Inverse Cauchy problem for fractional telegraph equations with distributions

Abstract

The inverse Cauchy problem for the fractional telegraph equation $$u^{(\alpha)}_t-r(t)u^{(\beta)}_t+a^2(-\Delta)^{\gamma/2} u=F_0(x)g(t), \;\;\; (x,t) \in {\rm R}^n\times (0,T],$$ with given distributions in the right-hand sides of the equation and initial conditions is studied. Our task is to determinate a pair of functions: a generalized solution $u$ (continuous in time variable in general sense) and unknown continuous minor coefficient $r(t)$. The unique solvability of the problem is established.

Inverse Source Problem for a Semilinear Fractional Diffusion-Wave Equation Under a Time-Integral Condition

“…Äîñòàòíi óìîâè ¹äèíîñòi ðîçâ'ÿçêó îáåðíåíî¨êðàéîâî¨çàäà÷i äëÿ ïiâëiíiéíîãî ðiâíÿííÿ äðîáîâî¨äèôóçi¨ç íåâiäîìèì ìíîaeíèêîì, ùî çàëåaeèòü âiä ÷àñó, çíàéäåíî â [11] ïðè iíòåãðàëüíié çà ïðîñòîðîâèìè çìiííèìè óìîâi ïåðåâèçíà÷åííÿ. Òàêîae ïðè òàêîãî âèãëÿäó äîäàòêîâié óìîâi â [12] îäåðaeàíî äîñòàòíi óìîâè îäíîçíà÷íî¨ðîçâ'ÿçíîñòi îáåð-íåíî¨çàäà÷i ç íåâiäîìèì ìîëîäøèì, çàëåaeíèì âiä ÷àñó, êîåôiöi¹íòîì äëÿ ïiâëiíiéíîãî òåëåãðàôíîãî ðiâíÿííÿ.…”

“…Çà ïðèïóùåíü òåîðåìè, âðàõîâóþ÷è âëàñòèâîñòi ôóíêöi¨G 0 i ðåçóëüòàòè [2], îäåð-aeó¹ìî, ùî u ¹ ðîçâ'ÿçêîì êðàéîâî¨çàäà÷i (10), (11) òîäi é òiëüêè òîäi, êîëè âîíà çàäî-âîëüíÿ¹ ó C( Q) iíòåãðàëüíå ðiâíÿííÿ u(x, t) = t 0 dτ Ω G 0 (x, t, y, τ ) g 2 (y)F 01 (y, τ )u(y, τ ) +g(y)F 0 (y, τ, u 1 (y, τ )) dy, (x, t) ∈ Ω. (13) Çàñòîñîâóþ÷è äî îáîõ ÷àñòèí ðiâíÿííÿ (10) óìîâó ïåðåâèçíà÷åííÿ (12), îäåðaeó¹ìî…”

“…Âðàõîâóþ÷è ðåçóëüòàòè [2], îäåðaeó¹ìî, ùî êîaeíèé íåïåðåðâíèé â Q ðîçâ'ÿçîê iíòåãðàëüíîãî ðiâíÿííÿ (15) (ó íàøîìó âèïàäêó òðèâiàëüíèé) ¹ ðîçâ'ÿçêîì iç C 2,α ( Q) çàäà÷i (10), (11). Î÷åâèäíî, âií çàäîâîëüíÿ¹ óìîâó ïåðåâèçíà÷åííÿ (12). 2…”

Uniqueness of solution for the inverse problem of finding two minor coefficients in a semilinear time fractional telegraph equation

“…Some inverse problems to diffusion-wave equations with different unknown functions or parameters (source, order of partial derivative, elder or minor coefficient, boundary or initial data) were investigated, for example, in [2]- [13]. In particular, in papers [2,4,5,7,8,9] the integral overdetermination conditions were used in the inverse source and coefficient problems for a time fractional diffusion equations.…”

“…We note that inverse problems for semilinear parabolic and ultraparabolic equations with one unknown function were investigated, for example, in [14,15] and for a semilinear time fractional telegraph equation in [7].…”