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In this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type. [ABSTRACT FROM AUTHOR]

The principal result of this paper is a reformulation of Weyl–Titchmarsh theory for (three-coefficient) regular and singular Sturm–Liouville operators for separated and coupled self-adjoint boundary conditions (if any) in terms of the (diagonal) Green's function and some of its (mixed) first quasi-derivatives. In particular, this Green's function approach now unifies the treatment of separated and coupled boundary conditions (the treatment of the latter appears to be new in this context). [ABSTRACT FROM AUTHOR]

In this paper we provide conditions under which the hitting-time problem for Brownian motion is equivalent to solving a heat equation with moving boundary and distributional initial conditions. Motivated by the hitting-time problem, we study the heat equation with absorbing moving boundaries. Using Fourier analysis we develop a procedure to solve this problem for a family of curves that includes the square root, quadratic, and cubic boundaries. As an application of our results, and using Sturm-Liouvile theory, we compute the density of the hitting time of a Brownian motion to a family of quadratic boundaries. [ABSTRACT FROM AUTHOR]

A theorem on the maximum regularity of solutions of the nonlinear Sturm–Liouville equation with greatly growing and rapidly oscillating potential in the space L2(ℝ)(ℝ=(−∞,∞))$$ {L}_2\left(\mathrm{\mathbb{R}}\right)\kern0.1em \left(\mathrm{\mathbb{R}}=\left(-\infty, \infty \right)\right) $$ is proved in this paper. Two‐sided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear Sturm–Liouville equation are also obtained. As is known, the obtained estimates give the opportunity to choose approximation apparatus that guarantees the minimum possible error. [ABSTRACT FROM AUTHOR]

In this study, we consider a boundary value problem generated by the Sturm-Liouville equation with a frozen argument and with non-separated boundary conditions on a time scale. Firstly, we present some solutions and the characteristic function of the problem on an arbitrary bounded time scale. Secondly, we prove some properties of eigenvalues and obtain a formulation for the eigenvalues-number on a finite time scale. Finally, we give an asymptotic formula for eigenvalues of the problem on another special time scale: T = [α, δ1] ∪ [δ2, β]. [ABSTRACT FROM AUTHOR]

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ∫ a b f (x) d μ (x) = ∑ i = 1 n w i f (x i) where f belongs to H 1 (μ). Here, μ belongs to a class of continuous probability distributions on [ a , b ] ⊂ R and ∑ i = 1 n w i δ x i is a discrete probability distribution on [ a , b ]. We show that H 1 (μ) is a reproducing kernel Hilbert space with a continuous kernel K , which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although K has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a T -system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature. We derive several results for the Poincaré quadrature weights and the associated worst-case error. When μ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as b − a 2 3 n − 1 for large n. By comparison with known results for H 1 (0 , 1) , this shows that the Poincaré quadrature is asymptotically optimal. For a general μ , we provide an efficient numerical procedure, based on finite elements and linear programming. Numerical experiments provide useful insights: nodes are nearly evenly spaced, weights are close to the probability density at nodes, and the worst-case error is approximately O (n − 1) for large n. [ABSTRACT FROM AUTHOR]

In this work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations. [ABSTRACT FROM AUTHOR]

This paper is on the solution of a nonhomogeneous Sturm-Liouville fuzzy problem with fuzzy forcing function. The problem is studied by a different solution method. Example is solved. [ABSTRACT FROM AUTHOR]

The dynamic characteristics of rod fastening rotor supported by a finite journal bearing are investigated in this study. To model the dynamic behaviors of the bearing-rotor system, the oil film force of bearing is calculated by approximately solving the Reynolds equation with the variables separation method and Sturm–Liouville theory, and then a motion equation is developed with consideration of the contact and gyro effects of the disks of the rotor. To solve the motion equation with small error and excellent stability, an improved Newmark method is proposed. On this basis, the dynamics characteristics of the rod fastening rotor are analyzed for different rotor speeds, disk eccentricities, shaft bearing stiffness, and contact stiffness. And the orbits of the rod fastening rotor and integral rotor are compared. The numerical results indicate that the analytical solution of the oil film force has higher computational efficiency than the finite difference method. The rod fastening rotor shows higher stability than the integral rotor, and exhibits rich dynamic behaviors, such as periodic, qusi-periodic, period-2, period-4, and period-6. [ABSTRACT FROM AUTHOR]

In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.

The theoretical bases of this paper are the theory of spectral analysis and the theory of singular and regular perturbations. We obtain an approximate price of Ornstein-Uhlenbeck double barrier options with multidimensional stochastic diffusion as expansion in eigenfunctions using infinitesimal generators of a $(l+r+1)$-dimensional diffusion in Hilbert spaces. The theorem of accuracy estimation of options prices approximation is established. We also obtain explicit formulas for derivatives price based on the expansion in eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations.

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively.

New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form −u″+ℓ(ℓ+1)x2+q(x)u=ω2u are obtained. The integral kernel is represented as a Fourier‐Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to ω. For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones previously obtained by the authors for large values of ω and ℓ and for noninteger values of ℓ. The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley‐Wiener theorem, and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy. [ABSTRACT FROM AUTHOR]

This study aimed at presenting a steady-state analytical solution for the two-dimensional heat conduction in a cylindrical segment made of functionally graded materials. It is acquired by taking advantage of the Fourier transform and separation of variables rather than numerical methods. Sturm–Liouville theory is employed to find the proper and adequate Fourier transformation. Continuous variations along the radial and circumferential directions based on the power-law function are taken into account, and non-homogeneous boundary conditions are applied to the problem. The obtained formulation is verified by the available solutions. Through solving an illustrative example, the temperature distribution is deliberated for a combination of boundary conditions. It is to be emphasized that mathematical robustness and generality of the solution are its primary advantage which is not often seen in the previously published literature. [ABSTRACT FROM AUTHOR]

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of Röhrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected boundary conditions are also eliminated. As partial data, we take two spectra, the set of the jth elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth continuous, and noncontinuous, respectively. [ABSTRACT FROM AUTHOR]

There is c > 0 such that for all f ∈ C [ 0 , π ] with at most d − 1 roots inside (0 , π) ∑ 1 ≤ n ≤ d | 〈 f , sin ⁡ (n x) 〉 | ≥ κ − κ 2 log ⁡ κ ‖ f ‖ L 2 where κ = c ‖ ∇ f ‖ L 2 ‖ f ‖ L 2 . This quantifies the Sturm-Hurwitz Theorem and connects a purely topological condition (number of roots) to the Fourier spectrum. It is also one of few estimates on Fourier coefficients from below. The result holds more generally for eigenfunctions of regular Sturm-Liouville problems − (p (x) y ′ (x)) ′ + q (x) y (x) = λ w (x) y (x) on (a , b). Sturm-Liouville theory shows the existence of a sequence of solutions (ϕ n) n = 1 ∞ that form an orthogonal basis of L 2 (a , b) with respect to w (x) d x. Sturm himself proved that if f : (a , b) → R is a finite linear combinations of ϕ n having d − 1 roots inside (a , b) , then f cannot be orthogonal to A = span { ϕ 1 , ... , ϕ d }. We prove a lower bound on the size of the projection ‖ π A f ‖. [ABSTRACT FROM AUTHOR]

In this study, we consider a boundary value problem generated by a second‐order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem. [ABSTRACT FROM AUTHOR]

This paper introduces a systematic study for analytic aspects of discrete spectra methods for functions supported on some compact domains of S E (2) , according to Sturm–Liouville theory. We then apply these discrete spectra methods to approximate convolution of functions supported on these compact domains. We shall also investigate different aspects of the presented theory in the cases of zero-value boundary condition and derivative boundary condition. The paper is concluded by some Plancherel formulas associated to Sturm–Liouville theory and special cases of boundary conditions. [ABSTRACT FROM AUTHOR]

We construct a one-parameter family of Laplacians on the Sierpiński gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the fact that all these Laplacians satisfy a version of spectral decimation that builds a precise catalog of eigenvalues and eigenfunctions for any choice of the parameter. We give a number of applications of this spectral decimation. We also prove analogous results for fractal Laplacians on the unit interval, and this yields an analogue of the classical Sturm-Liouville theory for the eigenfunctions of these one-dimensional Laplacians. [ABSTRACT FROM AUTHOR]

We consider the normal form of Sturm-Liouville differential equations with separable boundary conditions. For this problem we know that the potential function is determined uniquely by two spectra and that if the potential is symmetric, then it is determined uniquely by just one spectrum. In this paper firstly we generalize symmetric potential and then investigate change of needed data to determine potential function uniquely.

Highlights • New orthogonal UWB pulses. • Sturm–Liouville theory. • High utilization efficiency. Abstract The problem of designing UWB pulses which meet specific spectrum requirements is usually treated by filtering common pulses such as Gaussian doublets, modified Hermite polynomials and wavelets. When there is the need to have a number of orthogonal pulses (e.g. , in a multiuser scenario), a naive approach is to filter all the members of an orthogonal set, which is likely to destroy their orthogonality property. In this paper, we study the design of a set of pulses that simultaneously satisfy the orthogonality property and spectrum requirements. Our design is based on the eigenfunctions of Sturm–Liouville boundary value problems. Indeed, we introduce Sturm–Liouville differential equations for which the eigenfunctions meet the FCC mask constraints. Computer simulation results show that all such waveforms occupy almost 55% of the allowed spectrum (utilization efficiency). A comparison of the proposed method with some conventional techniques of orthogonal UWB pulse generation will demonstrate the advantages of the new proposal. [ABSTRACT FROM AUTHOR]

A method for solving the inverse scattering problem on the line is proposed. It is based on a Fourier‐Laguerre series representation of the integral transmutation kernel. Substitution of the representation into the Gel'fand‐Levitan‐Marchenko equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential. [ABSTRACT FROM AUTHOR]

We analyze two weak random operators, initially motivated from processes in random environment. At first glance, these operators are ill-defined, but using bilinear forms, one can deal with them in a rigorous way. This point of view can be found, for instance, in [A. V. Skorohod, Random Linear Operators, Math. Appl. (Sov. Ser.), D. Reidel Publishing, Dordrecht, 1984], and it remarkably helps to carry out specific calculations. In this paper, we find explicitly the inverse of such weak operators by providing the closed forms of the so-called Green kernel. We show how this approach helps to analyze the spectra of the operators. In addition, we provide the existence of strong operators associated to our bilinear forms. Important tools that we use are the Sturm–Liouville theory and the stochastic calculus. [ABSTRACT FROM AUTHOR]

The paper considers the nonlinear eigenvalue problem for the equation y″(x)=λ-α|y(x)|2qy(x) with boundary conditions y(0)=y(h)=0 and y′(0)=p, where α, q, and p are positive constants, λ is a real spectral parameter. It is proved that the nonlinear problem has infinitely many isolated negative as well as positive eigenvalues, whereas the corresponding linear problem (for α=0) has only an infinite number of negative eigenvalues. Negative eigenvalues of the nonlinear problem reduce to the solutions to the corresponding linear problem as α→+0; positive 'nonlinear' eigenvalues are nonperturbative. Asymptotical inequalities for the eigenvalues are found. Periodicity of the eigenfunctions is proved and the period is found, zeros of the eigenfunctions are determined, and a comparison theorem is proved. [ABSTRACT FROM AUTHOR]

Highlights • A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves is proposed. • The fluid film forces calculated by the proposed method agree well with the results obtained by the finite-difference method. • The method facilitates the accurate calculation of the fluid film force of a journal bearing and saves computational costs. • The effects of the bearing parameters on the nonlinear fluid film forces are analyzed. Abstract A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves under the cavitation boundary condition is proposed. The pressure distribution of the Reynolds equation of a finitely long journal bearing with axial grooves is expressed as a particular solution and a homogeneous solution. The particular solution and the homogeneous solution are separated, respectively, into an additive form and a multiplicative form by the method of separation of variables. The circumferential separable function of the homogeneous solution can be expanded on the basis of the infinite series of trigonometric functions. The pressure distribution of the particular solution is obtained by the Sommerfeld transformation. The termination positions of the fluid film are determined by the continuity condition. The analytical expressions for the nonlinear fluid film forces of a finitely long journal bearing with two axial grooves are obtained. The fluid film forces calculated by the proposed method agree well with the results obtained by the finite-difference method. The effects of the bearing parameters on the nonlinear fluid film forces are analyzed. [ABSTRACT FROM AUTHOR]

Recent studies have suggested that it is feasible to recover a physical measure from a risk-neutral measure. Given a market state variable modeled as a Markov process, the key concept is to extract a unique positive eigenfunction of the generator of the Markov process. In this work, the feasibility of this recovery theory is examined. We prove that, under a restrictive integrability condition, recovery is feasible if and only if both endpoints of the state variable are limit-point. Several examples with explicit positive eigenfunctions are considered. However, in general, a physical measure cannot be recovered from a risk-neutral measure. We provide a financial and mathematical rationale for such recovery failure.

The problem of natural oscillations of a rotating rod of variable cross section is considered. It is believed that the rod is rigidly attached at one end to a rotor with a constant speed orthogonal to the axis of rotation, the other end is assumed to be free. Flexural movements occur in the plane of rotation or perpendicular to it. To determine the natural oscillations, the Euler rod oscillation model is used, taking into account, in addition to the tensile force, the bond reaction force caused by the movements of the neutral axis. Using an original numerical-analytical method, the lower frequencies were calculated for the power and exponential laws of the cross section change. [ABSTRACT FROM AUTHOR]

The theoretical bases of this paper are the theory of spectral analysis and the theory of singular and regular perturbations. We obtain an approximate price of Ornstein-Uhlenbeck double barrier options with multidimensional stochastic diffusion as expansion in eigenfunctions using infinitesimal generators of a (l + r + 1)-dimensional diffusion in Hilbert spaces. The theorem of accuracy estimation of options prices approximation is established. We also obtain explicit formulas for derivatives price based on the expansion in eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new extensions of the Jacobi and Laguerre polynomials as the eigenfunctions of two nonclassical Sturm-Liouville problems. We prove some important properties of these operators such as: These sets of functions are orthogonal with respect to a positive definite inner product defined over the compact intervals [-1, 1] and [0, ∞), respectively and also these sequences form two new orthogonal bases for the corresponding Hilbert spaces. Finally, the spectral and Rayleigh-Ritz methods are carry out using these basis functions to solve some examples. Our numerical results are compared with other existing results to confirm the efficiency and accuracy of our method. [ABSTRACT FROM AUTHOR]

We reveal some geometric properties of energy minimal diffeomorphisms defined on an annulus, whose existence was established in works by Iwaniec et al. (2011) and Kalaj (2014). [ABSTRACT FROM AUTHOR]

We reveal some geometric properties of energy minimal diffeomorphisms defined on an annulus, whose existence was established in works by Iwaniec et al. (2011) and Kalaj (2014). [ABSTRACT FROM AUTHOR]