Narrow Results

In this paper, we solve explicitly and analyze rigorously inhomogeneous initial-boundary-value problems (IBVP) for several fourth-order variations of the traditional diffusion equation and the associated linearized Cahn–Hilliard (C-H) model (also Kuramoto–Sivashinsky equation), formulated in the spatiotemporal quarter-plane. Such models are of relevance to heat-mass transfer phenomena, solid-fluid dynamics and the applied sciences. In particular, we derive formally effective solution representations, justifying a posteriori their validity. This includes the reconstruction of the prescribed initial and boundary data, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula is utilized to rigorously deduce the solution’s regularity and asymptotic properties near the boundaries of the domain, including uniform convergence, eventual (long-time) periodicity under (eventually) periodic boundary conditions, and null noncontrollability. Importantly, this analysis is indispensable for exploring the (non)uniqueness of the problem’s solution and a new counter-example is constructed. Our work is based on the synergy between: (i) the well-known Fokas unified transform method and (ii) a new approach recently introduced for the rigorous analysis of the Fokas method and for investigating qualitative properties of linear evolution partial differential equations (PDE) on semi-infinite strips. Since only up to third-order evolution PDE have been investigated within this novel framework to date, we present our analysis and results in an illustrative manner and in order of progressively greater complexity, for the convenience of readers. The solution formulae established herein are expected to find utility in well-posedness and asymptotics studies for nonlinear counterparts too.

This research addresses a multi-item newsvendor problem with an equality resource constraint. We focus on retail industries with insufficient development power, where shelves must be filled with a variety of goods at the beginning of every period to make the selling space attractive. Technically, the total occupation volume of the goods must be adjusted to the shelf space. Using resource terminology, these elements correspond to resource consumption and resource capacity, respectively. Thus, we consider a constraint in which the total resource consumption is adjusted to a given resource capacity. We derive explicit solutions for several scenarios regarding resource capacity. These give important managerial implications that are helpful for both managers and front-line workers. We also consider two approaches for obtaining an approximate optimal solution for an arbitrary resource capacity, one of which can be implemented with ease in an ordering support system.

With the aid of symbolic computation, we use the Lie group method to find the Lie point symmetries of (3+1)-dimensional Burgers system, and reduce the system with the obtained symmetries. As a result, many kinds of new symmetry reductions have been presented. And some explicit solutions are given due to the Riccati equation method.

The novel coupling Benjamin–Ono system is obtained with the dark parameterization approach.By solving these coupling equations, three typical explicit solutions including the traveling wave solution are constructed through the special property of the anyon-$n$ algebra. Several novel types of localized excitations are depicted with some solutions.

A generalized $3\times 3$ matrix spectral problem is investigated to generate two five-component nonlinear integrable systems, which involve an arbitrary smooth function. These systems are proven integrable in the sense of Lax pair. As the reduction cases, a four-component reaction diffusion equation and a four-component modified Korteweg-de Vries (mKdV) equation are solved by Darboux transformation approach.

In this work, we study the modified Volterra lattice. Applying the gauge transformation of the associated $2\times 2$ matrix spectral problems, we establish the N-fold Darboux transformation (DT), and then construct a few explicit solutions in terms of determinants upon using the obtained DT. Moreover, all the results are illustrated by the graphs of the solitonic evolution profiles of the aforementioned solutions. Finally, infinitely many conservation laws for the modified Volterra lattice are proposed. The obtained results of this research might be applied to the research on nonlinear phenomena in physics or engineering areas.

A new coupled Burgers equation and a new coupled KdV equation which are associated with $3\times 3$ matrix spectial problem are investigated for complete integrability and covariant property. For integrability, Lax pair and conservation laws of the two new coupled equations with four potentials are established. For covariant property, Darboux transformation (DT) is used to construct explicit solutions of the two new coupled equations.

New simulation approaches to evaluating path-dependent options without matrix inversion issues nor Euler bias are evaluated. They employ three main contributions: (1) stochastic approximation replaces regression in the LSM algorithm; (2) explicit weak solutions to stochastic differential equations are developed and applied to Heston model simulation; and (3) importance sampling expands these explicit solutions. The approach complements Heston [(1993) A closed-form solutions for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies6, 327–343] and Broadie & Kaya [(2006) Exact simulation of stochastic volatility and other affine jump diffusion processes, Operations Research54 (2), 217–231] by handling the case of path-dependence in the option’s execution strategy. Numeric comparison against standard Monte Carlo methods demonstrates up to two orders of magnitude speed improvement. The general ideas will extend beyond the important Heston setting.

In this paper, explicit local buckling analysis of orthotropic plates subjected to uniaxial compression with two loaded edges simply-supported and two unloaded edges supported by combined vertical and rotational restraining springs is presented. Based on the total potential energy function, the eigenvalue problem is formulated by treating the buckled shape functions as the admissible functions that satisfy the boundary conditions of the rectangular plates. Closed-form and approximate local buckling solutions of the combined rotationally- and vertically-restrained orthotropic plates, as well as explicit formulas for the critical buckling load and critical aspect ratio under the uniform compression, are obtained. By adjusting the stiffness of the rotational and vertical restraining springs, explicit local buckling solutions are established for eight simple cases of boundary conditions. To verify the explicit solutions, numerical analyses of orthotropic plates using the exact transcendental and finite element methods are conducted, for which reasonable agreement has been obtained between the explicit and numerical solutions, particularly for the simplified cases. The explicit solution obtained in this study can be used to facilitate the buckling analysis of composite laminated structures with different boundary conditions or joint connections as parts of stiffened and thin-walled structures by treating them as discrete plates with restrained boundary conditions.

Thermoelastic damping (TED) has been discerned as a definite source of intrinsic energy loss in miniaturized mechanical elements. The size-dependent structural and thermal behavior of these small-sized structures has been proven through experimental observations. As a first attempt, this article exploits nonlocal strain gradient theory (NSGT) and nonlocal dual-phase-lag (NDPL) heat conduction model simultaneously to acquire a mathematical formulation and analytical solution for TED in nanobeams that can accommodate size effect into both structural and heat transfer fields. For this purpose, the coupled equations of motion and heat conduction are firstly extracted via NSGT and NDPL model. Next, by deriving the distribution of temperature from heat conduction equation and substituting it in the motion equation, the unconventional thermoelastic frequency equation is established. By deriving the real and imaginary parts of the frequency from this equation and employing the definition of quality factor, an explicit solution is given for approximating TED value. The veracity of the proposed model is checked by comparing it with the solutions reported in the literature for specific and simpler cases. A diverse set of numerical results is then presented to appraise the influence of some factors like structural and thermal nonlocal parameters, strain gradient length scale parameter, geometrical parameters, mode number and material on the amount of TED. According to the results, use of NDPL model yields a smaller value for TED than DPL model, but prediction of NSGT about the magnitude of TED, in addition to the relative amounts of its two scale parameters, strongly depend on other factors such as aspect ratio, vibration mode and material type.

In this letter the explicit form of the evolution operator of the Tavis–Cummings model with three and four atoms is given. This is an important progress in quantum optics or mathematical physics.

Shape-memory alloy (SMA) structures, either dense or porous, are widely used because of their functional characteristics arising from shape-memory effect and pseudoelasticity. Hence, the precise modeling of these materials is essential due to design issues. In this study, the solution algorithm based on an explicit phenomenological constitutive model is investigated to analyze pressurized cylinders. The proposed method is extracted following return mapping scheme to find the evolution of phase transformation in these structures. The formulation is validated by comparing uniaxial tension–compression curves during loading, unloading and thermal actuation against published literature. By the aid of this model, assuming elastic and phase transformation strains, a comprehensive study is presented to inspect stress, strain and displacement fields along the thickness of internally and externally pressurized thick-walled cylinders. To this end, the governing nonlinear equation in elastic–inelastic zones is discretized by the efficient method of generalized differential quadrature (GDQ) and the resultant relations are solved numerically. The influences of internal and external pressure on stress distribution along the cylinder wall during loading–unloading for different temperatures including phase transformation and saturation states are discussed. Also, the closed-form solution for critical pressure at the onset of the phase transformation is studied.

The Black-Merton-Scholes equation plays a fundamental role in the option pricing theory. Our main purpose is to derive an explicit formula for its solution, using simple tools from operator semigroups. The paper includes also an expository treatment of how the equation arises.