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International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 27, 2015 14:30–14:55, Дифференциальные уравнения II, Moscow, Steklov Mathematical Institute of RAS
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On a Volterra equation of the second kind with “incompressible” kernel
M. T. Jenaliyeva, M. M. Amangaliyevaa, M. T. Kosmakovab, M. I. Ramazanovc a Institute of Mathematics and Mathematical Modeling
b Al-Farabi Kazakh National University
c E. A. Buketov Karaganda State University
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This page: | 262 | Materials: | 80 |
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Abstract:
Solving the boundary value problems of the heat equation in
noncylindrical domains degenerating at the initial moment leads to
the necessity of research of the singular Volterra integral
equations of the second kind, when the norm of the integral operator
is equal to 1. The paper deals with the singular Volterra integral
equation of the second kind, to which by virtue of 'the
incompressibility' of the kernel the classical method of successive
approximations is not applicable. It is shown that the corresponding
homogeneous equation when $|\lambda|>1$ has a continuous spectrum,
and the multiplicity of the characteristic numbers increases
depending on the growth of the modulus of the spectral parameter
$|\lambda|$. By the Carleman-Vekua regularization method [1] the initial equation is
reduced to the Abel equation. The eigenfunctions of the equation are
found explicitly. Similar integral equations also arise in the study
of spectral-loaded heat equations [2].
When solving model problems for parabolic equations in domains with
moving boundary the singular integral equations of the following
form arise:
\begin{equation}
\label{316:eq1}
\varphi(t)-\lambda\int\limits_{0}^{t}K(t,\tau)
\varphi(\tau)\,d\tau=f(t),\qquad t>0,
\end{equation}
where
$$
K(t,\tau)=\frac{1}{2a\sqrt{\pi}} \biggl\{\frac{t+\tau}
{(t-\tau)^{3/2}} \exp
\biggl(-\frac{(t+\tau)^{2}} {4a^{2}(t-\tau)}
\biggr)+\frac{1}{(t-\tau)^{1/2}}
\exp\biggl(-\frac{t-\tau}{4a^{2}}\biggr)\biggr\}.
$$
The kernel $ K(t, \tau )$ has the following properties:
- 1) $K(t,\tau )\ge 0$ and continuously at $0<\tau < t < +\infty$;
- 2) $\lim_{t\to t_{0} } \int_{t_{0} }^{t} K(t,\tau)\, d\tau =0$, $t_{0} \ge \varepsilon >0$;
- 3) $\lim_{t\to 0} \int_{0}^{t}K(t, \tau )\, d\tau
=1$, $\lim_{t\to +\infty } \int_{0}^{t}K(t, \tau )\, d\tau =1$.
The kernel $K(t,\tau)$ is summable with weight function
$t^{-1/2}$.
Problem.
To find the solution $\varphi(t)$ of integral equation
\eqref{316:eq1} satisfying the condition $\sqrt{t}\cdot \varphi(t)\in
L_\infty(0,\infty)$ for any given function $\sqrt{t}\cdot f(t)\in
L_\infty(0,\infty)$ and each given complex spectral parameter
$\lambda\in\mathcal{C}$.
The following theorem holds.
Theorem.
The nonhomogeneous integral equation
\eqref{316:eq1} is solvable in the class $\sqrt{t}\cdot\varphi(t)\in
L_\infty(0,\infty)$ for any right-hand side $\sqrt{t}\cdot f(t)\in
L_\infty(0;\infty)$ and for each $|\lambda|>1.$ The corresponding
homogeneous equation has $(N_{1} +N_{2}+1)$ eigenfunctions
$$
\varphi_k(t)= \frac{1}{\sqrt{t}}
\exp\biggl(\frac{p_{k}}{t}-\frac{t}{4a^{2}}\biggr)
+\frac{\lambda \sqrt{\pi}}{2a} \exp
\biggl(\frac{\lambda^2-1}{4a^2}t-\frac{\lambda
\sqrt{-p_k}}{a}\biggr)\cdot
\mathrm{erfc}\biggl(\frac{2a\sqrt{-p_{k}}-\lambda t}{2a\sqrt{t}}
\biggr),
$$
and the general solution of integral equation
\eqref{316:eq1} can be written as
$$
\varphi(t)=F(t)+\frac{\lambda^2}{4a^2} \int_0^t
\exp \biggl(\frac{\lambda^2(t-\tau)}{4a^2}\biggr) F(t)\,d\tau +
\sum_{k=-N_1}^{N_2} C_k \varphi_k(t),
$$
where
\begin{gather*}
N_{1}=\biggl[\frac{\ln|\lambda|+\arg \lambda}{2\pi}\biggr], \qquad
N_{2}=\biggl[\frac{\ln|\lambda|-\arg \lambda}{2\pi}\biggr],
\\
F(t)=\widetilde{f}_2(t)-\frac{\lambda}{2a\sqrt{\pi}} \int_0^t
\frac{\widetilde{f}_2(\tau)}{\sqrt{t-\tau}}\,d\tau,
\end{gather*}
and the function
$\sqrt{t}\cdot\exp\{-t/(4a^2)\}\cdot\widetilde{f}_{2}(t)\in
L_\infty(0,\infty)$ is defined by equality
$$
\widetilde{f}_2(t)=\widetilde{f}(t)+\lambda\int_0^t r(t,\tau)
\widetilde{f}(\tau)\,d\tau.
$$
Supplementary materials:
abstract.pdf (161.5 Kb)
Language: English
References
-
I. N. Vekua, Generalized analytic functions, FIZMATLIT, Moscow, 1988
-
M. M. Amangaliyeva, D. M. Akhmanova, M. T. Jenaliyev, M. I. Ramazanov, “Boundary value problems for a spectrally loaded heat operator with load line approaching the time axis at zero or infinity”, Differentsialniye uravneniya, 47:2 (2011), 231–243
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