Solution Of Partial Differential Equations Research Articles

Analytical investigation of space–time shifted nonlocal stochastic Sasa–Satsuma equation using the enhanced modified extended tanh-expansion method for stochastic solitary waves Solutions

In this paper, discover exact solutions of the space–time shifted nonlocal stochastic Sasa–Satsuma equations using the enhanced modified extended tanh-expansion method. This method is a sophisticated mathematical approach for analytical and approximate solution of a nonlinear partial differential equations. The space–time shifted nonlocal stochastic Sasa–Satsuma equations define the behavior of two complex-valued functions, u and v, in a nonlinear system. These equations are complicated to solve analytically because they contain a number of components that describe space–time shifting, nonlocal interactions, and stochastic problems. To increase reliability and effectiveness of the solutions, this method couples the extended tanh-function method with an enhanced method. In order to demonstrate how white noise affects the solutions to the space–time shifted nonlocal stochastic Sasa–Satsuma equations, we also use Matlab to create 3D surfaces and contour graphs of exact solutions.

Unraveling Drug Delivery from Cyclodextrin Polymer-Coated Breast Implants: Integrating a Unidirectional Diffusion Mathematical Model with COMSOL Simulations.

Breast cancer ranks among the most commonly diagnosed cancers worldwide and bears the highest mortality rate. As an integral component of cancer treatment, mastectomy entails the complete removal of the affected breast. Typically, breast reconstruction, involving the use of silicone implants (augmentation mammaplasty), is employed to address the aftermath of mastectomy. To mitigate postoperative risks associated with mammaplasty, such as capsular contracture or bacterial infections, the functionalization of breast implants with coatings of cyclodextrin polymers as drug delivery systems represents an excellent alternative. In this context, our work focuses on the application of a mathematical model for simulating drug release from breast implants coated with cyclodextrin polymers. The proposed model considers a unidirectional diffusion process following Fick's second law, which was solved using the orthogonal collocation method, a numerical technique employed to approximate solutions for ordinary and partial differential equations. We conducted simulations to obtain release profiles for three therapeutic molecules: pirfenidone, used for preventing capsular contracture; rose Bengal, an anticancer agent; and the antimicrobial peptide KR-12. Furthermore, we calculated the diffusion profiles of these drugs through the cyclodextrin polymers, determining parameters related to diffusivity, solute solid-liquid partition coefficients, and the Sherwood number. Finally, integrating these parameters in COMSOL multiphysics simulations, the unidirectional diffusion mathematical model was validated.

Some New Families of Exact Solitary Wave Solutions for Pseudo-Parabolic Type Nonlinear Models

The objective of the current study is to provide a variety of families of soliton solutions to pseudo-parabolic equations that arise in nonsteady flows, hydrostatics, and seepage of fluid through fissured material. We investigate a class of such equations, including the one-dimensional Oskolkov (1D OSK), the Benjamin-Bona-Mahony (BBM), and the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation. The Exp (-ϕξ)-expansion method is used for new hyperbolic, trigonometric, rational, exponential, and polynomial function-based solutions. These solutions of the pseudo-parabolic class of partial differential equations (PDEs) studied here are new and novel and have not been reported in the literature. These solutions depict the hydrodynamics of various soliton shapes that can be utilized to study the nature of traveling wave solutions of other nonlinear PDE’s.

Approximate Solution of Linear and Nonlinear Partial Differential Equations Using Picard’s Iterative Method

This work discusses a new description of Picard’s iterative method (PIM) to get the approximate or sometimes exact solution of (linear and nonlinear) partial differential equations ,Provided that by examples to illustrate the reliability and ease of this method.

Solutions of linear and non-linear partial differential equations by means of tensor product theory of Banach space

In this article, we introduce an analytical method for solving both non-separable linear and non-linear partial differential equations, for which separation of variables method does not work. This method is based on the theory of tensor product in Banach spaces coupled with some properties of atoms operators. We provide some illustrative examples.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/28/abstr.html

An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations

In recent years, machine learning algorithms, especially deep learning, have shown promising prospects in solving Partial Differential Equations (PDEs). However, as the dimension increases, the relationship and interaction between variables become more complex, and existing methods are difficult to provide fast and interpretable solutions for high-dimensional PDEs. To address this issue, we propose a genetic programming symbolic regression algorithm based on transfer learning and automatic differentiation to solve PDEs. This method uses genetic programming to search for a mathematically understandable expression and combines automatic differentiation to determine whether the search result satisfies the PDE and boundary conditions to be solved. To overcome the problem of slow solution speed caused by large search space, we propose a transfer learning mechanism that transfers the structure of one-dimensional PDE analytical solution to the form of high-dimensional PDE solution. We tested three representative types of PDEs, and the results showed that our proposed method can obtain reliable and human-understandable real solutions or algebraic equivalent solutions of PDEs, and the convergence speed is better than the compared methods. Code of this project is at https://github.com/grassdeerdeer/HD-TLGP.

SNN-PDE: Learning Dynamic PDEs from Data with Simplicial Neural Networks

Dynamics of many complex systems, from weather and climate to spread of infectious diseases, can be described by partial differential equations (PDEs). Such PDEs involve unknown function(s), partial derivatives, and typically multiple independent variables. The traditional numerical methods for solving PDEs assume that the data are observed on a regular grid. However, in many applications, for example, weather and air pollution monitoring delivered by the arbitrary located weather stations of the National Weather Services, data records are irregularly spaced. Furthermore, in problems involving prediction analytics such as forecasting wildfire smoke plumes, the primary focus may be on a set of irregular locations associated with urban development. In recent years, deep learning (DL) methods and, in particular, graph neural networks (GNNs) have emerged as a new promising tool that can complement traditional PDE solvers in scenarios of the irregular spaced data, contributing to the newest research trend of physics informed machine learning (PIML). However, most existing PIML methods tend to be limited in their ability to describe higher dimensional structural properties exhibited by real world phenomena, especially, ones that live on manifolds. To address this fundamental challenge, we bring the elements of the Hodge theory and, in particular, simplicial convolution defined on the Hodge Laplacian to the emerging nexus of DL and PDEs. In contrast to conventional Laplacian and the associated convolution operation, the simplicial convolution allows us to rigorously describe diffusion across higher order structures and to better approximate the complex underlying topology and geometry of the data. The new approach, Simplicial Neural Networks for Partial Differential Equations (SNN PDE) offers a computationally efficient yet effective solution for time dependent PDEs. Our studies of a broad range of synthetic data and wildfire processes demonstrate that SNN PDE improves upon state of the art baselines in handling unstructured grids and irregular time intervals of complex physical systems and offers competitive forecasting capabilities for weather and air quality forecasting.

SHoP: A Deep Learning Framework for Solving High-Order Partial Differential Equations

Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is widely used to approximate the solutions of PDEs. However, existing works are incapable of solving high-order PDEs due to insufficient calculation accuracy of higher-order derivatives, and the final network is a black box without explicit explanation. To address these issues, we propose a deep learning framework to solve high-order PDEs, named SHoP. Specifically, we derive the high-order derivative rule for neural network, to get the derivatives quickly and accurately; moreover, we expand the network into a Taylor series, providing an explicit solution for the PDEs. We conduct experimental validations four high-order PDEs with different dimensions, showing that we can solve high-order PDEs efficiently and accurately. The source code can be found at https://github.com/HarryPotterXTX/SHoP.git.

Neural Oscillators for Generalization of Physics-Informed Machine Learning

A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

The Lie point symmetry criteria and formation of exact analytical solutions for Kairat-II equation: Paul-Painlevé approach

The renowned non-linear Kairat-II equation, which expresses the surface curves, is examined meticulously in this work. There has never been a study before that discussed conserved quantities, sensitivity analysis, Hamiltonian function, Lie symmetry invariance criteria, and invariant solutions of Kairat-II equation The Lie invariance criteria are taken into consideration by the symmetry generators. The suggested technique leads to in a four-dimensional Lie algebra, where the translation point symmetries in space and time correlate with the conservation of mass and the energy, respectively, and remaining point symmetries are dilation and scaling. Firstly, we use symmetry reduction of Lie subalgebras to obtain closed-form invariant solutions. In specific reduction cases, we transform the Kairat-II equation into a spectrum of non-linear ordinary differential equations, which have the benefit of being able to offer an extensive selection of closed-form solitary wave solutions. For this equation, the Cauchy problem cannot be solved by the inverse scattering transform, therefore, traveling wave exact solutions are generated using the analytical Paul-Painlevé approach. In both two and three dimensions, the graphical behavior of specific solutions is presented for specific quantities of physical factors of examined equation. Utilizing conservation laws multipliers, the study culminates by determining an extensive set of local conservation laws for the non-linear Kairat-II equation that are applicable to arbitrary constant coefficients. Instead of focusing on the physical aspects of conservation laws, this study computed the conserved quantities using a mathematical perspective that can be used to identify potential symmetries of partial differential equations, guiding readers toward the solutions of partial differential equations. The existence of the Hamiltonian function is presented. The model’s sensitivity to various starting conditions is highlighted by the sensitivity analysis.

Plane Stress Problems for Isotropic Incompressible Hyperelastic Materials

The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If however the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean material.

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

Weak-PDE-LEARN: A weak form based approach to discovering PDEs from noisy, limited data

We introduce Weak-PDE-LEARN, a Partial Differential Equation (PDE) discovery algorithm that can identify non-linear PDEs from noisy, limited measurements of their solutions. Weak-PDE-LEARN uses an adaptive loss function based on weak forms to train a neural network, U, to approximate the PDE solution while simultaneously identifying the governing PDE. This approach yields an algorithm that is robust to noise and can discover a range of PDEs directly from noisy, limited measurements of their solutions. We demonstrate the efficacy of Weak-PDE-LEARN by learning several benchmark PDEs.

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing in space, (ii) multigroup-in-energy, and (iii) discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10−5.

Explicit solitary wave profiles and stability analysis of biomembranes and nerves

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system’s astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields.

Exact solutions for nonlinear partial differential equations via a fusion of classical methods and innovative approaches

This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form utt+a(x,t)uxt+b(t)ut=α(x,t)+G(u)(ut+a(x,t)ux)e-∫b(t)dt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{tt}+a(x,t)u_{xt}+b(t)u_{t}=\\alpha (x,t)+ G(u)(u_{t}+a(x,t)u_{x})e^{-\\int b(t)dt}$$\\end{document} and utm(utt+a(x,t)uxt)+b(t)utm+1=e-(m+1)∫b(t)dt(ut+a(x,t)ux)F(u,ute∫b(t)dt).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{t}^{m}(u_{tt}+a(x,t)u_{xt})+b(t)u_{t}^{m+1}=e^{-(m+1)\\int b(t)dt}(u_{t}+a(x,t)u_{x}) F(u,u_{t}e^{\\int b(t)dt}).$$\\end{document} We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.

Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes

Training nonlinear parametrizations such as deep neural networks to numerically approximate solutions of partial differential equations is often based on minimizing a loss that includes the residual, which is analytically available in limited settings only. At the same time, empirically estimating the training loss is challenging because residuals and related quantities can have high variance, especially for transport-dominated and high-dimensional problems that exhibit local features such as waves and coherent structures. Thus, estimators based on data samples from un-informed, uniform distributions are inefficient. This work introduces Neural Galerkin schemes that estimate the training loss with data from adaptive distributions, which are empirically represented via ensembles of particles. The ensembles are actively adapted by evolving the particles with dynamics coupled to the nonlinear parametrizations of the solution fields so that the ensembles remain informative for estimating the training loss. Numerical experiments indicate that few dynamic particles are sufficient for obtaining accurate empirical estimates of the training loss, even for problems with local features and with high-dimensional spatial domains.

Optimal Choice of the Auxiliary Equation for Finding Symmetric Solutions of Reaction–Diffusion Equations

This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case.

Taylor Polynomials in a High Arithmetic Precision as Universal Approximators

Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.

Utilizing a third order difference scheme in numerical method of lines for solving the heat equation

Obtaining the numerical solution for partial differential equations (PDEs) is a fundamental task in computational mathematics and scientific computing. In this work, we propose a third order numerical method of lines (MOL) based on difference schemes for the efficient and accurate approximation of heat equation. The method combines the flexibility of MOL with the superior accuracy of higher order difference schemes. The proposed approach begins by discretizing the spatial domain, and then applies higher order difference schemes to approximate the spatial derivatives in the PDE. The MOL framework is utilized to transform the resultant set of ordinary differential equations to a system of algebraic equations, facilitating their solution through an appropriate numerical method of choice. We have utilized the Adams method for this purpose. We analyze the stability of the proposed method and establish conditions under which it yields accurate and reliable solutions. We present the theoretical foundations of the method and discuss its implementation. We also present numerical results that demonstrate the procedure’s precision and effectiveness. The results show that the method is able to achieve high accuracy with relatively few grid points, and the results are compared with those obtained by some existing numerical schemes accessible in the literature.