Solution Of Partial Differential Equations Research Articles

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Can physics-informed neural networks beat the finite element method?

Partial differential equations (PDEs) play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of PDEs. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear PDEs: Poisson in 1D, 2D and 3D, Allen-Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

An innovative approach to approximating solutions of fractional partial differential equations

The RPS-M (residual power series method) is a valuable technique for solving F-PDEs (fractional partial differential equations). However, the derivative of the residual function to obtain the coefficients of the series is required in RPS-M. This makes the application of the classical RPS-M limited to a certain extent due to the complexity of the derivation of the residual function for higher iterations. To overcome this obstacle, in this study, we present a simplified version of this approach with the help of Laplace transform that requires less computation and offers higher accuracy. This modified method does not require derivation as well as limit of the residual function to estimate the unknown coefficients of the series solution. To demonstrate its effectiveness, we apply the proposed method to nonlinear F-PDEs to obtain their semi-analytical solution. The obtained solutions exhibit excellent agreement when compared to results obtained using other established approaches. We have also provided the convergence analysis of the obtained solution. Furthermore, by comparing the outcomes for various values of the non-integer order σ, we observe that as the value approaches an integer order, the solution converges towards the exact solution.

A novel method for response probability density of nonlinear stochastic dynamic systems

AbstractThis paper presents a novel method for analyzing high-dimensional nonlinear stochastic dynamic systems. In particular, we attempt to obtain the solution of the Fokker–Planck–Kolmogorov (FPK) equation governing the response probability density of the system without using the FPK equation directly. The method consists of several important components including the radial basis function neural networks (RBFNN), Feynman–Kac formula and the short-time Gaussian property of the response process. In the area of solving partial differential equations (PDEs) using neural networks, known as physics-informed neural network (PINN), the proposed method presents an effective alternative for obtaining solutions of PDEs without directly dealing with the equation, thus avoids evaluating the derivatives of the equation. This approach has a potential to make the neural network-based solution more efficient and accurate. Several highly challenging examples of nonlinear stochastic systems are presented in the paper to illustrate the effectiveness of the proposed method in comparison to the equation-based RBFNN approach.

Weak solution for biharmonic equation with Navier boundary conditions

The aim of this paper is to study the existence and uniqueness of weak solutions of certain nonlinear second order elliptic partial differential equations.

Inferring Stochastic Rates from Heterogeneous Snapshots of Particle Positions.

Many imaging techniques for biological systems-like fixation of cells coupled with fluorescence microscopy-provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Deep operator networks for bioheat transfer problems with parameterized laser source functions

To numerically characterize optothermal interactions at the laser-tissue interface during tissue imaging or surgery, the bioheat transfer partial differential equation (PDE) needs to be solved with a laser excitation source function. Considering various possible experimental scenarios, the PDEs become parametric and need to be solved for several optical and laser parameters. Classical numerical solvers that are used for performing such simulations are computationally expensive and time consuming. In this work, we use the recent deep operator network (DeepONet) framework for learning the mapping between infinite dimensional function spaces of the parameterized laser source functions and the solutions of the parameterized bioheat transfer PDE. We propose a new neural network architecture, Res-DeepONet, for training a purely physics-informed DeepONet with laser heating functions comprising of Gaussian beam profiles. We also showcase the versatility and robustness of our network architecture within purely data-driven and hybrid training frameworks for applications where labeled ground truth data may be accessible. The proposed neural network architecture exhibits excellent generalization performance on unseen Gaussian source functions with 10× higher accuracy when compared to traditional feedforward network architectures. Furthermore, zero-shot performance studies highlight the exceptional extrapolation capabilities of our network with 50× higher accuracy compared to traditional DeepONet architectures at inference times that are two orders of magnitude faster than classical solvers.

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

Automated intelligent design of modified uni-traveling carrier photodectors

This paper introduces an automatic intelligent design method for the modified uni-traveling carrier photodetector (MUTC-PD). The conventional photodetector design process often relies on the numerical solution of complex nonlinear partial differential equations to simulate and optimize device performance, which is not only computationally intensive but also inefficient. To overcome this challenge, we apply the charge control principle to calculate the photodetector bandwidth, which improves the computational speed by a factor of approximately 1800 compared to the numerical solution of nonlinear partial differential equations. To further optimize the structure of the photodetector, we incorporate the Velocity Varying Climbing Particle Swarm Optimization (VVCPSO) algorithm. This is an improved algorithm based on the traditional particle swarm algorithm, which is able to quickly find the optimal solution in a complex parameter space. By applying the VVCPSO algorithm, we successfully fine-tuned the photodetector structure and obtained structural parameters with optimal performance. Our thorough verification process confirms that the proposed method is consistent with the results of ATLAS simulation software. Automated design has resulted in a high-performance MUTC-PD with a responsivity of 0.52A/W and a bandwidth of 60 GHz (@-3 V) at a mesa diameter of 16µm. Compared to the pre-optimized device, the bandwidth is increased to three times the original. By reducing the mesa diameter to 4µm, the bandwidth can be further increased to 82 GHz (@-3 V). The proposed method's calculation speed is fast enough, enabling extensive parameter studies to optimize device performance.

Unsteady MHD dusty fluid flow over a cone in the vicinity of porous medium: a numerical study

This research presents an unsteady free convection magnetohydrodynamics (MHD) dusty fluid flow over a vertical cone enclosed inside a porous medium. The Crank–Nicolson approach is used to get the numerical solutions for these non-linear partial differential equations (PDEs). The velocity and temperature distributions are graphed numerically for a wide range of physical parameters to highlight important characteristics of the solutions. The results showed that increasing the porosity parameter leads to a rise in the velocity profile for the fluid and dust phases, while doing the reverse with the rise of the magnetic parameter, fluid-particle interaction parameter, and mass concentration of the particle phase parameter. Additionally, it is shown that the temperature profile rises as the magnetic parameter and the mass concentration of particle phase parameter grow, whereas the fluid-particle interaction parameter and porosity parameter show the opposite tendency.

Analyzing the time-fractional (3 + 1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions

The paper focuses on investigating the time-fractional (3 + 1)-dimensional cubic and quantic nonlinear Schrödinger equation. We adopt the novel Kudryashov method to generate a distinct class of optical solutions for the current conformable fractional derivative problem. Our method explores various solution forms, including dark, wave, mixed dark-bright, and singular solutions. The soliton solutions we construct are visually represented to illustrate the influence of the fractional order derivative. Further, we elucidate the influence of solution parameters on the wave envelope, providing clear interpretations through 2D graphics presentations. The results underscore the efficacy of our approach in discovering exact solutions for nonlinear partial differential equations, especially in cases where alternative methods prove ineffective. The significance of the present paper lies in its contribution to advancing the understanding of the behavior of optical solutions in nonlinear systems, providing valuable insights for both theoretical and practical applications. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons.

Exact solutions of the Landau–Ginzburg–Higgs equation utilizing the Jacobi elliptic functions

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations.

Experimental validation of a design model for latent thermal energy storage extended with sensible heat

Latent thermal energy storage can be a key technology for a green energy transition by matching fluctuating heat demand and supply. In order to implement a storage system, it needs to be designed which requires estimating the outlet temperature of a system for a given geometry and time history of the heat transfer fluid’s mass flow rate and inlet temperature. Currently, design methods are either overly simplistic, focusing solely on e.g. the phase change time or requiring the solution of partial differential equations which can be computationally expensive. The present paper proposes a novel approach where a latent thermal energy storage system is decomposed into a heat transfer fluid vessel, a sensible storage system and a storage system with only latent heat. Computationally inexpensive models are available for all three of these sub heat exchangers. A heat exchanger model is obtained by connecting the sub heat exchangers in parallel. This novel approach is used to model an industrial scale shell and tube latent thermal energy storage heat exchanger. The predicted outlet temperature is compared to the measured outlet temperature and the design model obtains good agreement.

Numerical modelling of continuous casting of round billets with turbulence in the melt

The present work aims to solve the continuous casting benchmark problem in axisymmetry with turbulence in the melt by the meshless method. The physical model of the liquid-solid system that involves mass, momentum and energy conservation is formulated in the mixture continuum approximation. The melt is assumed Newtonian and incompressible. A k-epsilon Reynolds averaged Navier-Stokes turbulence model is implemented with Abe-Kondoh-Nagano closures. The mushy region is modelled as a Darcy porous media with the Kozeny-Carman permeability model. The solution of partial differential equations is implemented locally using collocation with radial basis functions and explicit time-stepping. The velocity pressure coupling is performed using the fractional step method. The validation of the model is assessed by comparing its outcomes with the results of the finite volume method. A sensitivity study of the varying casting speed and temperature on the velocity, temperature, and solid fraction fields is shown.

LordNet: An efficient neural network for learning to solve parametric partial differential equations without simulated data

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson’s equation and (2D and 3D) Navier–Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40× faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

Optical solutions for a quintic derivative nonlinear Schrödinger equation using symmetry analysis

In this paper, the derivative nonlinear Schrödinger equation with time-dependent-coefficient coefficients (DNLS) has been studied. This model is widely used in the hydrodynamic wave packets studies and optical fiber optics. The DNLS will be solved using a traveling wave transformation and Lie point symmetry analysis which is considered as a powerful tool that can be used in finding exact solutions of nonlinear partial differential equations (NLPDEs). The Lie symmetry technique provides a general way to obtain invariant solutions for NLPDEs. Some new exact solutions of the DNLS are obtained in the form of hyperbolic functions, Jacobi elliptic functions and Weierstrass elliptic function. The obtained waves have the forms of bright, dark and rogue waves soliton solutions which are considered as new solutions of the studied equation. The results established in this paper are new and extend the numbers of previously published results.

Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

Parameter‐robust higher‐order time‐accurate computational method for singularly perturbed time‐dependent semilinear convection‐diffusion PDEs with discontinuous data

This article is concerned with a class of singularly perturbed semilinear parabolic convection‐diffusion partial differential equations (PDEs) with discontinuous source function. Solutions of these PDEs usually exhibit a weak interior layer at one side of the point of discontinuity along with a boundary layer at one side of the spatial domain. We begin our study by proving existence of the analytical solution of the considered nonlinear PDE by means of the upper and lower solutions approach; and the ‐uniform stability of the analytical solution is established by using the comparison principle for the continuous nonlinear operator. In order to realize the asymptotic behavior of the analytical solution, we derive a priori bounds of the solution derivatives via decomposition of the solution into the smooth and the layer components. For an efficient numerical solution of the nonlinear PDE, the time‐derivative is approximated by the Crank–Nicolson method on an equidistant mesh, and we approximate the spatial derivative by a finite difference scheme on a suitable layer‐adapted mesh. We establish the comparison principle for the nonlinear difference operator to prove the ‐uniform stability of the discrete solution and further construct a suitable decomposition of the discrete solution for pursuing the convergence analysis. The computational method is proven to be parameter‐robust with second‐order time accuracy in the discrete supremum norm. The theoretical estimate is finally verified by the numerical experiments.

Solving a class of multi-scale elliptic PDEs by Fourier-based mixed physics informed neural networks

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations (PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks (dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be publicly available at https://github.com/Blue-Giant/FMPINN.

New analytical technique for prototype closed form solutions of certain nonlinear partial differential equations

This work proposes a novel modified (G’/G2)-expansion method to construct the exact travelling wave solutions of three famous nonlinear partial differential equations namely, nanoionic currents along microtubules equation, Benney-Luke equation and the generalized Hirota-Satsuma coupled KdV system. In the result, these solutions can be divided into different types i.e. trigonometric, rational and exponential solutions. The computational tool Maple is used to arrive at these solutions. Applying the novel modified (G’/G2)-expansion approach yields results that are trustworthy, easy to understand, and useful in a variety of disciplines including physics, biology, finance, engineering, statistics, and more. Software is used to create the two- and three-dimensional graphs. The comparison of the obtained solutions is also given in the form of Tables 1, 2 and 3, to show the novelty of our obtained solutions.