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.

scholarly journals Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

2021 ◽  
pp. 117-127
Author(s):  
Durdimurod K. Durdiev ◽  
Zhavlon Z. Nuriddinov
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Cauchy Problem ◽  
Direct Problem ◽  
Contractive Mapping ◽  
Equivalent System ◽  
Direct And Inverse Problems ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
The Right

The multidimensional parabolic integro-differential equation with the time-convolution in- tegral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. In this paper it is studied the inverse problem consisting in finding of a time and spatial dependent kernel of the integrated member on known in a hyperplane xn = 0 for t > 0 to the solution of direct problem. With use of the resolvent of kernel this problem is reduced to the investigation of more convenient inverse problem. The last problem is replaced with the equivalent system of the integral equations with respect to unknown functions and on the bases of contractive mapping principle it is proved the unique solvability to the direct and inverse problems

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.

UNIQUE SOLUTIONS OF DISCONTINUOUS O.D.E.'S IN BANACH SPACES

2006 ◽  
Vol 04 (03) ◽  
pp. 247-262 ◽  
Author(s):  
ALBERTO BRESSAN ◽  
WEN SHEN
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Right Hand ◽  
Directional Variation ◽  
The Cauchy Problem

We consider the Cauchy problem for an ordinary differential equation with discontinuous right-hand side in an L∞ space. Under the assumptions that the vector field is directionally continuous with bounded directional variation, we prove that the O.D.E. has a unique Carathéodory solution, which depends Lipschitz continuously on the data.

4.—A Cauchy Problem for an Ordinary Integro-differential Equation

1974 ◽  
Vol 72 (1) ◽  
pp. 39-55 ◽  
Author(s):  
E. A. Catchpole
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Cauchy Problem ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Integro Differential Equation ◽  
Linearly Independent ◽  
The Cauchy Problem ◽  
Integral Equation Approach ◽  
Necessary And Sufficient

SynopsisIn this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.

On Chaplygin’s theorem for an implicit differential equation of order n

2021 ◽  
pp. 225-233
Author(s):  
Sarra Benarab
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Differential Inequality ◽  
Sufficient Conditions ◽  
Implicit Differential Equation ◽  
Smooth Functions ◽  
Implicit Differential Equations ◽  
The Cauchy Problem ◽  
Partially Ordered Spaces

We consider the Cauchy problem for the implicit differential equation of order n g(t,x,(x,) ̇…,x^((n)) )=0,t ∈ [0; T],x(0)= A. It is assumed that A=(A_0,…,A_(n-1) )∈R^n, the function g:[0,T] × R^(n+1)→ R is measurable with respect to the first argument t∈[0,T], and for a fixed t, the function g(t,∙)×R^(n+1)→ R is right continuous and monotone in each of the first n arguments, and is continuous in the last n+1-th argument. It is also assumed that for some sufficiently smooth functions η,ν there hold the inequalities ν^((i) ) (0)≥ A_i ≥ η^((i) ) (0),i= (0,n-1,) ̅ ν^((n) ) (t)≥ η^((n) ) (t),t∈[0; T]; g(t; ν(t),ν ̇(t),...,ν^((n) ) (t) )≥ 0,g(t,η(t),η ̇(t),…,η^((n)) (t))≤ 0,t∈[0; T]. Sufficient conditions for the solvability of the Cauchy problem are derived as well as estimates of its solutions. Moreover, it is shown that under the listed conditions, the set of solutions satisfying the inequalities η^((n) ) (t)≤x^((n) ) (t)≤ν^((n) ) (t), is not empty and contains solutions with the largest and the smallest n -th derivative. This statement is similar to the classical Chaplygin theorem on differential inequality. The proof method uses results on the solvability of equations in partially ordered spaces. Examples of applying the results obtained to the study of second-order implicit differential equations are given.

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