, and the sign “\(+\)” an approximation from above.
| \(x\) | \(t\) | \(u_{1}\) | \(\bar u_{1}\) | \(U_{1}\) | \(u_{1}^{(1)}\) |
|---|---|---|---|---|---|
| \multicolumn{4}{c}{Errors (in %)} | |||||
| 0,1 | 0,2 | \(+0,063\) | \(-0,087\) | \(-0,670\) | \(+0,662\) |
| 0,2 | 0,4 | \(+0,095\) | \(-0,251\) | \(-1,81\) | \(+1,07\) |
| 0,3 | 0,6 | \(+0,077\) | \(-0,470\) | \(-3,27\) | \(+1,105\) |
| 0,4 | 0,8 | \(+0,010\) | \(-0,727\) | \(-5,02\) | \(+0,624\) |
| 0,5 | 1,0 | \(-0,100\) | \(-1,009\) | \(-7,04\) | \(-0,140\) |
| 0,6 | 1,2 | \(-0,249\) | \(-1,306\) | \(-9,29\) | \(-1,19\) |
| 0,7 | 1,4 | \(-0,433\) | \(-1,613\) | \(-11,78\) | \(-2,48\) |
| 0,8 | 1,6 | \(-0,646\) | \(-1,923\) | \(-14,49\) | \(-3,96\) |
| 0,9 | 1,8 | \(-0,888\) | \(-2,232\) | \(-17,40\) | \(-5,60\) |
| 1,0 | 2,0 | \(-1,15\) | \(-2,537\) | \(-20,53\) | \(-7,37\) |
It is clear from this table that \(u_1\) represents \(u\) better than \(\bar u_1\). Moreover, not only \(u_1\), but also \(u_1^{(1)}\) leads to a more accurate result than the function \(U_1\), which is the first approximation found by the classical method of successive approximations.
References
- Sokolov Yu. D. Ukrainian Mathematical Journal, 9, No. 1, 82–100, 1957.
- Sokolov Yu. D. Ukrainian Mathematical Journal, 10, No. 2, 193–208, 1958.
- Sokolov Yu. D. Ukrainian Mathematical Journal, 12, No. 2, 181–195, 1960.
- Sokolov Yu. D. Ukrainian Mathematical Journal, 15, No. 1, 58–70, 1963.
- Chernyshenko E. A. Ukrainian Mathematical Journal, 6, No. 3, 305–313, 1954.
- Luchka A. Yu. Theory and Application of the Method of Averaging Functional Corrections. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1963.
- Kurpel N. S. Ukrainian Mathematical Journal, 15, No. 3, 309–314, 1963.
- Tivonchuk V. I. Reports of the Academy of Sciences of the Ukrainian SSR, No. 8, 1014–1018, 1964.
- Tivonchuk V. I. Reports of the Academy of Sciences of the Ukrainian SSR, No. 10, 1281–1284, 1964.
- Tivonchuk V. I. Reports of the Academy of Sciences of the Ukrainian SSR, No. 12, 1559–1563, 1964.
Received by the editors
June 19, 1965
Institute of Mathematics, Academy of Sciences of the Ukrainian SSR