abstract differential equations
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Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 474
Author(s):  
Luciano Abadias ◽  
Edgardo Alvarez ◽  
Rogelio Grau

We investigate the semi-linear, non-autonomous, first-order abstract differential equation x′(t)=A(t)x(t)+f(t,x(t),φ[α(t,x(t))]),t∈R. We obtain results on existence and uniqueness of (ω,c)-periodic (second-kind periodic) mild solutions, assuming that A(t) satisfies the so-called Acquistapace–Terreni conditions and the homogeneous associated problem has an integrable dichotomy. A new composition theorem and further regularity theorems are given.


Author(s):  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov ◽  
Nataliya V. Mayko

AbstractWe represent the solution {u(t)} of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials. The approximate solution is the truncated sum of N (the discretization parameter) summands. We give the error estimate of these approximations depending on N and the distance of t to the initial point of the time interval or of the spatial argument x to the boundary of the spatial domain.


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