Implicit Differential Equations Research Articles

HYSTERESIS OF IMPLICIT EQUATIONS IN HYPOPLASTICITY FOR SOIL MATERIALS WITH GRANULAR HARDNESS DEGRADATION

AbstractWe study a hypoplastic model for soil and granular materials stemming from geomechanical engineering which further incorporates effects of degradation of the granular hardness, therefore allowing for the description of environmental weathering. The governing system is described by a nonlinear system of transcendental-differential equations for stress and strain rate, which is investigated with respect to its long-time dynamic. Under deviatoric stress control, two different solutions of the underlying, implicit differential equations are constructed analytically. The spherical components of stress and strain rate converge asymptotically to an attractor and lead to the sparsification of material states. Whereas under cyclic loading-unloading carried out in a numerical simulation, finite ratcheting of the deviatoric strain rate is observed in the form of a square spiral.

Bending of Lloyd’s mirror to eliminate the period chirp in the fabrication of diffraction gratings

We present a new technique to prevent the detrimental period chirp that appears in optical gratings fabricated by laser interference lithography (LIL). The idea is to bend the Lloyd’s mirror in the lithographic setup to eliminate the period chirp already at the step of the grating’s exposure. A new mathematical model was developed to describe the required bending geometry of the mirror. It is shown that this geometry can be described by multiple cross-sections of the mirror, each obtained by the solution of an implicit first-order differential equation. The proposed approach is illustrated on the basis of a concrete example. By slightly bending the Lloyd’s mirror (by ≈ 3.5 mm of maximum deflection over an area of 142 mm × 215 mm) the period chirp of the exposed grating can be eliminated completely.

On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions

Abstract This study investigates the existence of solutions for implicit fractional differential equations with fractional-order integral boundary conditions. We create the required conditions to ensure unique solution and Ulam-Hyers-Rassias stability. We also give examples to highlight the major findings.

The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

The inverse problem for a class of implicit differential equations and the coisotropic embedding theorem

Modeling of implicit multi term fractional delay differential equation: Application in pollutant dispersion problem

This work explores a new abstract model using multi-term fractional differential operator and delay effect. The model is defined by the existence of a delay parameter and insertion of multi term fractional differential operators in the input function. For the solution of the newly formulated problem, we utilized fixed-point theorems, which provide a powerful analytical tool for establishing the existence and uniqueness of solutions in functional spaces. Additionally, we investigate the stability properties of the implicit multi-term fractional delay differential equation (IMTFDDE) and employ the concept of Ulam-Hyers stability to assess their behaviour. The Ulam-Hyers stability framework is a contemporary approach that characterizes the sensitivity of solutions in functional spaces. Furthermore, we use an example to show how to apply our results. Lastly, for the real implementation of this research endeavour, we provided a real model devoted to modeling the dispersion of pollutants in a river while accounting for delayed effects and strip boundary circumstances.

The inverse problem within free Electrodynamics and the coisotropic embedding theorem

In this paper, we present the coisotropic embedding theorem as a tool to provide a solution for the inverse problem of the calculus of variations for a particular class of implicit differential equations, namely the equations of motion of free Electrodynamics.

Bringing metasurfaces to analytical lens design: stigmatism and specific ray mapping.

We propose a method to design the exact phase profile of at least one metasurface in a stigmatic singlet that can be made to implement a desired ray mapping. Following the generalized vector law of refraction and Fermat's principle, we can obtain exact solutions for the required lens shape and phase profile of a phase gradient metasurface to respect particular ray conditions (e.g., Abbe sine) as if it were a freeform refractive element. To do so, the method requires solving an implicit ordinary differential equation. We present comparisons with Zemax simulations of illustrative designed lenses to confirm the anticipated optical behaviour.

A multivariate model of deep slurry infiltration in slurry shield tunneling

There are multiple factors that influence slurry infiltration processes, making it necessary to consider these factors when comprehensively analyze the process of deep slurry infiltration. Firstly, this paper proposes a theoretical model for multivariate deep slurry infiltration and introduces a control equation governing the variation of the slurry infiltration domain. This equation accurately reflects changes in the infiltration boundary during the slurry infiltration process and meticulously responds to variations in pore water pressure, slurry retention rate, and pressure conversion efficiency-critical factors in the slurry infiltration process. Then, the Lax method was used to solve the corresponding control equations of the model, realizing the effective solution of the implicit partial differential equation system. The computational results of the theoretical model were compared with the indoor test results. Finally, the validated model was used to analyze the deep slurry infiltration characteristics. The results indicate that the theoretical model can reasonably simulate the deep slurry infiltration process, and the use of dynamic computational domain to describe deep slurry infiltration process is more in line with engineering practice. Furthermore, this model has guiding significance for the division of deep slurry infiltration stage and slurry surface infiltration stage.

Problem Involving the Riemann–Liouville Derivative and Implicit Variable-Order Nonlinear Fractional Differential Equations

The problem of boundary values for implicit differential equations with nonlinear fractions involving the variable order and the Riemann–Liouville derivative is examined in this article along with its existence and stability. Specifically, the locally solvability, which is equivalent to the existence of solutions, is related to the symmetry of a transformation of a nonlinear equations system. To demonstrate the reliability of the found results, we design an example.

Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

Implicit Caputo-Fabrizio fractional differential equations with delay

"This article deals with some existence and uniqueness results for several classes of implicit fractional differential equations with delay. Our results are based on some fixed-point theorems. To illustrate our results, we present some examples in the last section. Keywords: Caputo-Fabrizio fractional order derivative, implicit, delay, fixed point."

Implicit fractional differential equations: Existence of a solution revisited

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: for all with the initial condition , where is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution.

Caputo-Fabrizio type fractional differential equations with non-instantaneous impulses: Existence and stability results

In the current article, we establish the existence and uniqueness of solutions to the initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses, including the Caputo-Fabrizio fractional derivative. This derivative encapsulates non-local and memory-dependent effects in the system's behavior. The problem's complexity arises from its implicit nature, where the relationships between variables are not directly specified. This inherent intricacy is further compounded by the introduction of non-instantaneous impulses, signifying abrupt, intermittent changes within the system at specific junctures, which in turn trigger sudden transitions. At its core, the initial value problem is concerned with deciphering a system's behavior based on its initial conditions, necessitating a solution that adheres to the given equation while considering the presence of non-instantaneous impulses. To derive the existence results, the study leverages established mathematical tools, specifically, the standard Banach's fixed point theorem and Darbo's fixed point theorem connected to Kuratowski's measure of noncompactness (KMN). Furthermore, the Hyers-Ulam stability (HU-stability) of the given problem is discussed, which, in turn, enhances the utility and reliability of these solutions in practical applications. In order to demonstrate the applicability of the discovered results for various values of δ, we construct some examples towards the end. The practical implications of these results extend to a wide range of fields, offering improved modeling, control, and understanding of complex systems.

On deformable fractional impulsive implicit boundary value problems with delay

This paper deals with some existence and uniqueness results for a class of deformable fractional differential equations. These problems encompassed nonlinear implicit fractional differential equations involving boundary conditions and various types of delays, including finite, infinite, and state-dependent delays. Our approach to proving the existence and uniqueness of solutions relied on the application of the Banach contraction principle and Schauder’s fixed-point theorem. In the last section, we provide different examples to illustrate our obtained results.

Existence and stability results for implicit impulsive convex combined Caputo fractional differential equations

This paper deals with the existence and uniqueness results for a class of impulsive implicit fractional initial value problems of the convex combined Caputo fractional derivative. The arguments are based on Banach's contraction principle, Schauder's and Mönch's fixed point theorems. We will also establish the Ulam stability and give some examples to illustrate our results.

On a Control Problem for a System of Implicit Differential Equations

We consider the differential inclusion F(t,x,x˙)∋0 with the constraint x˙(t)∈B(t), t∈[a,b], on the derivative of the unknown function, where F and B are set-valued mappings, F:[a,b]×Rn×Rn×Rm⇉ is superpositionally measurable, and B:[a,b]⇉Rn is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form f(t,x,x˙,u)=0, x˙(t)∈B(t), u(t)∈U(t,x,x˙), t∈[a,b].

Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving $$\varrho $$-Fractional Derivative with Anti Periodic Boundary Conditions

Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving $$\varrho $$-Fractional Derivative with Anti Periodic Boundary Conditions

Modeling and analysis of an implicit fractional order differential equation with multiple first-order fractional derivatives and non-local boundary conditions

Modeling and analysis of an implicit fractional order differential equation with multiple first-order fractional derivatives and non-local boundary conditions

On a Control Problem for a System of Implicit Differential Equations

On a Control Problem for a System of Implicit Differential Equations

Bumble bee pollination and the wildflower/crop trade-off: When do wildflower enhancements improve crop yield?

Populations of wild insect pollinators such as bumble bees are threatened worldwide, which compromises pollinator-dependent crop yields. Intentionally planting wildflower patches in agricultural landscapes can support these populations and increase the pollination of nearby crops via the “spillover effect” (i.e., the exporter hypothesis), but may also distract bees from the crops and reduce their pollination via the “Circe principle” (i.e., the aggregation hypothesis). Considering the potentially high costs of these management strategies and the necessity to support wild insect pollinators in the Anthropocene, there is a pressing need to provide simulation tools that can inform best practices for wildflower plantings in agro-ecosystems. We developed a spatially implicit ordinary differential equations (ODEs) model specifically designed to determine the optimal wildflower-to-crop ratio as a function of wildflower patch (i) attractiveness, (ii) nutritional benefits, and (iii) blooming period relative to the crop. The model represents the population dynamics of a bumble bee colony and floral resources (crop and wildflower) in the landscape and nest during one harvesting season. We conduct a full factorial simulation experiment to identify the optimal characteristics of the wildflower patch (i.e., blooming period, attractiveness, relative abundance) that maximise crop yield via the enhancement of the number of bees pollinating crop flowers in a fictional blueberry farm. Our results suggest that providing highly attractive and nutritive wildflower resources before and not during the crop blooming season is the most beneficial strategy. When both flower types are in competition, pollination services can decrease, either when wildflowers are too attractive, or if they provide less benefits to the bees than the crop due to a trade-off between resources quality versus quantity.