scholarly journals Determining of unknown functions of different arguments in minor coefficient and right-hand side of semilinear ultraparabolic equation

2020 ◽  
Vol 12 (2) ◽  
pp. 317-332
Author(s):  
N.P. Protsakh ◽  
V.M. Flyud
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Uniqueness Of Solution ◽  
Ultraparabolic Equation ◽  
Right Hand

In this paper, we consider the inverse problem for semilinear ultraparabolic equation. The equation has two unknown functions of different arguments in its minor coefficient and in right-hand side function. The sufficient conditions of the existence and the uniqueness of solution on some interval $[0,T],$ where $T$ depends on the coefficients of the equation, are obtained.

Reconstruction of a convolution operator from the right-hand side on the semiaxis

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Anatoly F. Voronin
Keyword(s):  
Inverse Problem ◽  
Integral Operator ◽  
Sufficient Conditions ◽  
Stability Theorem ◽  
Explicit Formulas ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient ◽  
Operator Kernel ◽  
Problem Solvability

AbstractIn this paper, a Volterra integral equation of the first kind in convolutions on the semiaxis when the integral operator kernel and the right-hand side of the equation have a bounded support is considered. An inverse problem of reconstructing the solution to the equation and the integral operator kernel from values of the right-hand side is formulated. Necessary and sufficient conditions for the inverse problem solvability are obtained. A uniqueness and stability theorem is proved. Explicit formulas for reconstruction of the solution and kernel are obtained.

scholarly journals Inverse problem for semilinear Eidelman type equation

2020 ◽  
Vol 53 (1) ◽  
pp. 48-58
Author(s):  
N.P. Protsakh ◽  
O. E. Parasiuk-Zasun
Keyword(s):  
Inverse Problem ◽  
Weak Solution ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Initial Boundary ◽  
Time Dependent ◽  
Integral Type ◽  
Dependent Function ◽  
Right Hand

The inverse problem for semilinear Eidelman type equation with unknown time dependent function in its right-hand side is considered in this paper. The initial, boundary and integral type overdetermination conditions are posed. The sufficient conditions of the existence and the uniqueness of weak solution for the problem are obtained.

scholarly journals An inverse problem for the second-order integro-differential pencil

2019 ◽  
Vol 50 (3) ◽  
pp. 223-231 ◽  
Author(s):  
Natalia P. Bondarenko
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Spectral Parameter ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Uniqueness Of Solution ◽  
Convolution Kernel ◽  
Sturm Liouville ◽  
Necessary And Sufficient

We consider the second-order (Sturm-Liouville) integro-differential pencil with polynomial dependence on the spectral parameter in a boundary condition. The inverse problem is solved, which consists in reconstruction of the convolution kernel and one of the polynomials in the boundary condition by using the eigenvalues and the two other polynomials. We prove uniqueness of solution, develop a constructive algorithm for solving the inverse problem, and obtain necessary and sufficient conditions for its solvability.

scholarly journals Determining of right-hand side of higher order ultraparabolic equation

2017 ◽  
Vol 15 (1) ◽  
pp. 1048-1062 ◽  
Author(s):  
Nataliya Protsakh
Keyword(s):  
Inverse Problem ◽  
Existence And Uniqueness ◽  
Higher Order ◽  
Integral Type ◽  
Ultraparabolic Equation ◽  
Right Hand ◽  
Uniqueness Of The Solution ◽  
Time Variables

Abstract In the paper the conditions of the existence and uniqueness of the solution for the inverse problem for higher order ultraparabolic equation are obtained. The equation contains two unknown functions of spatial and time variables in its right-hand side. The overdetermination conditions of the integral type are used.

scholarly journals Inverse problem for $2b$-order differential equation with a time-fractional derivative

2019 ◽  
Vol 11 (1) ◽  
pp. 107-118 ◽  
Author(s):  
A.O. Lopushansky ◽  
H.P. Lopushanska
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Cauchy Problem ◽  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Generalized Solution ◽  
Right Hand ◽  
The Cauchy Problem ◽  
The Right

We study the inverse problem for a differential equation of order $2b$ with the Riemann-Liouville fractional derivative of order $\beta\in (0,1)$ in time and given Schwartz type distributions in the right-hand sides of the equation and the initial condition. The problem is to find the pair of functions $(u, g)$: a generalized solution $u$ to the Cauchy problem for such equation and the time dependent multiplier $g$ in the right-hand side of the equation. As an additional condition, we use an analog of the integral condition $$(u(\cdot,t),\varphi_0(\cdot))=F(t), \;\;\; t\in [0,T],$$ where the symbol $(u(\cdot,t),\varphi_0(\cdot))$ stands for the value of an unknown distribution $u$ on the given test function $\varphi_0$ for every $t\in [0,T]$, $F$ is a given continuous function. We prove a theorem for the existence and uniqueness of a generalized solution of the Cauchy problem, obtain its representation using the Green's vector-function. The proof of the theorem is based on the properties of conjugate Green's operators of the Cauchy problem on spaces of the Schwartz type test functions and on the structure of the Schwartz type distributions. We establish sufficient conditions for a unique solvability of the inverse problem and find a representation of anunknown function $g$ by means of a solution of a certain Volterra integral equation of the second kind with an integrable kernel.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

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