Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any $t>0,$ the density (with respect to the $(d+1)$-dimensional Lebesgue measure) of the pair $\big(M_t,X_t\big)$ is a weak solution of a Fokker–Planck partial differential equation on the closed set $\big\{(m,x)\in \mathbb{R}^{d+1},\,{m\geq x^1}\big\},$ using an integral expansion of this density.

In this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis–Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.

We present a Hopf boundary point lemma for the difference between two Hölder continuously differentiable functions, each weak solutions to a divergence-form quasilinear equation, under mild boundedness assumptions on the coefficients of this equation.

We give a Hopf boundary point lemma for weak solutions of linear divergence form uniformly elliptic equations, with Hölder continuous top-order coefficients and lower-order coefficients in a Morrey space.

Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

This paper gives an equivalent form of Picard’s theorem via entire solutions of the functional equation ${{f}^{2}}\,+\,{{g}^{2}}\,=\,1$ and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

Adaptive numerical methods for solving partial differential equations (PDEs) that control the movement of grid points are called moving mesh methods. In this paper, these methods are examined in the case where a separate PDE, that depends on a monitor function, controls the behavior of the mesh. This results in a system of PDEs: one controlling the mesh and another solving the physical problem that is of interest. For a class of monitor functions resembling the arc length monitor, a trade off between computational efficiency in solving the moving mesh system and the accuracy level of the solution to the physical PDE is demonstrated. This accuracy is measured in the density of mesh points in the desired portion of the domain where the function has steep gradient. The balance of computational efficiency versus accuracy is illustrated numerically with both the arc length monitor and a monitor that minimizes certain interpolation errors. Physical solutions with steep gradients in small portions of their domain are considered for both the analysis and the computations.

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only.

We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions that are applied in order to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations.

For 0 < α ≤ 2 and 0 < H < 1, anα-time fractional Brownian motion is an iterated processZ =  {Z(t) = W(Y(t)), t ≥ 0} obtained by taking a fractional Brownian motion {W(t), t ∈ ℝ} with Hurst index0 < H < 1 and replacing the time parameter with astrictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that suchprocesses have natural connections to partial differential equations and, whenY is a stable subordinator, can arise as scaling limit of randomlyindexed random walks. The existence, joint continuity and sharp Hölder conditions in theset variable of the local times of a d-dimensionalα-time fractional Brownian motionX = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ...,Xd(t)), where t ≥ 0 andX1, ..., Xdare independent copies of Z, are investigated. Our methods rely on thestrong local nondeterminism of fractional Brownian motion.

A scheme is developed to study numerical solution of the time-fractional shock wave equation and wave equation under initial conditions by the homotopy perturbation method (HPM). The fractional derivatives are taken in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical results are illustrated through the graph.

In this paper we consider an initial-value problem for the nonlinear fourth-order partial differential equation ut+uux+γuxxxx=0, −∞<x<∞, t>0, where x and t represent dimensionless distance and time respectively and γ is a negative constant. In particular, we consider the case when the initial data has a discontinuous expansive step so that u(x,0)=u0(>0) for x≥0 and u(x,0)=0 for x<0. The method of matched asymptotic expansions is used to obtain the large-time asymptotic structure of the solution to this problem which exhibits the formation of an expansion wave. Whilst most physical applications of this type of equation have γ>0, our calculations show how it is possible to infer the large-time structure of a whole family of solutions for a range of related equations.

A generalization of Bailey's general epidemic model is considered. In this generalized model, it is assumed that the probability of any particular susceptible becoming infected during the small time interval (t, t + Δt) is α(X(t))Δt + o(Δt), for some function a, where X(t) is the proportion of infected individuals in the entire population, the probability that an infected individual is infected for at least a length of time t is F(t), and recovered individuals are permanently immune from further attack. In this paper, central limit theorems are obtained for the proportion of infected individuals and the proportion of susceptibles in the entire population.

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.

A technique is derived providing for the analysis of busy period processes in the queueing models M/Mn/1 and Mn/M/1; queueing models with state dependent service and arrival rates respectively. A single treatment covers both models and no separate analysis is required.