Geodesic
Integro-differential equation associated with the Riemann--Carleman boundary value problem
Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 73-81

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We consider a linear integro-differential equation on a closed curve located on the complex plane. The coefficients of the equation have a special structure. The equation is first reduced to the mixed Riemann–Carleman boundary value problem for analytic functions. Next, two differential equations are solved in areas of the complex plane with additional conditions. The conditions for the solvability of the original equation are indicated explicitly. When they executed, the solution is given in closed form. An example is given.
Keywords: integro-differential equation, hypersingular integral, generalized Sokhotsky formulas, Riemann–Carleman boundary problem, linear differential equation. 
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     author = {A. P. Shilin},
     title = {Integro-differential equation associated with the {Riemann--Carleman} boundary value problem},
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     publisher = {mathdoc},
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     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a6/}
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A. P. Shilin. Integro-differential equation associated with the Riemann--Carleman boundary value problem. Trudy Instituta matematiki, Tome 32 (2024) no. 2, pp. 73-81. http://geodesic.mathdoc.fr/item/TIMB_2024_32_2_a6/