One-Dimensional Inverse Problems of Finding the Kernel of Integrodifferential Heat Equation in a Bounded Domain

Durdiev, D. K.; Zhumaev, Zh. Zh.

doi:10.1007/s11253-022-02026-0

Cited by 15 publications

(3 citation statements)

References 5 publications

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“…Now we substitute the expression (19) for q(t) into ( 6) and arrive at an integral equation for u(x, t):…”

Section: Study Of the Inverse Problemmentioning

confidence: 99%

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

Durdiev,

Durdiev

2024

EJMCA

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In this study, we consider the inverse problem of determining the time-dependent beam stiffness coefficient q(t) and transverse bending vibrations u(x,t) of a homogeneous beam using the finite difference method. The problem is governed by a fourth-order partial differential equation, which is discretized in space and time using numerical techniques to obtain an accurate and stable solution. An additional condition in the form of an integral is also included to determine q(t). The results obtained for different discretization grid points are compared, and a simple test example is presented to demonstrate the accuracy and agreement of the numerical solutions with analytical solutions. The paper provides a comprehensive overview of the methodology used to solve the inverse problem, and presents a detailed analysis of the results obtained.

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“…Now we substitute the expression (19) for q(t) into ( 6) and arrive at an integral equation for u(x, t):…”

Section: Study Of the Inverse Problemmentioning

confidence: 99%

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

Durdiev,

Durdiev

2024

EJMCA

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Section: Introductionmentioning

confidence: 97%

“…Among the works close to our problem we mention [32]- [35]. In [32] the uniqueness theorem for solution of kernel determination problem for one-dimensional heat conduction equation was proven. Papers [33]- [35] dealt with the inverse problems of determining the kernel depending on a time variable 𝑡 and (𝑛 − 1)-dimensional spatial variable 𝑥 ′ = (𝑥 1 , .…”

Section: Introductionmentioning

confidence: 97%

Inverse problem on determining two kernels in integro-differential equation of heat flow

Durdiev,

Jumayev,

Atoev

2023

Ufimsk. Mat. Zh.

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We study the inverse problem on determining the energy-temperature relation 𝜒(𝑡) and the heat conduction relation 𝑘(𝑡) functions in the one-dimensional integrodifferential heat equation. The direct problem is an initial-boundary value problem for this equation with the Dirichlet boundary conditions. The integral terms involve the time convolution of unknown kernels and a direct problem solution. As an additional information for solving inverse problem, the solution of the direct problem for 𝑥 = 𝑥 0 and 𝑥 = 𝑥 1 is given. We first introduce an auxiliary problem equivalent to the original one. Then the auxiliary problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to the unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions, we prove the main result of the article, which a local existence and uniqueness theorem for the inverse problem.

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Convolution Kernel Determining Problem for an Integro-Differential Heat Equation With Nonlocal Initial-Boundary and Overdetermination Conditions

Durdiev¹,

Zhumaev²,

Atoev³

2023

J Math Sci

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One-Dimensional Inverse Problems of Finding the Kernel of Integrodifferential Heat Equation in a Bounded Domain

Cited by 15 publications

References 5 publications

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

Inverse problem on determining two kernels in integro-differential equation of heat flow

Convolution Kernel Determining Problem for an Integro-Differential Heat Equation With Nonlocal Initial-Boundary and Overdetermination Conditions

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