Theory Of Differential Equations Research Articles

Optimal investment in defined contribution pension schemes with forward utility preferences

Optimal investment strategies of an individual worker during the accumulation phase in the defined contribution pension scheme have been well studied in the literature. Most of them adopted the classical backward model and approach, but any pre-specifications of retirement time, preferences, and market environment models do not often hold in such a prolonged horizon of the pension scheme. Pre-commitment to ensure the time-consistency of an optimal investment strategy derived from the backward model and approach leads the supposedly optimal strategy to be sub-optimal in the actual realizations. This paper revisits the optimal investment problem for the worker during the accumulation phase in the defined contribution pension scheme, via the forward preferences, in which an environment-adapting strategy is able to hold optimality and time-consistency together. Stochastic partial differential equation representation for the worker's forward preferences is illustrated. This paper constructs two of the forward utility preferences and solves the corresponding optimal investment strategies, in the cases of initial power and exponential utility functions.

On the periodic and antiperiodic aspects of the Floquet normal form

Floquet's Theorem is a celebrated result in the theory of ordinary differential equations. Essentially, the theorem states that, when studying a linear differential system with T-periodic coefficients, we can apply a, possibly complex, T-periodic change of variables that transforms it into a linear system with constant coefficients. In this paper, we explore further the question of the nature of this change of variables. We state necessary and sufficient conditions for it to be real and T-periodic. Failing those conditions, we prove that we can still find a real change of variables that is “partially” T-periodic and “partially” T-antiperiodic. We also present applications of this new form of Floquet's Theorem to the study of the behavior of solutions of nonlinear differential systems near periodic orbits.

First-Stage Dynamics of the Immune System and Cancer

The innate immune system is the first line of defense against pathogens. Its composition includes barriers, mucus, and other substances as well as phagocytic and other cells. The purpose of the present paper is to compare tissues with regard to their immune response to infections and to cancer. Simple ideas and the qualitative theory of differential equations are used along with general principles such as the minimization of the pathogen load and economy of resources. In the simplest linear model, the annihilation rate of pathogens in any tissue should be greater than the pathogen’s average replication rate. When nonlinearities are added, a stability condition emerges, which relates the strength of regular threats, barrier height, and annihilation rate. The stability condition allows for a comparison of immunity in different tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response. The way the tissue responds to an infection shows a correlation with the way it responds to cancer. The results of this paper are formulated in the form of precise statements in such a way that they could be checked by present-day quantitative immunology.

Problem of Determining the Density of Sources in a Multidimensional Heat Equation with the Caputo Time Fractional Derivative

In this paper, we propose a new formula for representing the solution of the third initial-boundary value problem for multidimensional fractional heat equation with the Caputo derivative. This formula is obtained by the continuation method used in the theory of partial differential equations with integer derivatives. The Green’s function of the problem is also constructed in terms of the Fox H- function. Involving the results of solving a direct problem and the overdetermination condition, a uniqueness theorem for the definition of the spatial part of the multidimensional source function is proved.

Optimal control strategies for water, sanitation, and hygiene in mitigating spread of waterborne diseases

Improving access to water, sanitation and hygiene (WASH) can help to eliminate the root cause of waterborne disease transmission. WASH has the potential to function as a sustained and effective mechanism for controlling enteric diarrheal disease (EDD) over the long term. To address this question, we formulate a mathematical model to study the impact of WASH services on reducing the transmission of waterborne diseases. The model is analyzed using the stability theory of differential equations in order to determine the threshold condition in disease control. We utilize optimal control theory to find the right balance in WASH interventions, ensuring a decrease in waterborne disease over a specific planning period. The total number of EDD cases is minimized when there are specific financial restrictions. To analyze the outcomes, we employed numerical simulations and conducted a sensitivity analysis on different weights in the cost index. Furthermore, we compared the numerical simulations of the optimal solutions and performed multiple regression analyses to gain insights into the impact of control strategies. Our findings reveal that wastewater and sewage treatment (WST) control has the most significant impact. When multiple controls are necessary, WST and drinking water supply (DWS) control provide the most effective combination. These results have significant implications for controlling infectious diseases, especially in developing countries where budgets are typically limited. Implementing these strategies could potentially reduce disease transmission and improve public health outcomes.

Application of Lyapunov–Poincaré method of small parameter for Nash and Berge equilibrium designing in one differential two-player game

The Poincaré small parameter method is actively used in celestial mechanics, as well as in the theory of differential equations and in its important section called optimal control. In this paper, the mentioned method is used to construct an explicit form of Nash and Berge equilibrium in a differential positional game with a small influence of one of the players on the rate of change of the state vector.

On the solutions of universal differential equation

The objective of this work is to give expressions for solutions of universal differential equation and finally, as application, provides the unique grouplike solution, by dévissage, of Knizhnik-Zamolodchikov equation satisfying asymptotic conditions, i.e. solution of KZn can be obtained using solution of KZ n−1 and the noncommutative generating series of hyperlogarithms. These expressions are obtained using the algebraic combinatorics on noncommutative formal series with holomorphic coefficients and a Picard-Vessiot theory of noncommutative differential equations.

Mean-variance reinsurance and asset liability management with common shock via non-Markovian stochastic factors

This paper explores a reinsurance and asset liability management problem in a factor model with dependent risks, considering economic factors as non-Markovian processes impacting the appreciation rate, volatility, and value of liabilities. It encompasses various stochastic volatility (SV) models like Hull-White, Heston, 3/2, and 4/2 models as special cases. The model addresses dependent risks through a common shock in claim number processes. The insurer's aim is to minimize variance in terminal net wealth while achieving a certain expected terminal net wealth, balancing reinsurance, new business, and asset allocation. The paper employs linear-quadratic (LQ) control and backward stochastic differential equation (BSDE) theory to derive both the efficient strategy and the efficient frontier. To illustrate our results, numerical examples are provided in two special cases, the 3/2 and 4/2 SV models.

Professor Alexandru Șubă on his 70th birthday

Alexandru Șubă is University Professor, Doctor Habilitatus in Mathematical and Physical Sciences. He is a Moldovan mathematician and a remarkable leader of the Moldovan School of Differential Equations, who contributed a lot to the qualitative theory of differential equations and to the education of new generations of highly-qualified specialists. On December 2, 2023, Professor Alexandru Șubă celebrated his 70th anniversary.

Professor Alexandru Șubă on his 70th birthday

Alexandru Șubă is University Professor, Doctor Habilitatus in Mathematical and Physical Sciences. He is a Moldovan mathematician and a remarkable leader of the Moldovan School of Differential Equations, who contributed a lot to the qualitative theory of differential equations and to the education of new generations of highly-qualified specialists. On December 2, 2023, Professor Alexandru Șubă celebrated his 70th anniversary.

Approximation of curve-based sleeve functions in high dimensions

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function—the profile—is recovered, and second, the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximates the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.

Limit Directions of Julia Sets of Entire Functions and Complex Differential Equations

The limiting directions of Julia sets of infinite order entire functions are studied by combining the theory of complex dynamic system and the theory of complex differential equations, in which the lower bound of the measure of limiting direction of Julia set of entire solutions of complex differential equations is obtained.

Reproducing kernel Banach space defined by the minimal norm property and applications to partial differential equation theory

Abstract It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any z ∈ U {z\in U} , in the set V z := { f ∈ V ∣ f ⁢ ( z ) = 1 } {V_{z}:=\{f\in V\mid f(z)=1\}} , if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality K ⁢ ( z , w ) = K ⁢ ( w , z ) ¯ {K(z,w)=\overline{K(w,z)}} does not have to be true without assumptions that K ⁢ ( z , w ) ≠ 0 , K ⁢ ( w , z ) ≠ 0 {K(z,w)\neq 0,K(w,z)\neq 0} . We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory.

A technique for modeling and optimizing the design of wave energy conversion devices

With the continuous development of society and the economy, energy shortages have become increasingly prominent. As an important marine renewable energy source, wave energy holds significant theoretical significance due to its widespread distribution and vast storage potential. In this paper, we employ differential equation theory, numerical optimization methods, and optimization techniques to model and design algorithms for investigating the energy conversion mechanisms of wave energy devices. Due to the complex situation, we decided to simplify that into two scenarios. As to the first scenario, the essence of energy output in wave energy devices lies in the relative velocity between the floater and the oscillator. Firstly, the product of the damping force and the instantaneous relative velocity is integrated over time and divided by the period to obtain the average output power. Maximizing the average output power, with the damping coefficient as the decision variable, establishes a nonlinear optimization model. Secondly, a stepwise optimization approach is applied to control the damping coefficient, resulting in optimal damping coefficients of 33600, . Finally, a genetic algorithm is employed to validate the results as to the next scenario. After adding rotational dampers and torsion springs to the wave energy device, the relative velocity and relative angular velocity between the floater and the oscillator jointly contribute to energy output. The objective is to maximize the sum of the average output power of linear dampers and rotational dampers. The decision variables are the linear damping coefficient and the rotational damping coefficient. A nonlinear optimization model is established. A stepwise optimization approach is applied to optimize both types of coefficients, and the results are validated using a genetic algorithm. The optimal solution is found to have a linear damping coefficient of 60135.7781 N·s/m and a rotational damping coefficient of 4244.90358 N·s·m for the rotational dampers.

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.

Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.

Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed type

This article is a survey of Cathleen Morawetz’s contributions to the mathematical theory of transonic flows, shock waves, and partial differential equations of mixed elliptic-hyperbolic type. The main focus is on Morawetz’s fundamental work on the nonexistence of continuous transonic flows past profiles, Morawetz’s program regarding the construction of global steady weak transonic flow solutions past profiles via compensated compactness, and a potential theory for regular and Mach reflection of a shock at a wedge. The profound impact of Morawetz’s work on recent developments and breakthroughs in these research directions and related areas in pure and applied mathematics are also discussed.

Catalyst: Fast and flexible modeling of reaction networks.

We introduce Catalyst.jl, a flexible and feature-filled Julia library for modeling and high-performance simulation of chemical reaction networks (CRNs). Catalyst supports simulating stochastic chemical kinetics (jump process), chemical Langevin equation (stochastic differential equation), and reaction rate equation (ordinary differential equation) representations for CRNs. Through comprehensive benchmarks, we demonstrate that Catalyst simulation runtimes are often one to two orders of magnitude faster than other popular tools. More broadly, Catalyst acts as both a domain-specific language and an intermediate representation for symbolically encoding CRN models as Julia-native objects. This enables a pipeline of symbolically specifying, analyzing, and modifying CRNs; converting Catalyst models to symbolic representations of concrete mathematical models; and generating compiled code for numerical solvers. Leveraging ModelingToolkit.jl and Symbolics.jl, Catalyst models can be analyzed, simplified, and compiled into optimized representations for use in numerical solvers. Finally, we demonstrate Catalyst's broad extensibility and composability by highlighting how it can compose with a variety of Julia libraries, and how existing open-source biological modeling projects have extended its intermediate representation.

Oscillation criteria of fourth-order nonlinear semi-noncanonical neutral differential equations via a canonical transform

In this work first we transform the semi-noncanonical fourth order neutral delay differential equations into canonical type. This simplifies the investigations of finding the relationships between the solution and its companion function which plays an important role in the oscillation theory of neutral differential equations. Moreover, we improve these relationships based on the monotonic properties of positive solutions. We present new conditions for the oscillation of all solutions of the corresponding equation which improve the oscillation results already reported in the literature. Examples are provided to illustrate the importance of our main results. For moreinformation see https://ejde.math.txstate.edu/Volumes/2023/70/abstr.html

Accelerating Bayesian inference for stochastic epidemic models using incidence data

We consider the case of performing Bayesian inference for stochastic epidemic compartment models, using incomplete time course data consisting of incidence counts that are either the number of new infections or removals in time intervals of fixed length. We eschew the most natural Markov jump process representation for reasons of computational efficiency, and focus on a stochastic differential equation representation. This is further approximated to give a tractable Gaussian process, that is, the linear noise approximation (LNA). Unless the observation model linking the LNA to data is both linear and Gaussian, the observed data likelihood remains intractable. It is in this setting that we consider two approaches for marginalising over the latent process: a correlated pseudo-marginal method and analytic marginalisation via a Gaussian approximation of the observation model. We compare and contrast these approaches using synthetic data before applying the best performing method to real data consisting of removal incidence of oak processionary moth nests in Richmond Park, London. Our approach further allows comparison between various competing compartment models.