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UDC 517.948.33
ON A GENERALIZATION OF CHANDRASEKHAR’S EQUATION
V. A. KAKICHEV, V. S. ROGOZHIN
In the theory of radiative equilibrium, the nonlinear integral equation of Chandrasekhar is well known [1, 2]
\[ H(t)^{-1}=1-t\int_0^1 \psi(\tau)H(\tau)\,\frac{d\tau}{\tau+t},\qquad 0\leq t\leq 1. \tag{1} \]
Works [3, 4] are devoted to the investigation of this equation. In [5] Fox reduced the nonlinear equation (1) to a special linear integral equation with Cauchy kernel, the theory of which is set forth in the monographs [6, 7]. Fox’s results were supplemented by Busbridge [8].
In the present note Fox’s idea is applied to the investigation of one new class of nonlinear integral equations of the form
\[ H^{-1}(t)=1-K^{-p}(t)\int_0^a \psi(\tau)H(\tau)K(\tau+t)\,d\tau,\qquad 0\leq t\leq a, \tag{2} \]
where \(\psi(t)\) and \(K(t)\) are prescribed functions satisfying conditions formulated below; \(p\) is an integer equal to zero or one, and \(H(t)\) is the unknown function. In particular, for \(K(t)=t^{-1}\) and \(p=1\), equation (1) is obtained.
§ 1. In this paragraph we study an equation of the form (2), assuming that the following conditions are satisfied:
a) the kernel \(K(t)\) has the form
\[ K(t)=\frac{1}{t}+M(t),\qquad -a\leq t\leq a, \]
where \(M(t)\) is a continuous function on the interval \([-a,a]\), satisfying the Hölder condition, and moreover \(M(-t)=-M(t)\);
b) there exists a number \(\alpha\) such that, for \(p=0\) or \(p=1\), the relation
\[ \frac{K(t-\tau)K(t-\xi)}{K^p(t)} + \frac{K(\tau-\xi)K(\tau-t)}{K^p(\tau)} + \frac{K(\xi-t)K(\xi-\tau)}{K^p(\xi)} = \frac{\alpha}{[K(t)K(\tau)K(\xi)]^p}. \]
c) \(K(t)\) and \(\psi(t)\) satisfy the Hölder condition and are nonzero for \(0 \leq t \leq a\).
We give examples of kernels satisfying conditions a) and b).
1) \(K(t)=t^{-1},\quad M(t)=0\). Condition b) is fulfilled both for \(p=0\) and for \(p=1\), and in each of these cases \(\alpha=0\).
2) \(K(t)=\operatorname{cosec} t,\quad M(t)=2\displaystyle\sum_{k=1}^{\infty}\frac{2^{2k}-1}{(2k)!}B_{2k}t^{2k-1},\quad a=\pi,\quad B_{2k}\) are the Bernoulli numbers. Condition b) is fulfilled for \(p=0\) and for \(p=1,\ \alpha=0\).
3) \(K(t)=\operatorname{csch} t,\quad M(t)=-2\displaystyle\sum_{k=1}^{\infty}\frac{2^{2k}-1}{(2k)!}B_{2k}t^{2k-1},\quad a=\pi,\quad p=0,1,\)
\(\alpha=0\).
4) \(K(t)=\operatorname{ctg} t,\quad M(t)=\displaystyle\sum_{k=1}^{\infty}\frac{2^{2k}}{(2k)!}|B_{2k}|t^{2k-1},\quad a<\pi/2,\) for \(p=0\ \alpha=\)
\(=-1,\) for \(p=1\ \alpha=1,\)
5) \(K(t)=\operatorname{cth} t,\quad M(t)=\displaystyle\sum_{k=1}^{\infty}\frac{2^{2k}}{(2k)!}B_{2k}t^{2k-1},\quad a=\pi,\quad p=0,1,\quad \alpha=1.\)
Proceeding to the solution of equation (2), we rewrite it in the form
\[ 1-H^{-1}(t)=K^{-p}(t)\int_{0}^{a}\psi(\tau)H(\tau)K(\tau+t)\,d\tau . \tag{3} \]
Let the function \(H(t)\) be a solution of equation (2). Substitute it into the right-hand side of relation (3) and replace \(t\) by \(-t\). In this way, we extend the definition of the solution \(H(t)\) to \(-a\leq t<0\) by the equality
\[ 1-H^{-1}(-t)=(-1)^pK^{-p}(t)\int_{0}^{a}\psi(\tau)H(\tau)K(\tau-t)\,d\tau . \tag{4} \]
From (3) and (4) there follows the relation
\[ [1-H^{-1}(-t)][1-H^{-1}(t)] =1-H^{-1}(t)-H^{-1}(-t)+[H(t)H(-t)]^{-1}= \]
\[ =\frac{(-1)^p}{K^p(t)}\int_{0}^{a}\int_{0}^{a}\psi(\tau)\psi(\xi)H(\tau)H(\xi) \frac{K(\tau+t)K(\xi-t)}{K^p(t)}\,d\tau\,d\xi . \tag{5} \]
From properties a) and b) of the kernel \(K(t)\) we derive
\[ \frac{K(t+\tau)K(\xi-t)}{K^p(t)} =-\frac{K(t+\tau)K(t-\xi)}{K^p(t)} =\frac{(-1)^{p+1}\alpha}{[K(t)K(\tau)K(\xi)]^p}+ \]
\[ +K(\tau+\xi)\left[(-1)^p\frac{K(\tau+t)}{K^p(\tau)}+\frac{K(\xi-t)}{K^p(\xi)}\right]. \tag{6} \]
Taking equality (6) into account and putting
\[ \beta=\int_{0}^{a}\frac{\psi(\tau)}{K^p(\tau)}H(\tau)\,d\tau, \tag{7} \]
rewrite the right-hand side of equality (5) in the form
\[ -\frac{\alpha\beta^2}{K^{2p}(t)} +\frac{1}{K^p(t)}\int_0^a \psi(\tau)H(\tau)\frac{K(\tau+t)}{K^p(\tau)}\,d\tau \int_0^a \psi(\xi)H(\xi)K(\tau+\xi)\,d\xi+ \]
\[ +\frac{(-1)^p}{K^p(t)}\int_0^a \psi(\xi)H(\xi)\frac{K(\xi-t)}{K^p(\xi)}\,d\xi \int_0^a \psi(\tau)H(\tau)K(\tau+\xi)\,d\xi . \tag{8} \]
Taking formula (3) into account, we transform this expression to the form
\[ -\frac{\alpha\beta^2}{K^{2p}(t)} +\frac{1}{K^p(t)}\int_0^a \psi(\tau)H(\tau)K(\tau+t)\left[1-\frac{1}{H(\tau)}\right]\,d\tau+ \]
\[ +\frac{(-1)^p}{K^p(t)}\int_0^a \psi(\xi)H(\xi)K(\xi-t)\left[1-\frac{1}{H(\xi)}\right]\,d\xi = -\frac{\alpha\beta^2}{K^{2p}(t)}+1- \]
\[ -\frac{1}{H(t)} -\frac{1}{K^p(t)}\int_0^a \psi(\tau)K(\tau+t)\,d\tau +1-\frac{1}{H(-t)} \]
\[ -\frac{(-1)^p}{K^p(t)}\int_0^a \psi(\xi)K(\xi-t)\,d\xi . \]
Hence, and from (5), we have
\[ H(t)H(-t)\left[T(t)-\frac{\alpha\beta^2}{K^{2p}(t)}\right]=1, \tag{9} \]
where it is set that
\[ T(t)=1-\frac{1}{K^p(t)}\int_0^a \psi(\tau)\left[K(\tau+t)+(-1)^pK(\tau-t)\right]\,d\tau . \tag{10} \]
Eliminating \(H(-t)\) from (9) by means of (4), we obtain
\[ \left\{ T(t)-\frac{\alpha}{K^{2p}(t)} \left[\int_0^a \frac{\psi(\tau)}{K^p(\tau)}H(\tau)\,d\tau\right]^2 \right\}H(t)+ \]
\[ +\frac{(-1)^p}{K^p(t)}\int_0^a \psi(\tau)H(\tau)K(\tau-t)\,d\tau=1. \tag{11} \]
Taking (7) into account, we rewrite the equation obtained also in the form
\[ \left[T(t)-\frac{\alpha\beta^2}{K^{2p}(t)}\right]H(t) +\frac{(-1)^p}{K^p(t)}\int_0^a \psi(\tau)H(\tau)K(\tau-t)\,d\tau=1. \tag{12} \]
Thus, the original nonlinear equation (2) has been reduced to the nonlinear equation (12). However, if condition b), imposed on the kernel
\(K(t)\), is satisfied for \(a=0\), then the nonlinear term
\[ \beta^2=\left[\int_0^a \frac{\psi(\tau)}{K^p(\tau)}H(\tau)\,d\tau\right]^2 \]
drops out of equation (12), and the nonlinear equation (2) is reduced to the linear equation (12). For Chandrasekhar’s equation (1) this fact was first established by Fox [5]. If, however, \(a\ne 0\), then one may formally solve equation (12), regarding \(\beta\) as an arbitrary constant quantity, and then determine \(\beta\) from equation (7). Already in the simplest case of Chandrasekhar’s equation it turns out that the solution of the derived linear equation may fail to satisfy the original nonlinear equation. The question of the equivalence of these equations is studied in the paper of Busbridge [8], in which, however, there are still no final results. In the present paper we shall not touch upon the study of the equivalence of equations (2) and (12).
Thus, let \(a=0\); then (12) will be a linear singular integral equation with Cauchy kernel, the theory of which is well developed (see [6, 7]). In the case when \(M(t)=0\), i.e. \(K(t)=t^{-1}\), this equation is written in the form
\[ T(t)H(t)+(-1)^p t^p\int_0^a \frac{\psi(\tau)H(\tau)}{\tau-t}\,d\tau=1. \tag{13} \]
A singular equation of this kind is called characteristic, or elementary, and is solved in closed form by reducing it to a Riemann boundary-value problem. For this purpose one introduces the function \(F(z)\), analytic in the whole plane of the complex variable \(z\), except for the segment \([0,a]\), and vanishing at infinity, by the formula
\[ F(z)=-\frac{1}{2\pi i}\int_0^a \frac{\psi(\tau)H(\tau)}{\tau-z}\,d\tau. \tag{14} \]
The limiting values of this function on the upper and lower shores of the cut \(0<t<a\), which we denote by \(F^+(t)\) and \(F^-(t)\), respectively, will be the solution of the Riemann boundary-value problem
\[ F^+(t)=G(t)F^-(t)+g(t), \tag{15} \]
where
\[ G(t)=\frac{T(t)-(-1)^p\pi i\,\psi(t)K^{-p}(t)} {T(t)+(-1)^p\pi i\,\psi(t)K^{-p}(t)}, \]
\[ \tag{16} g(t)=\frac{f(t)\psi(t)} {T(t)+(-1)^p\pi i\,\psi(t)K^{-p}(t)}. \]
Since \(K^{-1}(0)=0\), it follows that \(G(0)=1\). Further, let us take into account that in a neighborhood of the point \(t=a\) the function \(T(t)\), by virtue of the properties of an integral of Cauchy type ([6], p. 91), has the form
\[ T(t)=(-1)^{p+1}\frac{\psi(a)}{K^p(a)}\ln(t-a)+T_*(t), \tag{17} \]
where \(T_*(t)\) satisfies a Hölder condition for values close to \(a\). Hence it follows that \(G(a)=1\). Thus, at each of the ends of theสล็อตออนไลน์
of the interval \([0,a]\), \(G(t)\) is a real positive number, and this means that the points \(t=0\) and \(t=a\) are ends of automatic boundedness of the solution (special ends) (see [6], p. 314, [7], p. 481). The index of problem (15) will be equal to zero and, thus, the nonlinear integral equation
\[ H^{-1}(t)=1-t^{-p}\int_0^a \frac{\psi(\tau)H(\tau)}{\tau+t}\,d\tau \qquad (p=0 \text{ or } p=1) \tag{18} \]
will be reduced to the linear singular integral equation (13), solvable in closed form in a unique way. We note that for \(p=1\) equation (18) becomes Chandrasekhar’s equation, and our result coincides with Fox’s result cited above.
If \(M(t)\) does not vanish identically, then the original equation (2) is reduced to the complete linear singular equation
\[ T(t)H(t)+\frac{(-1)^p}{K^p(t)} \int_0^a \frac{\psi(\tau)H(\tau)}{\tau-t}\,d\tau+ \]
\[ +\frac{(-1)^p}{K^p(t)} \int_0^a \psi(\tau)H(\tau)M(\tau-t)\,d\tau=1. \tag{19} \]
It is easy to see that the points \(t=a\) and \(t=0\) will again be ends of automatic boundedness and, consequently, the index of equation (19) is equal to zero. Solvability theorems for equations of this type (Noether theorems), in the case where the index of the problem is zero, reduce to Fredholm theorems, which makes it possible to draw certain conclusions concerning the solvability of equation (19). Suppose, for example, that the homogeneous equation corresponding to equation (19) has only trivial solutions; then equation (19) is uniquely solvable. If, however, the indicated homogeneous equation has nontrivial solutions, then, denoting by \(h_j(t)\), \(j=1,2,\ldots,q\), the solutions of the adjoint homogeneous equation
\[ T(t)h(t)+(-1)^{p+1}\psi(t)\int_0^a \frac{h(\tau)}{K^p(\tau)}K(\tau-t)\,d\tau=0, \tag{20} \]
we obtain that, for equation (19) to be solvable, it is necessary and sufficient that the conditions
\[ \int_0^a h_j(t)\,dt=0,\qquad j=1,2,\ldots,q. \tag{21} \]
be satisfied.
Since the points \(t=0\) and \(t=a\) were special ends for the original equation (19), they will also be such for equation (20), so that the solutions \(h_j(t)\) entering condition (21) will be bounded for \(t=0\) and \(t=a\).
Although in the present case equation (19), generally speaking, is not solved in closed form, it is nevertheless significantly simpler than the original nonlinear equation (2).
The indicated scheme is formally carried over also to the nonlinear case \((\alpha \ne 0)\). If in this case \(M(t)\equiv0\), then, regarding \(\beta\) as a temporarily known constant—
we find in closed form the solution \(H(t,\beta)\) of equation (12), and then determine, if possible, \(\beta\) from the equation
\[ \beta=\int_{0}^{a}\frac{\psi(\tau)}{K^{p}(\tau)}H(\tau,\beta)\,d\tau . \tag{22} \]
An investigation of this equation in the general case cannot be carried out because of the complexity of the structure of the function \(H(t,\beta)\). We note only that the unsolvability of equation (22) entails the unsolvability of equation (2).
If, however, \(M(t)\) is not identically equal to zero, then, regarding equation (12) for fixed \(\beta\) as linear, we arrive at solvability conditions of the form
\[ \int_{0}^{a} h_j(t,\beta)\,d\tau=0,\qquad j=1,2,\ldots,q, \tag{23} \]
where \(h_j(t,\beta)\) is a complete system of solutions of the homogeneous equation adjoint to (12),
\[ \left[T(t)-\frac{\alpha\beta^2}{K^{2p}(t)}\right]h(t) +(-1)^{p+1}\int_{0}^{a}\frac{h(\tau)}{K^p(\tau)}K(\tau-t)\,d\tau=0. \tag{24} \]
In each particular case it is necessary to investigate jointly equation (22) and the system of solvability conditions (23).
§ 2. A somewhat different type of nonlinear integral equations admitting linearization is obtained if one assumes that the functions \(K(t)\) and \(\psi(t)\) satisfy conditions b) and c) of the preceding paragraph for \(a=2\pi\), while condition a) is replaced by the following: \(\tilde{\text{a}})\) \(K(t)\) is a \(2\pi\)-periodic function of the variable \(t\), representable in the form
\[ K(t)=\operatorname{ctg}\frac{t}{2}+M(t), \]
where \(M(t)\) is a function continuous on the interval \((-2\pi,2\pi)\), satisfying a Hölder condition, with \(M(-t)=-M(t)\). Examples of kernels satisfying conditions \(\tilde{\text{a}}\)) and b) are
1) \(K(t)=\operatorname{ctg}t/2,\quad M(t)\equiv 0,\) for \(p=0\), \(\alpha=-1\), for \(p=1\), \(\alpha=1,\)
2) \(K(t)=\operatorname{cosec}t/2,\quad M(t)=\operatorname{tg}t/4,\) for \(p=1\), \(\alpha=0.\)
Consider the nonlinear integral equation of the form
\[ \frac{1}{H(t)}=1-\frac{1}{K^p(t)}\int_{0}^{2\pi}\psi(\tau)H(\tau)K(t+\tau)\,d\tau, \tag{25} \]
where \(p=0\) or \(p=1\), assuming that conditions \(\tilde{\text{a}}\)), b), and c) are fulfilled for \(a=2\pi\).
By the method set forth in § 1, this equation is reduced to an integral equation of the form
\[ \left[T(t)-\frac{\alpha\beta^2}{K^{2p}(t)}\right]H(t) +\int_{0}^{2\pi}\psi(\tau)H(\tau)\operatorname{ctg}\frac{\tau-t}{2}\,d\tau+ \]
\[ + \int_0^{2\pi} \psi(\tau) H(\tau) M(t-\tau)\,d\tau = 1, \tag{26} \]
where \(T(t)\) and \(\beta\) have the same meaning as above.
Equation (26) is a nonlinear singular integral equation with Hilbert kernel. If \(\alpha = 0\), then this equation is linear, and the well-developed theory applies to it; an exposition of this theory may be found, for example, in [6].
Without going into details, we note that for \(\alpha = 0\) the function \(M(t)\) cannot be identically equal to zero, as is easily verified by a direct check of condition b); consequently, reduction of equation (25) to a characteristic equation solvable in closed form is impossible.
If \(\alpha \ne 0\), then the arguments are carried out according to the scheme described in § 1.
References
- Chandrasekhar S. Astrophys. J. 105, 164—203, 1947.
- Chandrasekhar S. Bull. Amer. Math. Soc., 53, 641—711, 1947.
- Crum M. M. Quart. J. Math., Oxford ser., 18, 244—352, 1947.
- Busbridge J. Quart. J. Math., Oxford ser., 8, 133—140, 1957.
- Fox C. Trans. Amer. Math. Soc., 99, 285—291, 1961.
- Muskhelishvili N. I. Singular Integral Equations. Fizmatgiz, Moscow, 1962.
- Gakhov F. D. Boundary Value Problems, Fizmatgiz, Moscow, 1963.
- Busbridge J. W. Trans. Amer. Math. Soc., 105, 112—117, 1962.
Received by the editors
February 13, 1965
Rostov State
University