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

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 Asymptotic Solutions of Scalar Integro-Differential Equations with Partial Derivatives and with Fast Oscillating Coefficients

2020 ◽  
Vol 13 (2) ◽  
pp. 287-302
Author(s):  
Burkhan Kalimbetov ◽  
Akisher Temirbekov ◽  
Abdimuhan Tolep
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Unique Solvability ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
Integral Term ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
Oscillating Coefficients

In the paper, ideas of the Lomov regularization method are generalized to the Cauchy problem for a singularly perturbed partial integro-differential equation in the case when the integral term contains a rapidly varying kernel. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.

scholarly journals Integro-differential equation with two times modifications

2011 ◽  
Vol 27 (2) ◽  
pp. 209-216
Author(s):  
VERONICA-ANA ILEA ◽  
◽  
DIANA OTROCOL ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Operator Theory ◽  
Step Method ◽  
Picard Operator ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
Weakly Picard Operator

We consider an integro-differential equation with two times modifications. Existence, uniqueness and monotony results of solution for the Cauchy problem are obtained using weakly Picard operator theory. In the last section we present a step method for this type of equation.

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.

scholarly journals REGULAR SOLUTION OF THE INVERSE PROBLEM WITH INTEGRAL CONDITION FOR A TIME-FRACTIONAL EQUATION

2020 ◽  
Vol 8 (2) ◽  
pp. 103-113
Author(s):  
H. Lopushanska ◽  
A. Lopushansky
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Fourier Transform ◽  
Approximate Solution ◽  
Vector Function ◽  
Unique Solvability ◽  
Fractional Derivatives ◽  
Direct And Inverse Problems ◽  
The Cauchy Problem ◽  
The Fourier Transform

Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob\-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Bbb R^n\] where $u$ is the unknown solution of the Cauchy problem, $\eta_1$ and $\Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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