Solution Of Partial Differential Equations Research Articles

New exact soliton wave solutions appear in optical fibers with Sardar sub equation and new auxiliary equation techniques

This paper comprehensively analyzes exact solutions for the nonlinear long-short wave interaction system within the optical field. Consider two general techniques in this field, the Sardar sub-equation method, and a new auxiliary-equation technique. These methods are applied to derive a wide range of soliton solutions for nonlinear partial differential equations. By transforming the original partial differential equation into an ordinary differential equation using an appropriate transformation, various types of solitary wave solutions are obtained. The novelty of this work lies in the application of two powerful analytical methods. The study significantly broadens the scope of these techniques and their applications, providing a diverse set of exact solutions. To enhance comprehension, the obtained solutions are visualized through 3D, 2D, contour, and density plots, offering clear insights into the dynamics of solitary waves. Long-short-wave interaction model has many applications in different kinds of areas such as in optical fiber communication, to understand the interaction between different wave components that can influence the transmission of signals. This model is used to study the interaction between ion-acoustic waves and electron plasma waves. This helps in understanding energy transfer and wave stability in plasma, which is essential for applications like fusion energy research and space plasma. This is important in coastal engineering for predicting wave behaviors that affect coastal structures, sediment transport, and tsunami dynamics.

Stability analysis of chaotic systems in latent spaces

Abstract Partial differential equations, and their chaotic solutions, are pervasive in the modelling of complex systems in engineering, science, and beyond. Data-driven methods can find solutions to partial differential equations with a divide-and-conquer strategy: The solution is sought in a latent space, on which the temporal dynamics are inferred (“latent-space” approach). This is achieved by, first, compressing the data with an autoencoder, and, second, inferring the temporal dynamics with recurrent neural networks. The overarching goal of this paper is to show that a latent-space approach can not only infer the solution of a chaotic partial differential equation, but it can also predict the stability properties of the physical system. First, we employ the convolutional autoencoder echo state network (CAE-ESN) on the chaotic Kuramoto-Sivashinsky equation for various chaotic regimes. We show that the CAE-ESN (i) finds a low-dimensional latent-space representation of the observations and (ii) accurately infers the Lyapunov exponents and covariant Lyapunov vectors (CLVs) in this low-dimensional manifold for different attractors. Second, we extend the CAE-ESN to a turbulent flow, comparing the Lyapunov spectrum to estimates obtained from Jacobian-free methods. A latent-space approach based on the CAE-ESN effectively produces a latent space that preserves the key properties of the chaotic system, such as Lyapunov exponents and CLVs, thus retaining the geometric structure of the attractor. The latent-space approach based on the CAE-ESN is a reduced-order model that accurately predicts the dynamics of the chaotic system, or, alternatively, it can be used to infer stability properties of chaotic systems from data.

Generative learning of the solution of parametric Partial Differential Equations using guided diffusion models and virtual observations

Generative learning of the solution of parametric Partial Differential Equations using guided diffusion models and virtual observations

Transfinite Patches for Isogeometric Analysis

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L2 error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type.

Advances in Topological Methods for Solving Nonlinear Partial Differential Equations

Higher order nonlinear partial differential equations (PDEs) are very difficult to compare or analyze and therefore there is restriction towards analytical or numerical appraisal. The following research aims at solving the problem of the solutions of Nonlinear PDEs through the application of topological fixed-point theory and degree theory, as well as the application of variational methods towards improving the existence, uniqueness, and stability of solutions to PDEs under all possible boundary conditions. The purpose of the study is to investigate the effectiveness of these methods in multifaceted and multi-grained systems. We used both fixed-point analyses to check the convergence of solutions and degree-theoretic methods for checking multiplicity of solutions, and variational structure for critical points of solutions. Newton-Raphson and finite element techniques were used in solving solutions with much ease for increased efficiency. A selection of findings shows that topological techniques are an efficient means of handling nonlinearity in PDEs; degree theory uncovers the multiplicity of solutions while variational techniques unveil the stability characteristics. This research advances the field of nonlinear analysis beyond the prior art by broadening the scope of topology in new domains and providing enhanced methods of computation of PDEs where it matters most: where traditional linear methods fail to apply. In the context of physics, engineering, and environmental studies nonlinear modeling is vital and the practical implication highlighted herein is useful for all. Possible future work might apply these methods to more general boundary conditions as well as combined analytical-numerical approaches.

Output tracking problem for a class of 3-D hyperbolic PDEs

The output tracking problem (OTP) refers to the procedure of determining an input control that ensures the system output follows a specified trajectory. In this paper, we investigate the OTP for controlled systems governed by a class of 3-D hyperbolic partial differential equations (PDEs), aiming for an exact alignment between the system output and the desired trajectory. Utilizing backstepping techniques, semigroup theory, the Hahn-Banach theorem, Laplace transforms, and Green's functions, our approach begins by establishing the OTP solution as a bounded feedback law, with Green's functions allowing the solution of a Cauchy-Euler equation. This is followed by deriving a detailed formulation for the complete state of the corresponding control problem. Finally, the approach provides an explicit formula for the OTP solution, using the desired trajectory and the solutions of specific kernel PDEs.

On Lie symmetries, soliton interaction nature and conservation laws of Broer-Kaup-Kupershmidt system in shallow water of uniform depth

Abstract The exact solutions of nonlinear partial differential equations are significant to analyze the soliton nature of physical phenomena. Therefore, the Lie symmetry method is introduced to obtain exact solutions of the Broer-Kaup-Kupershmidt system. It represents two-dimensional propagation of nonlinear and dispersive long gravity waves in shallow water with uniform depth. The present work is devoted to derive symmetry reductions, soliton solutions, and conserved vectors under one-parameter Lie group transformations. Meanwhile, the invariance property of Lie groups validates the reducibility of the system. Therefore, a continual process of reductions transmutes the test system into a system of ordinary differential equations. Thus, this integrability property leads to generalized exact solutions with parametric constraints. These solutions entail all three arbitrary functions, f 1(y), f 2(t), f 3(t) existing in infinitesimals and various free parameters. On account of existing functions and free parameters, the obtained solutions are more admirable than those of preceding works (Kassem and Rasheed 2019 Chinese J. Phys. 57 90–104, Kumar et al 2022 Math. Comput. Simulat. 196 319–35, Tanwar and Kumar 2024 J. Ocean Eng. Sci. 9 199–206) and helpful to describe significant physical nature under wide parametric ranges. The deductions of previous results on imposing the particular values to arbitrary functions and constants ensure the generalness and novelty of derived results. Moreover, adjoint equations and conserved quantities associated with Lagrangian formulation have been evolved via Lie symmetries. We numerically simulate these solutions to analyze their physical nature. Consequently, soliton fusion and fission, line solitons, doubly solitons, multisolitons, and bright and dark soliton nature are discussed, which validate the physical relevance of the solutions.

Optical soliton solutions of the M-fractional paraxial wave equation

This research used a modified and extended auxiliary mapping method to examine the optical soliton solutions of the truncated time M-fractional paraxial wave equation. We employed the truncated time M-fractional derivative to eliminate the fractional order in the governing model. The few optical wave examples of the paraxial wave condition can assume an insignificant part in depicting the elements of optical soliton arrangements in optics and photonics for the investigation of different actual cycles, including the engendering of light through optical frameworks like focal points, mirrors, and fiber optics. We identified the solution using a few free parameters regarding hyperbolic function form. We discovered periodic wave, bright and dark kink wave, bell wave, and singular soliton solution for the numerical values of the free parameters. To explain the behavior of various solutions, we have spoken the obtained solutions graphically for a physical explanation using MATLAB. The strategy introduced is fundamental and robust as a smart soliton solution for nonlinear partial differential equations, and it may play a crucial role in nonlinear optics, fiber optics, and communication systems.

Modelling and Structural Simulation of Thin-Wall Structure

This paper performs a non-linear analysis of a thin-walled structure. This simulation uses the finite element method to calculate actual conditions in a matrix calculation that is visualized. In the finite element method for numerical solutions of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved to ensure approximate solutions to the differential equations. A plate will be given different stiffener shapes with the same pressure value. There will be two boundary conditions applied at this thin-walled structure, displacement and fixed support will be applied at the side of the plate. The output obtained from this simulation are von mises stress, total deformation, elastic strain and strain energy. Each model of thin-walled structure will have various values of this output. And the outcome of this research will be helpful as the selected design of the best stiffener able to improve the performance of thin-walled structure for design and manufacture of marine and land transportation.

Geometry physics neural operator solver for solid mechanics

AbstractThis study developed Geometry Physics neural Operator (GPO), a novel solver framework to approximate the partial differential equation (PDE) solutions for solid mechanics problems with irregular geometry and achieved a significant speedup in simulation time compared to numerical solvers. GPO leverages a weak form of PDEs based on the principle of least work, incorporates geometry information, and imposes exact Dirichlet boundary conditions within the network architecture to attain accurate and efficient modeling. This study focuses on applying GPO to model the behaviors of complicated bodies without any guided solutions or labeled training data. GPO adopts a modified Fourier neural operator as the backbone to achieve significantly improved convergence speed and to learn the complicated solution field of solid mechanics problems. Numerical experiments involved a two‐dimensional plane with a hole and a three‐dimensional building structure with Dirichlet boundary constraints. The results indicate that the geometry layer and exact boundary constraints in GPO significantly contribute to the convergence accuracy and speed, outperforming the previous benchmark in simulations of irregular geometry. The comparison results also showed that GPO can converge to solution fields faster than a commercial numerical solver in the structural examples. Furthermore, GPO demonstrates stronger performance than the solvers when the mesh size is smaller, and it achieves over 3 and 2 speedup for a large degree of freedom in the two‐dimensional and three‐dimensional examples, respectively. The limitations of nonlinearity and complicated structures are further discussed for prospective developments. The remarkable results suggest the potential modeling applications of large‐scale infrastructures.

A Comparative Study of Partial Differential Equation Solving Methods and Their Applications

In this review, we undertake an in-depth survey of the traditional as well as modern methods used in finding solutions for partial differential equations (henceforth PDEs). We categorise these equations into three main kinds: elliptic, parabolic, and hyperbolic. We also give illustrative examples of these PDEs and discuss the applications of them in a range of fields. This range extends from fluid dynamics (hydrodynamics), as well as thermal (heat) conduction, to quantum mechanics. Our exploration features a number of analyses used in this regard such as variable splitting or defactorising in addition to the transforms invented by Fourier and Laplace. Not only this but also this survey takes in numerical methods ranging from grid-based (finite difference), mesh-based (finite element) to spectral. Also discussed in this paper is a range of special techniques that ranges from the variational techniques, Green's (fundamental solution) functions to perturbation (also known as Asymptotic expansion in addition to sketching the latest developments with respect to computational methods. This review also sheds light on current challenges that confront addressing complicated PDEs especially those nonlinear and multi-variable. In this regard, the paper calls for more research in order to develop more effective methods. The paper maps out the importance of PDE usages in real-life and its potential for more related discoveries in the future particularly with respect to areas such as machine learning and quantum computing.

Discretization-independent surrogate modeling of physical fields around variable geometries using coordinate-based networks

Abstract Numerical solutions of partial differential equations require expensive simulations, limiting their application in design optimization, model-based control, and large-scale inverse problems. Surrogate modeling techniques aim to decrease computational expense while retaining dominant solution features and characteristics. Existing frameworks based on convolutional neural networks and snapshot-matrix decomposition often rely on lossy pixelization and data-preprocessing, limiting their effectiveness in realistic engineering scenarios. Recently, coordinate-based multilayer perceptron networks have been found to be effective at representing 3D objects and scenes by regressing volumetric implicit fields. These concepts are leveraged and adapted in the context of physical-field surrogate modeling. Two methods toward generalization are proposed and compared: design-variable multilayer perceptron (DV-MLP) and design-variable hypernetworks (DVH). Each method utilizes a main network which consumes pointwise spatial information to provide a continuous representation of the solution field, allowing discretization independence and a decoupling of solution and model size. DV-MLP achieves generalization through the use of a design-variable embedding vector, while DVH conditions the main network weights on the design variables using a hypernetwork. The methods are applied to predict steady-state solutions around complex, parametrically defined geometries on non-parametrically-defined meshes, with model predictions obtained in less than a second. The incorporation of random Fourier features greatly enhanced prediction and generalization accuracy for both approaches. DVH models have more trainable weights than a similar DV-MLP model, but an efficient batch-by-case training method allows DVH to be trained in a similar amount of time as DV-MLP. A vehicle aerodynamics test problem is chosen to assess the method’s feasibility. Both methods exhibit promising potential as viable options for surrogate modeling, being able to process snapshots of data that correspond to different mesh topologies.

A unified theory for inertial manifolds, saddle point property and exponential dichotomy

A unified theory for inertial manifolds, saddle point property and exponential dichotomy

Modeling the structure and relaxation in glycerol-silica nanocomposites.

The relationship between the dynamics and structure of amorphous thin films and nanocomposites near their glass transition is an important problem in soft-matter physics. Here, we develop a simple theoretical approach to describe the density profile and the α-relaxation time of a glycerol-silica nanocomposite (S. Cheng et al., J. Chem. Phys., 2015, 143, 194704). We begin by applying the Derjaguin approximation, where we replace the curved surface of the particle with the planar one; thus, modeling the nanocomposite is reduced to that of a confined thin film. Subsequently, by employing the molecular dynamics (MD) simulation data of Cheng et al., we approximate the density profile of a supported liquid thin film as a stationary solution of a fourth-order partial differential equation (PDE). We then construct an appropriate density functional, from which the density profile emerges through the minimization of free energy. Our final assumption is that of a consistent, temperature-independent scaled density profile, ensuring that the free volume throughout the entire nanocomposite increases with temperature in a smooth, monotonic fashion. Considering the established relationship between glycerol relaxation time and temperature, we can employ Doolittle-type analysis ("naïve" free-volume model), to calculate the relaxation time based on temperature and film thickness. We then convert the film thickness into the interparticle distance and subsequently the filler volume fraction for the nanocomposites and compare our model predictions with experimental data, resulting in a good agreement. The proposed approach can be easily extended to other nanocomposite and film systems.

Comparing Solutions to the Third-Order Dispersive Partial Differential Equation

In previous decades, many of the practical problems arising in scientific fields such as mathematics, physics, chemistry, biology,and engineering have been related to nonlinear fractional partial differential equations. One of these nonlinear partial differential equations, the third-order dispersive partial differential equation, has been found to have a plethora of useful applications in different fields such as Newtonian fluid mechanics, optimal control, convection diffusion processes, hydrodynamics, and aerodynamics. A special class of solutions has been studied for the third-order dispersive partial differential equation including exact solutions and approximate solutions. The aim of this article is to compare were the Adomian decomposition method, the lines method, an exponential quartic spline and finite difference discretization method, and the non-polynomial spline methodwith the solution of the third-order dispersive partial differential equation. We will conduct a comparison of the stability of the two methods using the Von Neumann stability analysis. In addition, a numerical example will be presented to illustrate the accuracy of these methods.

Computational Approaches to Solving Some Partial Differential Equations and Neutrosophic Partial Differential with Variable Coefficients Using the Laplace Residual Power Series Method

We employ the Laplace Residual Power Series Method to approximate analytical solutions for differential equations and neutrosophic differential equations with associated parameters, including non-homogeneous equations and fractional formulas in partial differential equations (PDEs). This approach showcases the method's simplicity, effectiveness, and robustness in deriving analytical series solutions for PDEs that involve associated parameters, especially in the context of fractional differential equations. Several practical uses of LRPSM with an emphasis on non-homogeneous and partial differential equations and neutrosophic equations with fractions (PDEs). These applications are significant in a variety of scientific and engineering domains that simulate complicated dynamic system such as anomalous diffusion in physics, viscoelastic material modeling in engineering and signal processing.

Mitigating stop-and-go traffic congestion with operator learning

Mitigating stop-and-go traffic congestion with operator learning

Numerical Solution of Elliptic Partial Differential Equation: Method of Lines and Crank-Nicholson Method

The method of lines (MOL) is a solution procedure for solving partial differential equation (PDE) and the Crank-Nicholson method (CNM) is an implicit finite difference method, used to solve the elliptic equation and similar partial differential equations numerically. In this study, the numerical techniques were introduced to solve elliptic partial differential equation (PDE) together with initial and boundary conditions. The Poisson equation was considered as an elliptic PDE. Particularly, two methods were used to solve a two-dimensional Poisson equation numerically, such as the method of lines (MOL) and Crank-Nicholson method (CNM). The implementation of the solutions was done using Microsoft office spreadsheet and MATLAB programing language. Lastly, the numerical solutions were presented graphically which was obtained with method of lines along with Crank-Nicholson method.

Atomic solutions of some fractional partial differential equations

Atomic solutions of some fractional partial differential equations

Comparison of the combinations of Fourier pseudospectral method, finite volume method, Euler method and fourth order Runge-Kutta method used to solve Burgers equation

The Burgers equation is a mathematical model frequently used in Computational Fluid Dynamics. It is often employed to test and calibrate numerical methods, as it is one of the few nonlinear transport equations with an exact analytical solution. In this paper, numerical solutions are obtained using the Finite Difference Method (FDM) and the Fourier Pseudospectral Method (FPSM) for spatial discretization, combined with the Euler Method and the Fourth-Order Runge-Kutta Method (FRKM) for time discretization. The results are compared with the exact analytical solution in terms of accuracy, convergence rate, and computational cost. The findings indicate that the combination of the FDM and Euler Method achieves excellent computational efficiency when compared to the other approaches. Meanwhile, the combination of FPSM and FRKM demonstrates superior accuracy (achieving round-off errors) and a high order of convergence (spectral convergence order). Thus, combining methods with similar convergence rates and accuracy is the optimal strategy for obtaining efficient numerical solutions of partial differential equations (PDEs).