scholarly journals Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Pengbin Feng ◽  
Erkinjon T. Karimov
Keyword(s):  
Wave Equation ◽  
Mixed Type ◽  
Orthonormal System ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Initial Time ◽  
Inverse Source Problem ◽  
Ill Posed ◽  
Inverse Source Problems ◽  
Well Posed

AbstractIn the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

scholarly journals An inverse source problem for the stochastic wave equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoli Feng ◽  
Meixia Zhao ◽  
Peijun Li ◽  
Xu Wang
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Direct Problem ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Stochastic Wave Equation ◽  
Unique Mild Solution ◽  
Well Posed ◽  
Class Of Functions

<p style='text-indent:20px;'>This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.</p>

scholarly journals On solution uniqueness for Dezin problem analog for model parabolic hyperbolic type equation

2021 ◽  
Vol 21 (4) ◽  
pp. 15-17
Author(s):  
R.A. Kirzhinov ◽  
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Type Condition ◽  
Solution Uniqueness ◽  
Mixed Type Equation

In this paper investigated a problem with Dezin type condition for parabolic-hyperbolic mixed type equation. It is established a criterion for solution uniqueness to the problem.

The near-field singularity predicted by the spiral Green’s function in acoustics and electrodynamics

1991 ◽  
Vol 433 (1888) ◽  
pp. 451-459 ◽  
Keyword(s):  
Wave Equation ◽  
Point Source ◽  
Wave Speed ◽  
Near Field ◽  
Radial Component ◽  
Extended Source ◽  
Ill Posed ◽  
Field Singularity ◽  
Theory Of Generalized Functions ◽  
Well Posed

The retarded solution of the wave equation for a point source in circular motion, whose speed exceeds the wave speed, is singular on a spiralling tube-like surface that is at rest in the rest frame of the point source. When solving the wave equation for a corresponding extended source, therefore, we are faced with integrals over the volume of the source which are improper and need to be handled either with the aid of the theory of generalized functions or by Hadamard’s method of finite parts. In this paper, after isolating the finite part of the gradient of the retarded potential due to a rotating extended source, we calculate the asymptotic values of the coefficients in its Fourier representation and show that the radial component of this gradient does not remain finite at those points within the source which move with the wave speed, and so lie on the boundary of the domain of hyperbolicity of the equation of the mixed type to which the wave equation in this case reduces. This latter singularity arises because the problem in question, though well posed physically, is in fact mathematically ill posed.

scholarly journals OPTIMAL CONTROL FOR A CONTROLLED ILL-POSED WAVE EQUATION WITHOUT REQUIRING THE SLATER HYPOTHESIS

2020 ◽  
Vol 6 (1) ◽  
pp. 84
Author(s):  
Abdelhak Hafdallah
Keyword(s):  
Optimal Control ◽  
Wave Equation ◽  
Missing Data ◽  
Incomplete Data ◽  
Initial Condition ◽  
Control Sequence ◽  
Empty Interior ◽  
Ill Posed ◽  
Well Posed ◽  
Admissible Controls

In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with  ncomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.

Convergence and stability analysis of the half thresholding based few-view CT reconstruction

2020 ◽  
Vol 28 (6) ◽  
pp. 829-847
Author(s):  
Hua Huang ◽  
Chengwu Lu ◽  
Lingli Zhang ◽  
Weiwei Wang
Keyword(s):  
Noise Suppression ◽  
Reconstructed Image ◽  
Reference Image ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Reconstruction Algorithms ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Stability ◽  
Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

scholarly journals Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 573
Author(s):  
Marzia Sara Vaccaro ◽  
Francesco Paolo Pinnola ◽  
Francesco Marotti de Sciarra ◽  
Raffaele Barretta
Keyword(s):  
Boundary Conditions ◽  
Structural Engineering ◽  
Beam Deflection ◽  
Soil Reaction ◽  
Differential Problem ◽  
Reaction Field ◽  
Ill Posed ◽  
Elastic Responses ◽  
Benchmark Solutions ◽  
Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

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