The problem of determining the one-dimensional kernel of the electroviscoelasticity equation

We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.     

The Problem of Finding the One-Dimensional Kernel of the Thermoviscoelasticity Equation

The problem of determining the kernel h(t), t ∈ [0, T], appearing in the system of integro-differential thermoviscoelasticity equations is considered. It is assumed that the coefficients of the equations depend only on one space variable. The inverse problem is replaced by the equivalent system of integral equations for unknown functions. The contraction mapping principle with weighted norms is applied to this system in the space of continuous functions. A global unique solvability theorem is proved and an estimate of the stability of the solution of the inverse problem is obtained.     

Inverse Problem for an Integro-Differential Equation of Acoustics

We consider the hyperbolic integro-differential equation of acoustics. The direct problem is to determine the acoustic pressure created by a concentrated excitation source located at the boundary of a spatial domain from the initial boundary-value problem for this equation. For this direct problem, we study the inverse problem, which consists in determining the onedimensional kernel of the integral term from the known solution of the direct problem at the point x = 0 for t > 0. This problem reduces to solving a system of integral equations in unknown functions. The latter is solved by using the principle of contraction mapping in the space of continuous functions. The local unique solvability of the posed problem is proved.     

System of nonlinear integral equations for the unknown functions in a functional-differential hyperbolic equation

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.     

On the determination of the potential function from given orbits

The paper deals with the problem of finding the field of force that generates a given (N ? 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand’s problem. At the end we establish the relation between Bertrand’s problem and the solutions to the Heun differential equation. We illustrate our results by several examples.     

Well-posed solvability of an analytic Cauchy problem in spaces with an integral metric

We study the complex Cauchy problem for a system of linear differential equations in a class of analytic functions with an integral metric. For the case in which L_p is a weighted Lebesgue space, we obtain necessary and sufficient conditions for the local solvability of the problem.     

A Riemann-Hilbert Problem for the Moisil-Teodorescu System

In a bounded domain with smooth boundary in ?³ we consider the stationary Maxwell equations for a function u with values in ?³ subject to a nonhomogeneous condition (u, v)_x = u₀ on the boundary, where v is a given vector field and u₀ a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Determination of the spacetime from local time measurements

We consider an inverse problem for a Lorentzian spacetime (M, g), and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold \(\Sigma \subset M\) determine the \(C^\infty \)-jet of the metric in the Fermi coordinates associated to \(\Sigma \). We use this result to study the global determination of the spacetime (M, g) when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that (M, g) is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry—the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

Boundary value problems for two types of degenerate elliptic systems in <Emphasis Type="Italic">R</Emphasis><Superscript>4</Superscript>

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R ⁴ is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R ⁴ are derived.     

A global convergent derivative-free method for solving a system of non-linear equations

Finding all zeros of a system of \(m \in \mathbb {N}\) real non-linear equations in \(n \in \mathbb {N}\) variables often arises in engineering problems. Using Newtons’ iterative method is one way to solve the problem; however, the convergence order is at most two, it depends on the starting point, there must be as many equations as variables and the function F, which defines the system of nonlinear equations F(x)=0 must be at least continuously differentiable. In other words, finding all zeros under weaker conditions is in general an impossible task. In this paper, we present a global convergent derivative-free method that is capable to calculate all zeros using an appropriate Schauder base. The component functions of F are only assumed to be Lipschitz-continuous. Therefore, our method outperforms the classical counterparts.     

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Raki?, N. ?. Din?i? and D. S. Djordjevi? generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R# with a# = b if and only if (ab)* = ab, ba² = a, and ab² = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.     

Differential-geometric structure associated with Lagrangian,and its dynamic interpretation

The paper is devoted to investigation of differential-geometric structure associated with Lagrangian t depending on n functions of one variable L and their derivatives by means of Cartan–Laptev method. We construct a fundamental object of a structure associated with Lagrangian. We also construct a covector E _i (i = 1,..., n) embraced by prolonged fundamental object so that the system of equalities E _i = 0 is an invariant representation of the Euler equations for the variational functional. Due to this, there is no necessity to connect Euler equations with the variational problem. Moreover,we distinguish in an invariant way the class of special Lagrangians generating connection in the bundle of centroaffine structure over the base M. In the case when Lagrangian L is special, there exists a relative invariant Π defined on M which generates a covector field on M and fibered metric in the bundle of centroaffine structure over the base M.     

A spectral method for nonlinear elliptic equations

Let Ω be an open, simply connected, and bounded region in \(\mathbb {R}^{d}\), d ≥ 2, and assume its boundary ?Ω is smooth and homeomorphic to \(\mathbb {S}^{d-1}\). Consider solving an elliptic partial differential equation L u = f(?, u) over Ω with zero Dirichlet boundary value. The function f is a nonlinear function of the solution u. The problem is converted to an equivalent elliptic problem over the open unit ball \(\mathbb {B}^{d}\) in \(\mathbb {R}^{d}\), say \(\widetilde {L}\widetilde {u} =\widetilde {f}(\cdot ,\widetilde {u})\). Then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials \(\widetilde {u} _{n}\) of degree ≤ n that is convergent to \(\widetilde {u}\). The transformation from Ω to \(\mathbb {B}^{d}\) requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty } \left (\overline {\Omega }\right ) \) and assuming ?Ω is a C ^∞ boundary, the convergence of \(\left \Vert \widetilde {u} -\widetilde {u}_{n}\right \Vert _{H^{1}}\) to zero is faster than any power of 1/n. The error analysis uses a reformulation of the boundary value problem as an integral equation, and then it uses tools from nonlinear integral equations to analyze the numerical method. Numerical examples illustrate experimentally an exponential rate of convergence. A generalization to ?Δu + γ u = f(u) with a zero Neumann boundary condition is also presented.     

Direct and inverse problems of gas emission and the sorptive deformation of coal beds

Using the equations of state for fractured-porous media that describe the sorptioninduced deformation of coal, we develop a geomechanical model for radial gas influx to a borehole drilled in a coal bed with the concurrent evolution of stress field in the borehole environment. A numerical-and-analytical method is proposed for solving the corresponding system of equations for poroelastic media. A relation is found between the volume of slack withdrawn in the borehole (when opening up the gas-bearing seams), the sorption-and-storage capacities of coal, the permeability k, and the horizontal component σ _h of the natural stress field. We demonstrate the solvability of the inverse boundary-coefficient problem of determining k and σ _h on the basis of pressure in the closed borehole. We substantiate an express-method for estimating the permeability by the measurements of pressure in the borehole operating in the “pressure drop” mode.