Methods For Differential Equations Research Articles

Rotationally symmetric transverse magnetic vector wave propagation for nonlinear optics

In this paper the theory and simulation results are presented for 3D cylindrical rotationally symmetric spatial soliton propagation in a nonlinear medium using a modified finite-difference time-domain general vector auxiliary differential equation method for transverse magnetic polarization. The theory of 3D rotationally symmetric spatial solitons is discussed, and compared with two (1 + 1)D, termed “2D” for this paper, hyperbolic secant spatial solitons, with a phase difference of π (antiphase). The simulated behavior of the 3D rotationally symmetric soliton was compared with the interaction of the two antiphase 2D solitons for different source hyperbolic secant separation distances. Lastly, we offer some possible explanations for the simulated soliton behavior.

A Streamline Upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations Under Optimal Control

Abstract In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the Péclet number. In this situation, computational instabilities occur, both for steady and unsteady cases. A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. We will apply a finite element method discretization in an optimize-then-discretize approach. Concerning the parabolic case, a stabilized space-time framework will be considered and stabilization will also occur in both bilinear forms involving time derivatives. Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. It is the first time that Reduced Order Models are applied to stabilized parabolic problems in this setting. The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.

Runge-Kutta Method Based on the Linear Combination of Arithmetic Mean, Geometric Mean, and Centroidal Mean: A Numerical Solution for the Hybrid Fuzzy Differential Equation

Objectives: To find the absolute error of the Hybrid fuzzy differential equation with triangular fuzzy number since its membership function has a triangular form. Methods: The third order runge-kuttta method based on the linear combination of Arithmetic mean, Geometric mean and Centroidal mean is introduced to solve Hybrid fuzzy differential equation. Seikkala’s derivatives are considered, and Numerical example is given to demonstrate the effectiveness of the proposed method. Findings: The comparative study have been done between the proposed method and the existing third order runge-kutta method based on Arithmetic mean. The proposed method gives better error tolerance than the third order runge-kutta method based on Arithmetic mean. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Geometric Mean and Centroidal Mean using Khattri’s formula and it is solved by the third order runge-kutta method for the first order hybrid fuzzy differential equation. The error tolerance has been calculated for the proposed method and the Arithmetic mean and it is seen that the error tolerance is better for the proposed method. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala’s derivative, Third order runge-kutta method, Initial value problem

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations

The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations

A new method for calculating the soft anomalous dimension matrix for massive particle scattering

The general structure of infrared divergences in the scattering of massive particles is captured by the soft anomalous dimension matrix. The latter can be computed from a correlation function of multiple Wilson lines. The state-of-the-art two-loop result has a tantalizingly simple structure that is not manifest in the calculations. We argue that the complexity in intermediate steps of the known calculations comes from spurious, regulator-dependent terms. Based on this insight we propose a different infrared regulator that is associated to only one of the Wilson lines. We demonstrate that this streamlines obtaining the two-loop result: computing the required Feynman integrals via the differential equations method, only multiple polylogarithmic functions appear (to all orders in the dimensional regulator), as opposed to elliptic polylogarithms. We show that the new method is promising for higher-loop applications by computing a three-loop diagram of genuine complexity, and provide the answer in terms of multiple polylogarithms. The relatively simple symbol alphabet we obtain may be of interest for bootstrap approaches.

Numerical subgrid Bi-cubic methods of partial differential equations in image segmentation

Image segmentation is a core research in the image processing and computer vision. In this paper, we suggest a Bi-cubic spline phase transition potential and elaborate a Bi-Cubic spline phase transition potential development. In the image segmentation, we develop the new approach to apply the novel computational fluid dynamics in the boundary with subgrid. The numerical subgrid Bi-cubic method with Bi-Cubic spline for minimizing the piecewise constant energy functional is very efficient, robust and fast in the image segmentation with a multispecies multiphase segmentation models. The subgrid Bi-cubic spline is applied on the boundary with subgrid and the regular grid is applied on the non-boundary. The model generates a multispecies multiphase distribution with Bi-Cubic spline and we can extract the image segments with multispecies multiphase. Finally, we analyze the models and show the numerical results. Numerical results are presented with OCR (Optical Character Recognition) and the medical image.

Mathematical frameworks for investigating fractional nonlinear coupled Korteweg-de Vries and Burger’s equations

This article utilizes the Aboodh residual power series and Aboodh transform iteration methods to address fractional nonlinear systems. Based on these techniques, a system is introduced to achieve approximate solutions of fractional nonlinear Korteweg-de Vries (KdV) equations and coupled Burger’s equations with initial conditions, which are developed by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. As a result, the Aboodh residual power series and Aboodh transform iteration methods for integer-order partial differential equations may be easily used to generate explicit and numerical solutions to fractional partial differential equations. The results are determined as convergent series with easily computable components. The results of applying this process to the analyzed examples demonstrate that the new technique is very accurate and efficient.

Higher order numerical methods for fractional delay differential equations

Higher order numerical methods for fractional delay differential equations

Numerical method for singular drift stochastic differential equation driven by fractional Brownian motion

In this paper, we study the stochastic differential equation with singular drift coefficient driven by fractional Brownian motion with Hurst parameter H∈(12,1). We obtain that the optimal convergence rate of the backward Euler method is 1.0. The constant elasticity of variance model and the Aït-Sahalia interest rate model are performed as numerical experiments to validate our theoretical results.

Multi-stage Euler–Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains

Multi-stage Euler–Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains

Numerical Solution of Hybrid Fuzzy Differential Equation by using Third Order Runge-Kutta Method Based on Linear Combination of Arithmetic Mean, Root Mean Square and Centroidal Mean

Objectives: This article develops the third order Runge-Kutta method, which uses a linear combination of the arithmetic mean, root mean square, and centroidal mean, to solve hybrid fuzzy differential equations. Methods: Seikkala's derivative is taken into account, and a numerical example is provided to show the efficacy of the proposed method. The outcomes demonstrate that the suggested approach is an effective tool for approximating the solution of hybrid fuzzy differential equations. Findings: The comparative analysis was carried out using the third order Runge-Kutta method that is currently in use and is based on arithmetic mean, root mean square, and centroidal mean. Compared to other methods, the suggested method offers a more accurate approximation. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Root Mean Square, and Centroidal Mean using Khattri's formula. And the developed formula is used to solve the third order Runge-Kutta method for the first order hybrid fuzzy differential equation. All real life problems which can be modeled in to an initial value problem can be solved using this formula. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala's derivative, third order Runge-Kutta method, Arithmetic mean, Root mean square, Centroidal mean, Initial value problem

Fast and accurate evaluation of deep-space galactic cosmic ray fluxes with HelMod-4/CUDA

The accurate knowledge of cosmic ion fluxes is essential for fundamental physics, deep space missions, and exploration activities in the solar system. In the HelMod-4 model the Parker transport equation is solved using a Monte Carlo approach to evaluate the solar modulation effect on local interstellar spectra of Galactic Cosmic Rays (GCRs). This work presents the latest updates to the HelMod-4 model parameters, focusing on the descending phase of solar cycle 24. The updates are motivated by the latest high-precision measurements from the AMS-02 detector which revealed for the first time with high accuracy the features of GCR fluxes’ evolution during a period of positive interplanetary magnetic field polarity. Furthermore, we present HelMod-4/CUDA, a GPU-accelerated approach for solving the Parker equation in the heliosphere using a stochastic differential equation method. The code is an evolution of the HelMod-4 code, porting the algorithm to GPU architecture using the CUDA programming language. This approach achieves significant speedup compared to a CPU implementation. The HelMod-4/CUDA code has been validated by comparing its results with the most precise and updated experimental GCR spectra observed during high and low solar activity periods, both in the inner and outer heliosphere, at the Earth location, and outside the ecliptic plane. The comparison shows that HelMod-4 and HelMod-4/CUDA can be equivalently used to provide solar-modulated spectra with a similar degree of accuracy in reproducing observed data.

Cosmology meets cohomology

The cosmological polytope and bootstrap programs have revealed interesting connections between positive geometries, modern on-shell methods and bootstrap principles studied in the amplitudes community with the wavefunction of the Universe in toy models of FRW cosmologies. To compute these FRW correlators, one often faces integrals that are too difficult to evaluate by direct integration. Borrowing from the Feynman integral community, the method of (canonical) differential equations provides an efficient alternative for evaluating these integrals. Moreover, we further develop our geometric understanding of these integrals by describing the associated relative twisted cohomology. Leveraging recent progress in our understanding of relative twisted cohomology in the Feynman integral community, we give an algorithm to predict the basis size and simplify the computation of the differential equations satisfied by FRW correlators.

A Numerical Study of a Stabilized Hyperbolic Equation Inspired by Models for Bio-Polymerization

Abstract This report investigates a stabilization method for first order hyperbolic differential equations applied to DNA transcription modeling. It is known that the usual unstabilized finite element method contains spurious oscillations for nonsmooth solutions. To stabilize the finite element method the authors consider adding to the first order hyperbolic differential system a stabilization term in space and time filtering. Numerical analysis of the stabilized finite element algorithms and computations describing a few biological settings are studied herein.

Impact of weak nonlocal nonlinearity on modulational instability and chirp-free soliton trains in parabolic law optical fiber

This study considers the generation and evolution of chirp-free soliton trains in a focusing cubic-quintic nonlinear optical fiber media (by utilizing experimental parameters), under the influence of weak nonlocal nonlinearity. Analysis of modulational instability (MI) reveals that the MI gain exist only in the anomalous dispersion regime, with the quintic nonlinear term increasing the maximum gain and sideband frequencies. The nonlocal nonlinearity parameter generally suppresses the impact of MI, because the maximum gain is greatly reduced with increase in nonlocality. By using the subsidiary ordinary differential equation method, stationary periodic soliton trains are obtained; with the intensity of the optical beam greatly reduced and frequency increased as nonlocality is gradually stepped up. However, an increase in nonlocality instead reduces the frequency of the traveling optical signals; derived via an efficient transformation method. Our results strongly suggest that nonlocal nonlinearity has potential applications in the control of laser beams; as nonlinear periodic optical signals can be easily transformed to very weak quasi-plane waves.

On the Application of Fractional Derivative Operator Theory to the Electromagnetic Modeling of Frequency Dispersive Media

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

On the Solution of Blasius Boundary Layer Equations of Prandtl-Eyring Fluid Flow Past a Stretching Sheet

Objective: This paper investigates velocity profile for two-dimensional, incompressible, laminar forced convection flow of the fluid model for Prandtl-Eyring fluid past a stretching sheet in the presence of fluid parameters. Methods: The governing partial differential equation for the flow was transformed into non-linear ordinary differential equation by using the deductive one parameter group theoretic method and numerical solution of non-linear ordinary differential equation (ODE) is solved by MATLAB bvp4c solver. Findings: The solution of velocity profile obtained as a function of parameter and . The effect of the fluid parameter was discussed graphically. Novelty: The main goal of this article is to analyze boundary layer flow of Prandtl-Eyring fluid over a stretching surface. The conservation equations of mass, momentum are converted into non-linear ordinary differential equations along with boundary conditions using deductive one parameter group theoretic method and solved by MATLAB ODE solver. Comparisons with previously published works are made, and results show a high level of agreement. This type of research is applicable to extrusion, paper production, fiber glass production, hot rolling, condensation process, crystal growing, polymer sheets etc. Keywords: Boundary layer, laminar flow, Deductive one parameter Group theoretic method, Absolute invariant, Stretching Sheet, Prandtl-Eyring fluid

On Population Models with Delays and Dependence on Past Values

The current values of many populations depend on the past values of the population. In many cases, this dependence is caused by the time certain processes take. This dependence on the past can be introduced into mathematical models by adding delays. For example, the growth rate of a population depends on the population τ time units ago, where τ is the maturation time. For an epidemic, there is a time τ between the contact of an infected individual and a susceptible one, and the time the susceptible individual actually becomes infected. This time τ is also a delay. So, the number of infected individuals depends on the population at the time τ units ago. A second way of introducing this dependence on past values is to use non-local operators in the description of the model. Fractional derivatives have commonly been used to provide non-local effects. In population growth models, it can also be done by introducing a new compartment, the immature population, and in epidemic models, by introducing an additional exposed population. In this paper, we study and compare these methods of adding dependence on past values. For models of processes that involve delays, all three methods include dependence on past values, but fractional-order models do not justify the form of the dependence. Simulations show that for the models studied, the fractional differential equation method produces similar results to those obtained by explicitly incorporating the delay, but only for specific values of the fractional derivative order, which is an extra parameter. But in all three methods, the results are improved compared to using ordinary differential equations.

Many-Stage Optimal Stabilized Runge–Kutta Methods for Hyperbolic Partial Differential Equations

A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge–Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found for non-uniformly refined meshes and spatially varying wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N-storage many-stage Runge–Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. These methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.