Mathematical Physics

KIM ZE PKHEN

AN APPROXIMATE METHOD FOR SOLVING ONE TYPE OF SYSTEM OF LINEAR SINGULAR INTEGRAL EQUATIONS

(Presented by Academician N. N. Bogolyubov on 4 February 1963)

The work is devoted to an approximate method for solving a system of linear singular integral equations arising in the theory of dispersion relations.

An equation of the Chew–Low type for the process \(\pi + N \to 2\pi + N\) has the form \((^{1,2})\)

\[ u(t)=-\frac{1}{\pi}\int_{-1}^{+1}\frac{v(\tau)}{\tau-t}\,d\tau, \]

\[ \begin{aligned} v(t)=&\,[S_1(t)-N_1(-t)]\lambda+S_1(t)u(t)+S_2(t)v(t)-\\ &-N_1(-t)u(-t)+N_2(-t)v(-t), \end{aligned} \tag{1} \]

where \(-1\leq t\leq +1\); \(u(v), v(t)\) are the unknown vector-functions; \(S_1(t), S_2(t), N_1(t), N_2(t)\) are matrices belonging to \(H(\alpha)\) and, for \(t=\pm1\), have zero of order \(3/2\); \(S_1(t)=S_2(t)=N_1(t)=N_2(t)=0\) for \(t\leq0\). We seek solutions of the system of equations (1) in the Hölder class \(H(\alpha)\).

§ 1. Consider the system of equations (1) in \(L_2\).

Theorem 1. If \(\sqrt{n}\,(S_1+S_2+N_1+N_2)\leq q<1\), where

\[ S_\beta=\max_{i,j,t}|S_{\beta ij}(t)|,\qquad N_\beta=\max_{i,j,t}|N_{\beta ij}(t)|, \]

\[ \beta=1,2;\qquad i,j=1,2,\ldots,s;\qquad -1\leq t\leq +1, \]

then the system of equations (1) has a unique solution in the class \(H(\alpha)\), and the successive approximations

\[ \begin{aligned} v_{m+1}(t)=&\,[S_1(t)-N_1(-t)]\pi -S_1(t)\frac{1}{\pi}\int_{-1}^{+1}\frac{v_m(\tau)}{\tau-t}\,d\tau +S_2(t)v_m(t)+\\ &+N_1(-t)\frac{1}{\pi}\int_{-1}^{+1}\frac{v_m(\tau)}{\tau+t}\,d\tau +N_2(-t)v_m(-t),\qquad v_0(t)\in H(\alpha) \end{aligned} \]

converge in norm in \(L_2\).

Using the formula

\[ \left\|\frac{1}{\pi}\int_{-1}^{+1}\frac{\varphi(\tau)}{\tau-t}\,d\tau\right\|\leq \|\varphi(t)\| \quad (^{3,4}), \]

it is easy to prove Theorem 1.

§ 2. Consider the following Riemann boundary-value problem.

Problem 1. Find a vector-function \(\Phi(z)=(\Phi_1(z),\Phi_2(z),\ldots,\Phi_s(z))\), analytic in the plane cut along \([-1,+1]\), continuous everywhere up to the boundary except perhaps for only a finite number of points on \([-1,+1]\), where integrable singularities are allowed, tending to zero at infinity and satisfying the boundary condition

\[ \begin{aligned} &(P(t)+i\sqrt{1-t^2}\,Q(t))\Phi^+(t) -(P(t)-i\sqrt{1-t^2}\,Q(t))\Phi^-(t)=\\ &\qquad\qquad =2if(t), \end{aligned} \tag{2} \]

where \(P(t), Q(t)\) are matrices whose elements are polynomials; \(f(t)\) is a given

a vector-function satisfying the Hölder condition. \(\Phi^{\pm}(t)\) are the limiting values of \(\Phi(z)\), respectively for \(z \to t+0i,\ z \to t-0i,\ t\in[-1,+1]\).

Problem 1 is easily solved by the method of Muskhelishvili \((^5)\).

Theorem 2. If boundary-value problem 1 is solvable, then the solution is represented in the form
\[ \Phi(z)=X^{-1}(z)\left[\frac{1}{\pi}\int_{-1}^{+1}\frac{f(\tau)}{\tau-z}\,d\tau+R(z)\right], \tag{3} \]
where
\[ X(z)=P(z)+\sqrt{z^2-1}\,Q(z); \]
\(R(z)\) is a vector-function whose components are polynomials. \(R(z)\) is determined from the condition of analyticity of \(\Phi(z)\) in the plane with the cut \([-1,+1]\) and from its behavior at infinity. \(R(z)\) may include arbitrary coefficients.

Sec. 3. Consider the system of singular integral equations
\[ \overline{K}\varphi(t)\equiv P(t)\varphi(t)+\sqrt{1-t^2}\,Q(t)\frac{1}{\pi}\int_{-1}^{+1}\frac{\varphi(\tau)}{\tau-t}\,d\tau=f(t), \tag{4} \]
where \(\varphi(t)\equiv(\varphi_1(t),\varphi_2(t),\ldots,\varphi_s(t))\) is the sought vector-function; \(P(t), Q(t), f(t)\) are the same as in problem 1.

Putting
\[ \Phi(z)=\frac{1}{\pi}\int_{-1}^{+1}\frac{\varphi(\tau)}{\tau-z}\,d\tau, \]
the system of equations (4) can be reduced to problem 1 \((^5)\).

Theorem 3. The system of equations (4), in the sense of solvability, is equivalent to problem 1. If problem 1 is solvable, then the solution of the system of equations (4) is given in the form
\[ \varphi(t)=\Gamma f(t)+B(t)R(t), \tag{5} \]
where
\[ \Gamma f(t)\equiv A(t)f(t)+B(t)\frac{1}{\pi}\int_{-1}^{1}\frac{f(\tau)}{\tau-t}\,d\tau, \]
\[ A(t)\equiv \frac{1}{2}\left([X^{+}(t)]^{-1}+[X^{-}(t)]^{-1}\right), \]
\[ B(t)\equiv \frac{1}{2i}\left([X^{+}(t)]^{-1}-[X^{-}(t)]^{-1}\right). \]

Sec. 4. The system of equations (1) can be reduced to the following form
\[ a(t)\varphi(t)+b(t)\frac{1}{\pi}\int_{-1}^{+1}\frac{\varphi(\tau)}{\tau-t}=f(t), \tag{6} \]
where
\[ \varphi(t)=\left(\frac{v(t)+v(-t)}{2},\ \frac{v(t)-v(-t)}{2}\right), \]
\[ a(t)= \begin{pmatrix} I-c_1(t) & -d_2(t)\\ -c_2(t) & I-d_1(t) \end{pmatrix}, \]
\[ b(t)= \begin{pmatrix} b_2(t) & a_1(t)\\ b_1(t) & a_2(t) \end{pmatrix}, \]
\[ f(t)=(a_1(t)\lambda,\ a_2(t)\lambda), \]
\(I\) is the identity matrix,
\[ a_1(t)=\frac{S_1(t)+S_1(-t)-N_1(t)-N_1(-t)}{2}, \]
\[ a_2(t)=\frac{S_1(t)-S_1(-t)+N_1(t)-N_1(-t)}{2}, \]
\[ b_1(t)=\frac{S_1(t)+S_1(-t)+N_1(t)+N_1(-t)}{2}. \]

\[ b_2(t)=\frac{S_1(t)-S_1(-t)-N_1(t)+N_1(-t)}{2}, \]

\[ c_1(t)=\frac{S_2(t)+S_2(-t)+N_2(t)+N_2(-t)}{2}, \]

\[ c_2(t)=\frac{S_2(t)-S_2(-t)-N_2(t)+N_2(-t)}{2}, \]

\[ d_1(t)=\frac{S_2(t)+S_2(-t)^2-N_2(t)-N_2(-t)}{2}, \]

\[ d_2(t)=\frac{S_2(t)-S_2(-t)+N_2(t)-N_2(-t)}{2}, \]

\[ b_{ij}(\pm 1)=0 \]
and the order of the zero is \(3/2\).

It is then clear that any solution of system (1) is a solution of system (6), and conversely, from any solution of system (6), for \(\varphi_1,\varphi_2,\ldots,\varphi_s\) even and \(\varphi_{s+1},\varphi_{s+2},\ldots,\varphi_{2s}\) odd functions, a solution of system (1) is constructed by the formulas

\[ v_1(t)=\varphi_1(t)+\varphi_{s+1}(t),\quad v_2(t)=\varphi_2(t)+\varphi_{s+2}(t),\ldots,\quad v_s(t)=\varphi_s(t)+\varphi_{2s}(t). \]

In the case \(\det\|a(t)+ib(t)\|\ne0\) on \([-1;+1]\), the system of equations (6) is solved effectively.

We approximate \(a(t), b(t)\) by polynomials (this is possible \({}^{6}\)). Suppose that

\[ a(t)=P(t)+\xi(t),\qquad b(t)=\sqrt{1-t^2}\,Q(t)+\sqrt{1-t^2}\,\eta(t), \]

\[ \max|\xi_{ij}(t)|<\varepsilon,\qquad \max|\eta_{ij}(t)|<\varepsilon, \]

where \(\varepsilon>0\) is a sufficiently small number. Equation (6) can be rewritten in the form

\[ \overline K\varphi(t)=f(t)-S\varphi(t), \tag{7} \]

where

\[ S\varphi(t)\equiv \xi(t)\varphi(t)+\sqrt{1-t^2}\eta(t)\frac{1}{\pi}\int_{-1}^{+1}\frac{\varphi(\tau)}{\tau-t}\,d\tau, \]

\(\overline K\) is the same operator as in Sec. 3.

By virtue of Theorem 3, if equation (7) has a solution, then it must be

\[ \varphi(t)=\Gamma f(t)-\Gamma S\varphi(t)+B(t)R(t). \tag{8} \]

We choose \(\varepsilon\) so that the condition \(\|\Gamma S\|\le q<1\) is satisfied. Then, by Banach’s theorem, the solution of system (8) is given in the form

\[ \varphi(t)=\sum_{l=0}^{\infty}(-\Gamma S)^l\Gamma f(t)+\sum_{l=0}^{\infty}(-\Gamma S)^l B(t)R(t). \tag{9} \]

The coefficients in \(R(t)\) are undetermined.

Substituting (9) into (6), we can determine the coefficients \(R(t)\).

Thus, by solving equation (6), we can find all solutions of the original equation.

Received
30 I 1963

REFERENCES

\({}^{1}\) W. Zellner, JETP, 36, no. 4, 1109 (1959).
\({}^{2}\) Kim Tse Peng, W. Zoellner, Nuclear Physics, 34, 491 (1962).
\({}^{3}\) S. G. Mikhlin, UMN, 3, no. 3, 25, 29 (1949).
\({}^{4}\) E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Moscow, 1948.
\({}^{5}\) N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
\({}^{6}\) I. P. Natanson, Constructive Theory of Functions, 1946.