First-order Nonlinear Ordinary Differential Equations Research Articles

Soliton-comb generation in ring-shaped optical Kerr microresonators under thermal effects

Soliton combs are nonlinear wavetrains resulting from entanglements of localized waves which may be pulse or kink solitons, forming periodic structures called soliton crystals. In this work we examine the possible generation of soliton-comb structures in an optical Kerr microresonator in the ring configuration, considering the physical context where the laser stores heat in the ring cavity leading to sizable thermo-optical effects. To proceed with the study we consider an existing model proposed for this physical context, in which the optical field contribution to time evolution of the heat in the microresonator is described by the its average power. With help of appropriate variable changes the model is formulated in terms of a set of coupled first-order nonlinear ordinary differential equations, which are solved numerically paying particular attention to the impact of thermal effects on characteristic parameters of the soliton crystal structures, and also on the profile of time evolution of the temperature in the microresonator cavity. Simulations show that for the specific model considered, thermal effects do not oppose the formation of soliton crystals in the optical Kerr microresonator undergoing thermo-optical effects. Nevertheless the induced thermal detuning is likely to cause the average amplitude of the soliton crystal to increase or decrease, depending on variations of thermal coefficients in the heat equation.

EXACT SOLUTIONS FOR THE SINGULARLY PERTURBED RICCATI EQUATION AND EXACT WKB ANALYSIS

AbstractThe singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation $\hbar \partial _x f = af^2 + bf + c$ in the complex domain where $\hbar $ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent $\hbar $ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

A Higher-Order Galerkin Time Discretization and Numerical Comparisons for Two Models of HIV Infection.

Human immunodeficiency virus (HIV) infection affects the immune system, particularly white blood cells known as CD4+ T-cells. HIV destroys CD4+ T-cells and significantly reduces a human's resistance to viral infectious diseases as well as severe bacterial infections, which can lead to certain illnesses. The HIV framework is defined as a system of nonlinear first-order ordinary differential equations, and the innovative Galerkin technique is used to approximate the solutions of the model. To validate the findings, solve the model employing the Runge-Kutta (RK) technique of order four. The findings of the suggested techniques are compared with the results obtained from conventional schemes such as MuHPM, MVIM, and HPM that exist in the literature. Furthermore, the simulations are performed with different time step sizes, and the accuracy is measured at various time intervals. The numerical computations clearly demonstrate that the Galerkin scheme, in contrast to the Runge-Kutta scheme, provides incredibly precise solutions at relatively large time step sizes. A comparison of the solutions reveals that the obtained results through the Galerkin scheme are in fairly good agreement with the RK4 scheme in a given time interval as compared to other conventional schemes. Moreover, having performed various numerical tests for assessing the efficiency and computational cost (in terms of time) of the suggested schemes, it is observed that the Galerkin scheme is noticeably slower than the Runge-Kutta scheme. On the other hand, this work is also concerned with the path tracking and damped oscillatory behaviour of the model with a variable supply rate for the generation of new CD4+ T-cells (based on viral load concentration) and the HIV infection incidence rate. Additionally, we investigate the influence of various physical characteristics by varying their values and analysing them using graphs. The investigations indicate that the lateral system ensured more accurate predictions than the previous model.

Simulation of disequilibrium and chaos in aggregates of disposable income, wealth, and consumption in EU macroeconomics using nonlinear dynamic analysis

Economic disequilibrium theory (DT) more realistically represents modern macroeconomic systems than general equilibrium theory. DT coupled with applied mathematical economics and nonlinear dynamical analysis generates multi-dimensional phase spaces. Interdependencies of endogenous variables in state space create a flow of different and “parallel economic realities,” which depend on the initial conditions. By modeling variable changes using the nonlinear least squares (NLLS) method, we define the first-order nonlinear ordinary differential equation (NODE) system. The NODE system is impossible to solve analytically. The numerical solution and visualization requires the MATLAB software package, combined with its specialized applications pplane (two-dimensional (2D)) and MATCONT (three-dimensional (3D)). By analyzing the evolution of flow operators, we can predict the future qualitative behavior of the entire system, determine the model-optimal values, and perform inverse modeling for variables. The obtained data advocate better and more stable macroeconomic paths that economic policymakers can pursue. The proposed methodology’s boundaries have strong links to chaos theory. Chaotic behavior can arise after a certain number of periods. We found very high computation accuracy, transformation of discrete variables to continuous functions, and the implementation of high-order polynomial data fitting offset its effects in part and to some reasonable degree.

Soliton crystals in optical Kerr microresonators in the presence of thermo-optic effects

The influence of thermo-optic effects on shape profiles of soliton crystals in optical Kerr microresonators is investigated. The study rests on a model that consists of the Lugiato–Lefever equation, coupled to the one-dimensional heat diffusion equation with a source term proportional to the average power of the optical field. Using appropriate variable changes the model equations are transformed into a set of coupled first-order nonlinear ordinary differential equations. These equations are solved numerically with emphasis on the influence of thermo-optic effects on the amplitude and instantaneous frequency of the optical field, as well as on the temperature profile in the microresonator cavity. It is found that thermo-optic effects do not prevent soliton crystals from forming in optical Kerr microresonators, however, a strong thermal detuning will decrease the soliton-crystal amplitude. The model predicts a temperature profile in the microresonator cavity which is insensitive to the specific spatio-temporal profile of the soliton crystal propagating in the microresonator, a feature peculiar to the model.

Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method

In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.

Discounted optimal stopping problems in first-passage time models with random thresholds

AbstractWe derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

The Euler method is one of the oldest methods to solve differential equation problems. The Euler method produces the simplest solution. However, although is not computationally expensive, the Eulermethod is lack of accuracy. To improve the Euler method, the researcher proposed a new scheme for better accuracy. The Euler method equation and the mean method were combined to enhance this method. As the improvement basis, the researcher used the Centroidal mean and the midpoint method or Polygon to improve the Euler method. The combination of the Euler and Centroidal mean is known as Centroidal Polygon (CP) scheme. The CP scheme was used to solve first-order non-linear Ordinary Differential Equations (ODE). The researcher used SCILAB 6.0 software to solve the equation and the CP scheme was tested in three different step sizes (0.1,0.01, and 0.001). Aside from that, the researcher had compared the CP scheme with previous schemes such as ZulZamri's Polygon (P) scheme, Nurhafizah's Harmoni-Polygon (HP) scheme, and Nooraida's Cube-Polygon (CuP) scheme to ensure that the CP scheme is more accurate than previous research. When the maximum error is calculated by subtracting the scheme and exact solution, the results show that the CP scheme delivers the highest accuracy results in the shortest amount of time. The new enhanced, modified Euler method is useful for other researchers to achieve good accuracy at low computational cost as an alternative to the more computationally expensive methods.

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model.

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

Femtosecond laser inscriptions in Kerr nonlinear transparent media: dynamics in the presence of K-photon absorptions, radiative recombinations and electron diffusions

Femtosecond lasers interacting with Kerr nonlinear optical materials, propagate in form of filaments due to the balance of beam diffraction by self-focusing induced by the Kerr nonlinearity. Femtosecond laser filamentation is a universal phenomenon that belongs to a general class of processes proper to ultrashort lasers processing systems, associated with the competition between nonlinearity and dispersion also known to promote optical solitons. The present work considers a model describing femtosecond laser inscriptions in a transparent medium with Kerr nonlinearity. Upon inscription, the laser stores energy in the optical material which induces an electron plasma. The model consists of a cubic complex Ginzburg-Landau equation, in which an additional K-order nonlinear term takes into account K-photon absorption processes. The complex Ginzburg-Landau equation is coupled to a time first-order nonlinear ordinary differential equation, accounting for time evolution of the plasma density. The main objective of the study is to examine effects of the competition between multi-photon absorptions, radiative recombination and electron diffusion processes, on temporal profiles of the laser amplitude as well as of the plasma density. From numerical simulations, it is found that when the photon number (i.e. K) contributing to multiphoton ionization is large enough, taking the electron diffusion processes into account favours periodic structures in temporal profiles both of the laser and the plasma density. The pulse repetition rate in the optical soliton train is increased with increase of the electron diffusion coefficient, while the plasma density is a train of multi-periodic anharmonic wave patterns.

Optimal stopping problems for maxima and minima in models with asymmetric information

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normal-reflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations.

Equilibrium analysis of Mott memristor reveals criterion for negative differential resistance

Two-terminal electronic devices that exhibit voltage-controlled threshold switching (TS) via negative differential resistance (NDR) are important for many emerging applications. Pickett and Williams developed what has become a well-known physics-based model for nanoscale devices exhibiting NDR due to a reversible insulator-metal phase (Mott) transition. The Mott memristor model couples changes in electrical resistance and Joule heating to the phase of the material using one dynamic state variable, u, that describes the volume fraction of metal in the cross section of the device. The model formulation involves one nonlinear first-order ordinary differential equation and eight physical parameters. New equilibrium analysis reveals a simple condition that determines whether the model predicts NDR required for current–voltage (i–v) hysteresis in a voltage-controlled operation. We show that S-shaped NDR (also called current-controlled NDR) arises only above a critical ratio, Mc, of insulator to metal resistivity. Specifically, hysteresis in the i–v plane cannot occur below Mc=e2+1≈8.39 (i.e., e≈2.718…; Euler's number), but above this value hysteresis appears. This understanding enables tuning of hysteretic features, including threshold voltages for resistive switching, which benefit the use of TS memristors as memory storage elements, as well as excitable devices mimicking neural action potentials.

Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

AbstractWe present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

Reduced SIR Model of COVID-19 Pandemic

We propose a mathematical model of COVID-19 pandemic preserving an optimal balance between the adequate description of a pandemic by SIR model and simplicity of practical estimates. As base model equations, we derive two-parameter nonlinear first-order ordinary differential equations with retarded time argument, applicable to any community (country, city, etc.).The presented examples of modeling the pandemic development depending on two parameters: the time of possible dissemination of infection by one virus carrier and the probability of contamination of a healthy population member in a contact with an infected one per unit time, e.g., a day, is in qualitative agreement with the dynamics of COVID-19 pandemic. The proposed model is compared with the SIR model.

Formulation of a New Implicit Method for Group Implicit BBDF in Solving Related Stiff Ordinary Differential Equations

This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.

Topology optimization of structural frames considering material nonlinearity and time-varying excitation

An approach for the topology optimization of structures composed of nonlinear beam elements under time-varying excitation is presented. Central to this approach is a hysteretic beam finite element model that accounts for distributed plasticity and axial-moment interaction through appropriate hysteretic interpolation functions and yield/capacity function, respectively. Nonlinearity is represented via the hysteretic variables for curvature and axial deformations that evolve according to first order nonlinear ordinary differential equations (ODEs), referred to as evolution equations, and the yield function. Hence, the governing dynamic equilibrium equations and hysteretic evolution equations can thus be concisely presented as a system of first-order nonlinear ODEs that can be solved using general ODE solvers without the need for linearization. The approach is applied for the design of frame structures with an objective to minimize the total volume in the domain, such that the maximum displacement at specified node(s) satisfies a specified constraint (i.e., drift limit) for the given excitation. The maximum displacement is approximated using the p-norm and thus permits the completion of the analytical sensitivities required for gradient-based updating. Several numerical examples are presented to demonstrate the approach for the design of structural frames subjected to pulse, harmonic, and seismic base excitation. Topologies obtained using the suggested, nonlinear approach are compared to solutions obtained from topology optimization problems assuming linear-elastic material behavior. These comparisons show that although similarities between the designs exist, in general the nonlinear designs differ in composition and, importantly, outperform the linear designs when assessed by nonlinear dynamic analysis.

Dynamics of a Four-Dimensional System with Cubic Nonlinearities

In this paper, we investigate a three-parameter four-dimensional dynamical system, which is modeled by a set of four first-order nonlinear ordinary differential equations, each of which contains a crossed cubic term. Dynamical behaviors are characterized in the parameter space of the model. In fact, we use some cross-sections of a three-dimensional parameter-space, namely three related parameter planes, to locate regular and chaotic regions, as well as multistability regions. Lyapunov exponents spectra, bifurcation diagrams, and phase-space portraits are used to complete the analysis.

Applications of the Differential Transformation Method and Multi-Step Differential Transformation Method to Solve a Rotavirus Epidemic Model

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S; V; I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods