We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand–Levitan integral equation for boundary coefficients only.

We present a closed-form solution to a discounted optimal stopping zero-sum game in a model based on a generalised geometric Brownian motion with coefficients depending on its running maximum and minimum processes. The optimal stopping times forming a Nash equilibrium are shown to be the first times at which the original process hits certain boundaries depending on the running values of the associated maximum and minimum processes. The proof is based on the reduction of the original game to the equivalent coupled free-boundary problem and the solution of the latter problem by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are partially determined as either unique solutions to the appropriate system of arithmetic equations or unique solutions to the appropriate first-order nonlinear ordinary differential equations. The results obtained are related to the valuation of the perpetual lookback game options with floating strikes in the appropriate diffusion-type extension of the Black–Merton–Scholes model.

In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form

\[{-}M\Big(\big(a*(g\circ |u|)\big)(1)\Big)u''(t)=\lambda f\big(t,u(t)\big),\quad t\in(0,1), \]

$g$

$M$

$M$

$f$

The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.

We establish asymptotic formulas for all the eigenvalues of the linearization problem of the Neumann problem for the scalar field equation in a finite interval

\[ \begin{cases} \varepsilon^2u_{xx}-u+u^3=0, & 0< x<1,\\ u_x(0)=u_x(1)=0. \end{cases} \]

$\varepsilon ^2u_{xx}+u-u^3=0$

$-3$

$0$

$-3$

$0$

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

This paper consists of three parts: First, letting $b_1(z)$, $b_2(z)$, $p_1(z)$ and $p_2(z)$ be nonzero polynomials such that $p_1(z)$ and $p_2(z)$ have the same degree $k\geq 1$ and distinct leading coefficients $1$ and $\alpha$, respectively, we solve entire solutions of the Tumura–Clunie type differential equation $f^{n}+P(z,\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$, where $n\geq 2$ is an integer, $P(z,\,f)$ is a differential polynomial in $f$ of degree $\leq n-1$ with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation $f''-[b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}]f=0$ and prove that $\alpha =[2(m+1)-1]/[2(m+1)]$ for some integer $m\geq 0$ if this equation admits a nontrivial solution such that $\lambda (f)<\infty$. This partially answers a question of Ishizaki. Finally, letting $b_2\not =0$ and $b_3$ be constants and $l$ and $s$ be relatively prime integers such that $l> s\geq 1$, we prove that $l=2$ if the equation $f''-(e^{lz}+b_2e^{sz}+b_3)f=0$ admits two linearly independent solutions $f_1$ and $f_2$ such that $\max \{\lambda (f_1),\,\lambda (f_2)\}<\infty$. In particular, we precisely characterize all solutions such that $\lambda (f)<\infty$ when $l=2$ and $l=4$.

In this contribution, we present a modelling and simulation framework for parametrised lithium-ion battery cells. We first derive a continuum model for a rather general intercalation battery cell on the basis of non-equilibrium thermodynamics. In order to efficiently evaluate the resulting parameterised non-linear system of partial differential equations, the reduced basis method is employed. The reduced basis method is a model order reduction technique on the basis of an incremental hierarchical approximate proper orthogonal decomposition approach and empirical operator interpolation. The modelling framework is particularly well suited to investigate and quantify degradation effects of battery cells. Several numerical experiments are given to demonstrate the scope and efficiency of the modelling framework.

In this paper, we study the Friedrichs extensions of Sturm–Liouville operators with complex coefficients according to the classification of B. M. Brown et al. [3]. We characterize the Friedrichs extensions both by boundary conditions at regular endpoint and asymptotic behaviours of elements in the maximal operator domains at singular endpoint. Some of spectral properties are also involved.

We show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$. This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as

$$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$

$\boldsymbol {A}_0$

$\Theta $

$\mathbb {C}^d$

$\boldsymbol {A}_0$

$\mathcal {H}_{-1}(\boldsymbol {A}_0)$

$\Theta $

The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.

As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

We consider steady states with mass constraint of the fourth-order thin-film equation with van der Waals force in a bounded domain which leads to a singular elliptic equation for the thickness with an unknown pressure term. By studying second-order nonlinear ordinary differential equation,

\begin{equation*}h_{rr}+\frac{1}{r}h_{r}=\frac{1}{\alpha}h^{-\alpha}-p\end{equation*}

We develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance.

In this paper, we characterize jump phenomena of the $n$-th eigenvalue of self-adjoint discrete Sturm–Liouville problems in any dimension. For a fixed Sturm–Liouville equation, we completely characterize jump phenomena of the $n$-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in Hu et al. (2019, J. Differ. Equ. 266, 4106–4136) and Kong et al. (1999, J. Differ. Equ. 156, 328–354), the jump set here is involved with coefficients of the Sturm–Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing the jump areas. We study the singularity by partitioning and analysing the local coordinate systems, and provide a Hermitian matrix which can determine the areas’ division. To prove the asymptotic behaviour of the $n$-th eigenvalue, we generalize the method developed in Zhu and Shi (2016, J. Differ. Equ. 260, 5987–6016) to any dimension. As an application, by transforming the continuous Sturm–Liouville problem of Atkinson type to a discrete one, we determine the number of eigenvalues and obtain complete characterization of jump phenomena of the $n$-th eigenvalue for the Atkinson type.

This paper aims to investigate the existence of periodic solutions for $p$-Laplacian differential equations with jumping nonlinearity under the frame of half-eigenvalue. Based on the continuity theorem, some new results are obtained, which enrich and generalize the previous results.

We consider inverse nodal problems for the Sturm–Liouville operators on the tree graphs. Can only dense nodes distinguish the tree graphs? In this paper it is shown that the data of dense-nodes uniquely determines the potential (up to a constant) on the tree graphs. This provides interesting results for an open question implied in the paper.

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

The existence and multiplicity of T-periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree.