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Unique continuation is one of the most important properties for the solution of partial differential equation, which means the local information of the solution can determine the global one. In this paper, we discuss a special unique continuation for Helmholtz equations on a sphere in ℝ3$$ {\mathbb{R}}^3 $$, which is different with the classical unique continuation. The unique continuation holds only on the sphere and may not be extended to the domain in ℝ3$$ {\mathbb{R}}^3 $$. A Hölder type conditional stability of unique continuation is proved by complex extension method. Then a Tikhonov regularized scheme is proposed and the convergence rate is obtained by using the conditional stability estimate. Numerical examples are presented to show the performance of the scheme. It should be remarked here that our results may be applied to the problem of recovering a far field pattern with its local information. [ABSTRACT FROM AUTHOR]

The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output least-squares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in L²(Ω) for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the finite element analysis for the Bean–Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with L1-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D H(curl)-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution. [ABSTRACT FROM AUTHOR]

We extend the result of Kowalczyk, Martel, and Mu\~noz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133--2167] on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power nonlinear Klein--Gordon equation with p > 3, to the case 2 ≥ p > 5/3. . The result is attained performing new and refined estimates that allow us to close the argument for power law in the range 2 ≥ p > 5/3. . [ABSTRACT FROM AUTHOR]

The Riccati equation method is used to establish some global solvability criteria for a classes of Lane–Emdem–Fowler and Van der Pol type equations. Two oscillation theorems are proved. The results obtained are applied to the Emden–Fowler equation and to the Van der Pol type equation. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors u from images of a passive scalar field ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field ϕ and the measured scalar field ψ, under the constraint that ϕ is advected by the reconstructed velocity field u , which again is governed by the Navier–Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two-dimensional case, smooth solutions of the Navier–Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field ψ to the reconstructed velocity field u, on any interior subset, is Hölder continuous. [ABSTRACT FROM AUTHOR]

It is well known that the Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether they can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, the authors will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order. [ABSTRACT FROM AUTHOR]

For the time-fractional neutron diffusion equation with a Caputo derivative of order α ∈ (0 , 1 2) , we give the optimal error bounds of L1-type schemes under the spatial L ∞ -norm with lower regularity solution than typical | ∂ t l u (x , t) | ≤ C (1 + t 2 α - l) , l = 0 , 1 , 2 , where 2 α is the highest order of the time-fractional derivative. The unique solvability and the numerical stability of schemes will be exported via a simple restriction of coefficients with non-uniform time steps τ n < π 4 . The truncation error of term D 0 , t 2 α with low regularity | ∂ t l u (x , t) | ≤ C (1 + t α - l) , l = 0 , 1 , 2 , and the global error of full discretizations will be given. In addition, several numerical examples will be shown to test our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove a global Carleman estimate for the complex Ginzburg-Landau operator with a cubic nonlinear term in a bounded domain of Rn, n = 2, 3. As applications, we study state observation problems for the Ginzburg-Landau equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying two multiobjective problems for stochastic degenerate parabolic equations. The first one is a hierarchical control problem, in which the controls are classified into a pair of leaders and a pair of followers. For each pair of leaders, a Nash equilibrium is searched for a noncooperative game problem. The aim of the pair of leaders is to achieve null controllability of the system. The other multiobjective problem is an inverse initial problem under a Nash equilibrium strategy. In contrast to the classical inverse initial problem, an optimization problem for stochastic degenerate parabolic equations is first investigated. Then, the conditional stability of determining initial information is derived through terminal observation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simultaneous inversion of the source term and the initial data for a time-fractional diffusion equation based on the additional temperatures at two fixed times $ t=T_{1} $ and $ t=T_{2} $ . The inverse problem is formulated on the basis of the Fourier method as an operator equation of the first kind. For the overdetermined system of linear equations, we apply the conjugate gradient method with an appropriate stopping criterion to solve the least-squares problems. The method is not only able to converge quickly, but also it can solve large-scale linear equations. Based on the proposed results, it can be seen that the iteration numbers play a role of a regularization parameter. Four numerical examples are provided to demonstrate the effectiveness of the proposed method.

We consider the determination of an unknown spatial-dependent coefficient for the pseudoparabolic system with Dirichlet boundary condition from nonlocal inversion input data. Based on the positive property of the solution to the direct problem, the uniqueness and the Lipschitz conditional stability of this nonlinear inverse problem are addressed in Hölder space, by the principle of contraction mapping for specially chosen weight function describing the nonlocal inversion input. Then, an iterative algorithm is established in terms of the fixed point equation for the unknown potential to construct the approximate solution to this inverse problem, with rigorous convergence analysis and the error estimate on the iterative sequence, which also leads to the optimal approximation error for choosing the stopping rule of the iteration process. The proposed iterative algorithm is compared with classical optimization scheme with regularizing term, showing the advantages of our scheme. Some numerical results are presented to validate our proposed scheme. [ABSTRACT FROM AUTHOR]

We consider a nonlinear inverse problem for identifying an unknown time-dependent potential coefficient in a linear pseudoparabolic equation from nonlocal inversion input, which is quite different from the cases of parabolic equation due to the appearance of the mixed derivatives of third order in the governed equation. Based on the established positive property of the solution to the direct problem, the uniqueness and the Lipschitz conditional stability of this inverse problem are addressed by the principle of contraction mappings for specially chosen weight function in the average observation input data. Then, in terms of the fixed point equation, an iterative algorithm is established to construct the approximate solution to this inverse problem with explicit error estimate, which also leads to the optimal error bounds for specified choice strategy of the iteration times. Some numerical results are presented to validate our proposed reconstruction scheme. [ABSTRACT FROM AUTHOR]

The paper considers the inverse problem of determining the unknown coefficient on the right-hand side of the hyperbolic heat equation. An additional condition for finding the unknown coefficient, which depends on the time variable, is given in integral form. Theorems on the uniqueness, stability and existence of the solution are proved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a nonlinear inverse problem of identifying a time-dependent zeroth-order coefficient in a time-fractional diffusion-wave equation by using the measured data at a boundary point. We firstly prove the existence, uniqueness and regularity of the solution for the corresponding direct problem by using the contraction mapping principle. Then we try to give a conditional stability estimate for the inverse zeroth-order coefficient problem and propose a simple condition for the initial value and zeroth-order coefficient such that the uniqueness of the inverse coefficient problem is obtained. The Levenberg–Marquardt regularization method is applied to obtain a regularized solution. Based on the piecewise linear finite elements approximation, we find an approximate minimizer at each iteration by solving a linear system of algebraic equations in which the Fréchet derivative is obtained by solving a sensitive problem. Two numerical examples in one-dimensional case and two examples in two-dimensional case are provided to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simultaneous inversion of the source term and the initial data for a time-fractional diffusion equation based on the additional temperatures at two fixed times t = T 1 and t = T 2 . The inverse problem is formulated on the basis of the Fourier method as an operator equation of the first kind. For the overdetermined system of linear equations, we apply the conjugate gradient method with an appropriate stopping criterion to solve the least-squares problems. The method is not only able to converge quickly, but also it can solve large-scale linear equations. Based on the proposed results, it can be seen that the iteration numbers play a role of a regularization parameter. Four numerical examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in t and Lipschitz continuous in x. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jäh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hölder and logarithmic), arbitrarily closed, but not up to the final time. [ABSTRACT FROM AUTHOR]

The modeling of P-waves has essential applications in seismology. This is because the detection of the P-waves is the first warning sign of the incoming earthquake. Thus, P-wave detection is an important part of an earthquake monitoring system. In this paper, we introduce a linear computational cost simulator for three-dimensional simulations of P-waves. We also generalize our formulations and derivation for elastic wave propagation problems. We use the alternating direction method with isogeometric finite elements to simulate seismic P-wave and elastic propagation problems. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along the particular coordinate axis in the sub-steps. We show that the resulting problem matrix can be represented as a multiplication of three multidiagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. The resulting system of linear equations can be factorized in linear O(N) computational cost in every time step of the semi-implicit method. We use our method to simulate P-wave and elastic wave propagation problems. We derive the condition for the stability of seismic waves; namely, we show that the method is stable when t

The present paper is devoted to identifying the space-dependent source term and initial value simultaneously for a time-fractional diffusion equation. This inverse problem is reformulated on the basis of Fourier method as operator equations of the first kind. The construction of the solution to the inverse problem is derived by using the Landweber iteration method for the solution of the corresponding conjugate operator equation. In this paper, we also present error estimates between the exact solution and the regularized solution by the a priori and the a posteriori parameter choice rules. Numerical verification on the efficiency and accuracy of the proposed algorithm is performed by solving three numerical examples in one-dimensional and two-dimensional cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator, one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using a Galerkin finite element method in space and a finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem and present a thorough error analysis for the reconstructed potential. Our derived a priori error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this work, we first establish a local Carleman estimate for the ultrahyperbolic Schrödinger equation which arises in some theories of modern physics such as quantum mechanics and string theory. Next, we prove a Hölder stability estimate for the Cauchy Problem. In the proof, we introduce a suitable cut-off function and extend Cauchy data in a Sobolev space to reduce the problem to another problem for functions with compact supports and then we apply the Carleman estimate. [ABSTRACT FROM AUTHOR]

System theory methods are developed and applied to introduce a new analysis methodology based on the stability criteria of active two-ports, to the problem of thermo-acoustic instability in a combustion appliance. The analogy between thermo-acoustics of combustion and small-signal operation of microwave amplifiers is utilized. Notions of unconditional and conditional stabilities of an (active) two-port, representing a burner with flame, are introduced and analyzed. Unconditional stability of two-port means the absence of autonomous oscillation at any embedding of the given two-port by any passive network at the system's upstream (source) and downstream (load) sides. It has been shown that for velocity-sensitive compact burners in the limit of zero Mach number, the criteria of unconditional stability cannot be fulfilled. The analysis is performed in the spirit of a known criterion in microwave network theory, the so-called Edwards-Sinsky's criterion. Therefore, two methods have been applied to elucidate the necessary and sufficient conditions of a linear active two-port system to be conditionally stable. The first method is a new algebraic technique to prove and derive the conditional and unconditional stability criteria, and the second method is based on certain properties of Mobius (bilinear) transformations for combinations of reflection coefficients and scattering matrix of (active) two-ports. The developed framework allows formulating design requirements for the stabilization of operation of a combustion appliance via purposeful modifications of either the burner properties or/and of its acoustic embeddings. The analytical derivations have been examined in a case study to show the power of the methodology in the thermo-acoustics system application. [ABSTRACT FROM AUTHOR]

Consider an inverse problem of determining two stochastic source functions and the initial status simultaneously in a stochastic thermoelastic system, which is constituted of two stochastic equations of different types, namely a parabolic equation and a hyperbolic equation. To establish the conditional stability for such a coupling system in terms of some suitable norms revealing the stochastic property of the governed system, we first establish two Carleman estimates with regular weight function and two large parameters for stochastic parabolic equation and stochastic hyperbolic equation, respectively. By means of these two Carleman estimates, we finally prove the conditional stability for our inverse problem, provided the source in the elastic equation be known near the boundary and the solution be in an a priori bounded set. Due to the lack of information about the time derivative of wave field at the final time, the stability index with respect to the wave field at final time is found to be halved, which reveals the special characteristic of our inverse problem for the coupling system. [ABSTRACT FROM AUTHOR]

We consider the density field f(x) generated by a volume source μ(y) in D which is a domain in R3. For two disjoint segments γ, Γ1 on a straight line in R3 \ D, we establish a conditional stability estimate of Hölder type in determining f on Γ1 by data f on γ. This is a theoretical background for real-use solutions for the determination of air dose rates of radioactive substance at the human height level by high-altitude data. The proof of the stability estimate is based on the harmonic extension and the stability for line unique continuation of a harmonic function.

In this article we investigate an ill-posed problem for the system of inhomogeneous equations of parabolic type with changing the direction of time. We obtained an a priori estimate based on the given data. The theorems of uniqueness and conditional stability are proved on the set of correctness of the solution. An approximate solution to the problem is constructed by the regularization method. We calculate an estimate for efficiency of the norm of the difference between exact and approximate solutions. [ABSTRACT FROM AUTHOR]

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate. [ABSTRACT FROM AUTHOR]

In this paper, an ill-posed problem for a higher-order mixed-type differential equation is investigated on conditional correctness. A priori estimate for the solution of the problem is obtained and theorems on uniqueness and conditional stability in the corresponding set of well-posedness are proved. It is shown that the results obtained are applicable to the initial-boundary value problem for a system of mixed type equations. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity ∂ t 2 u − ∂ x 2 u + u − | u | p − 1 u + | u | q − 1 u = 0 , on [ 0 , ∞) × R , where 1 < q < p < ∞. The main result states the stability in the energy space H 1 (R) × L 2 (R) of the sums of decoupled solitary waves with different speeds, up to the natural instabilities. The proof is inspired by the techniques developed for the generalized Korteweg-de Vries equation and the nonlinear Schrödinger equation in a similar context by Martel, Merle and Tsai [14,15]. However, the adaptation of this strategy to a wave-type equation requires the introduction of a new energy functional adapted to the Lorentz transform. [ABSTRACT FROM AUTHOR]

In this article we study two inverse problems of recovering a space-time-dependent source component from the lateral boundary observation in a subdiffusion model. The mathematical model involves a Djrbashian--Caputo fractional derivative of order α ∈ (0, 1) in time, and a second-order elliptic operator with time-dependent coefficients. We establish a well-posedness and a conditional stability result for the inverse problems using a novel perturbation argument and refined regularity estimates of the associated direct problem. Further, we present a numerical algorithm for efficiently and accurately reconstructing the source component, and we provide several two-dimensional numerical results showing the feasibility of the recovery. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of determining the unknown coefficient on the right-hand side of a parabolic equation in a domain with moving boundaries. An additional condition for finding the unknown coefficient, which depends on the variable time, is given in integral form. A theorem on uniqueness and "conditional" stability of the solution is proved. [ABSTRACT FROM AUTHOR]

We consider a Cahn–Hilliard equation in a bounded domain Ω in ℝn over a time interval (0, T) and discuss the backward problem in time of determining intermediate data u(x, θ), θ ∈ (0, T), x ∈ Ω from the measurement of the final data u(x, T), x ∈ Ω. Under suitable a priori boundness assumptions on the solutions u(x, t), we prove a conditional stability estimate for the semilinear Cahn–Hilliard equation ∥u ⁢ (⋅, θ)∥ L2(Ω) ≤ C ∥u (⋅, T)∥H2(Ω) κ0, and a conditional stability estimate for the linear Cahn–Hilliard equation ∥u⁢ (⋅, θ)∥Hβ(Ω) ≤ C⁢ ∥u (⋅, T)∥ H2(Ω) κ1, where θ ∈ (0, T), β ∈ (0, 4) and κ0, κ1 ∈ (0 , 1). The proof is based on a Carleman estimate with the weight function e2seλt with large parameters s, λ ∈ ℝ+. [ABSTRACT FROM AUTHOR]

We prove a conditional stability estimate of log-type for determining unknown boundaries from a single Cauchy data taken on an accessible sub-boundary. Our approach relies on new interior and boundary estimates for elliptic equations which are derived from the Carleman estimate. Stability results for target identification of an acoustic sound-soft scatterer from one or several far-field patterns are also obtained. [ABSTRACT FROM AUTHOR]

In the paper we consider the linear inverse problem that consists in recovering the initial state in a first order evolution equation generated by a skew-adjoint operator. We studied the well-posedness of the inversion in terms of the observation operator and the spectra of the skew-adjoint generator. The stability estimate of the inversion can also be seen as a weak observability inequality. The proof of the main results is based on a new resolvent inequality and Fourier transform techniques which are of interest in themselves. [ABSTRACT FROM AUTHOR]

We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgood continuous in time. This result fills the gap between uniqueness and continuity results obtained so far. [ABSTRACT FROM AUTHOR]

The paper is concerned with weakly delayed linear discrete homogeneous planar systems with constant coefficients. By the method of Z -transform, formulas for the general solutions, dependent on the Jordan forms of the matrix of non-delayed linear terms, are derived and the influence is analyzed of the delay on the form of the general solutions. It is shown that, after several steps, the general solutions depend only on two arbitrary parameters which are linear combinations of the initial values. This property is used to prove results on conditional stability. Linear discrete homogeneous planar systems without delay are found to have the same general solutions as the initial one. The results are illustrated by examples. Previous results are analyzed, commented and improved. [ABSTRACT FROM AUTHOR]

In this work, we study non-local value problem for second order mixed type differential equation. We show this problems are ill-posed and prove the theorems of uniqueness and conditional stability of solutions in the set of correctness. [ABSTRACT FROM AUTHOR]

In this work, we study boundary value problem for second order mixed type nonhomogeneous differential equation with two degenerate lines. A priori estimate for the solution of the problem is obtained, theorems of uniqueness and conditional stability in the set of correctness are proved. [ABSTRACT FROM AUTHOR]

Weakly delayed planar linear discrete systems with multiple delays x (k + 1) = D x (k) + ∑ l = 1 n H l x (k − m l) , k = 0 , 1 , ... are considered where m 1 , m 2 , ... , m n are fixed integers, D , H 1 , ..., H n are nonzero 2 × 2 real constant matrices and x : { − m n , − m n + 1 , ... } → R 2. Formulas for general solutions are found and simplified, equivalent non-delayed planar linear discrete systems are constructed and conditional stability is analyzed. Results are illustrated by examples. • Weakly delayed discrete planar system. • Pasting of solutions. • Equivalent non-delayed system. • Conditional stability. • General solution. [ABSTRACT FROM AUTHOR]