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VIDEO LIBRARY |
International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
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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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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 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
The kernel Problem. To find the solution The following theorem holds. Theorem. The nonhomogeneous integral equation \eqref{316:eq1} is solvable in the class $$ \varphi_k(t)= \frac{1}{\sqrt{t}} \exp\biggl(\frac{p_{k}}{t}-\frac{t}{4a^{2}}\biggr) +\frac{\lambda \sqrt{\pi}}{2a} \exp \biggl(\frac{\lambda^2-1}{4a^2}t-\frac{\lambda \sqrt{-p_k}}{a}\biggr)\cdot \mathrm{erfc}\biggl(\frac{2a\sqrt{-p_{k}}-\lambda t}{2a\sqrt{t}} \biggr), $$ 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 $$ \widetilde{f}_2(t)=\widetilde{f}(t)+\lambda\int_0^t r(t,\tau) \widetilde{f}(\tau)\,d\tau. $$ Language: English References
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