One-dimensional Partial Differential Equations Research Articles

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.

Mathematical Simulation of Bio-Heat Transfer in Tissues Having Five Layers in the Presence of a Tumor Zone

A presence of tumor zones within biological tissues can be defined during the analysis of the skin surface temperature. This research is devoted to mathematical simulation of the time-dependent bio-heat transfer in tissues under a tumor influence. The one-dimensional partial differential equation of the Pennes model has been used for description of bio-heat transfer within the biological tissue with five layers, namely, epidermis, papillary dermis, reticular dermis, subcutaneous adipose tissue, and a muscle layer. The formulated boundary-value problem has been solved using the developed in-house computational code based on the finite difference schemes. The developed numerical algorithm has been verified using analytical and numerical solutions of other authors for the simpler test problem. As a result of this study, the temperature distributions have been obtained for the tissue in the presence of tumor zones in different layers of the skin. The influence of five layers of skin on the temperature distribution has been investigated, and the dependence for the skin surface temperature on the tumor zone location has been obtained. The obtained outcomes illustrate the effectiveness of this technique of cancer diagnosis and identify the optimal parameters for its application. Thus, this work represents an important step in the development of cancer diagnosis methods using thermography. The results obtained can be used to improve the accuracy of diagnosis and develop new treatment methods.

Trans-Net: A transferable pretrained neural networks based on temporal domain decomposition for solving partial differential equations

Physics-Informed Neural Networks (PINNs) has provided a novel direction for solving partial differential equations (PDEs) and has achieved significant advancements in the field of scientific computing. PINNs effectively incorporate the physical constraints of equations into the loss function, enabling neural networks to learn and approximate the behavior of physical systems by optimizing the loss function. According to the existing research, we propose a transferable pretrained neural networks framework based on temporal domain decomposition to solve partial differential equations. Specifically, we divide the domain into multiple subdomains according to time, each of which is solved using an individual neural network. Subsequently, the trained subnetwork and the predicted values at the common boundary are used as the pretrained model and initial values for the next subdomain, respectively. This not only improves the prediction accuracy of the subnetwork, but also reduces the training time. Finally, we present a series of classical numerical experiments including one-dimensional, two-dimensional, and three-dimensional partial differential equations. The experimental results indicate that the proposed method outperforms existing approaches in terms of accuracy and efficiency. Relevant data and code can be accessed on GitHub via https://github.com/karry-Zhang/Trans-Net.

Novel Injectable, Self-Healing, Long-Effective Bacteriostatic, and Healed-Promoting Hydrogel Wound Dressing and Controlled Drug Delivery Mechanisms.

Multivalent ion cross-linking has been used to form hydrogels between sodium alginate (SA) and hyaluronic acid (HA) in previous studies. However, more stable and robust covalent cross-linking is rarely reported. Herein, we present a facile approach to fabricate a SA and HA hydrogel for wound dressings with injectable, good biocompatibility, and high ductility. HA was first reacted with ethylenediamine to graft an amino group. Then, it was cross-linked with oxidized SA with dialdehyde to form hydrogel networks. The dressing can effectively promote cell migration and wound healing. To increase the antibacterial property of the dressing, we successfully loaded tetracycline hydrochloride into the hydrogel as a model drug. The drug can be released slowly in the alkaline environment of chronic wounds, and the hydrogel releases drugs again in the more acidic environment with wound healing, achieving a long-term antibacterial effect. In addition, one-dimensional partial differential equations based on Fickian diffusion with time-varying diffusion coefficients and hydrogel thicknesses were used to model the entire complex drug release process and to predict drug release.

Fourier-Gegenbauer pseudospectral method for solving time-dependent one-dimensional fractional partial differential equations with variable coefficients and periodic solutions

In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and periodic solutions. A main ingredient of our work is the use of the recently developed periodic RL/Caputo fractional derivative (FD) operators with sliding positive fixed memory length of Bourafaet al. (2021) or their reduced forms obtained by Elgindy (2023) as the natural FD operators to accurately model FPDEs with periodic solutions. The proposed method converts the IVP into a well-conditioned linear system of equations using the PS method based on Fourier collocations and Gegenbauer quadratures. The reduced linear system has a simple special structure and can be solved accurately and rapidly by using standard linear system solvers. A rigorous study of the computational storage requirements as well as the error and convergence of the proposed method is presented. The idea and results presented in this paper are expected to be useful in the future to address more general problems involving FPDEs with periodic solutions.

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

High-order spectral collocation method using tempered fractional Sturm–Liouville eigenproblems

This paper presents an accurate exponential tempered fractional spectral collocation method (TFSCM) to solve one-dimensional and time-dependent tempered fractional partial differential equations (TFPDEs). We use a family of tempered fractional Sturm–Liouville eigenproblems (TFSLP) as a basis and the fractional Lagrange interpolants (FLIs) that generally satisfy the Kronecker delta (KD) function at the employed collocation points. Firstly, we drive the corresponding tempered fractional differentiation matrices (TFDMs). Then, we treat with various linear and nonlinear TFPDEs, among them, the space-tempered fractional advection and diffusion problem, the time-space tempered fractional advection–diffusion problem (TFADP), the multi-term time-space tempered fractional problems, and the time-space tempered fractional Burgers’ equation (TFBE) to investigate the numerical capability of the fractional collocation method. The study includes a numerical examination of the produced condition number κ(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa (A)$$\\end{document} of the linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-norm error and exponential rate of spectral convergence.

A decoupled approach for determination of the joint probability density function of a high-dimensional nonlinear stochastic dynamical system via the probability density evolution method

The joint probability density function (PDF) of multiple response processes of a system is a crucial topic in the fields of science and engineering. It is adopted to describe the dependent uncertainty propagation in complex stochastic dynamical systems, including the displacement and velocity of one degree of freedom (DOF), and the nonlinear probabilistic dependencies between multiple variables. However, the partial differential equations (PDEs) governing the time evolution of joint PDFs for multiple stochastic processes are often high-dimensional and coupled, making them analytically or numerically intractable for systems with high dimensionality. The probability density evolution method (PDEM) has provided a new perspective from the standpoint of random events description of the principle of preservation of probability. For an n-dimensional system, if merely the joint PDF of m processes is needed, where m is not larger than n, then it is sufficient to solve the generalized density evolution equation (GDEE), i.e., the Li–Chen equation, in m dimensions. This approach is particularly convenient for cases where n is large while m is small. However, for cases where m is not less than two, it is still challenging to avoid solving high-dimensional PDEs. This paper proposes a new method for the determination of joint PDF of multiple responses based on the principle of preservation of probability. To capture the joint PDF of m processes, the m-dimensional PDE can be transformed into m one-dimensional PDEs, significantly reducing the computational complexity of numerical implementation. The proposed method gives an enhanced version to the PDEM, enabling it to not only determine the individual probabilistic response, but also efficiently determine the dependent probabilistic distribution of multiple responses of high-dimensional stochastic dynamical systems without solving the coupling PDE. It reduces the computational cost from exponential growth of m powers to linear growth. Some numerical examples are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Quantum algorithm for smoothed particle hydrodynamics

We present a quantum computing algorithm for the smoothed particle hydrodynamics (SPH) method. We use a normalization procedure to encode the SPH operators and domain discretization in a quantum register. We then perform the SPH summation via an inner product of quantum registers. Using a one-dimensional function, we test the approach in a classical sense for the kernel sum and first and second derivatives of a one-dimensional function, using both the Gaussian and Wendland kernel functions, and compare various register sizes against analytical results. Error convergence is exponentially fast in the number of qubits. We extend the method to solve the one-dimensional advection and diffusion partial differential equations, which are commonly encountered in fluids simulations. This work provides a foundation for a more general SPH algorithm, eventually leading to highly efficient simulations of complex engineering problems on gate-based quantum computers.

A novel fitted numerical scheme for singularly perturbed delay parabolic problems with two small parameters

In this paper, fitted operator finite difference methods are presented for two-parameter singularly perturbed one-dimensional parabolic partial differential equations with a delay in the time variable. Such boundary value problems are frequently encountered in the spatial diffusion of reactants and in control systems. To approximate the solution of the problem, we consider the Crank–Nicolson method for time discretization and the non-standard scheme for space discretization on uniform meshes. We prove that the proposed numerical method is uniformly convergent, having convergence of order two in time and one in space. Further, we present numerical results for two test problems in support of our theoretical findings and to demonstrate that the non-standard scheme is more effective than the hybrid scheme comprising central difference, upwind, and mid-point.

Solving a one-dimensional moving boundary problem based on wave digital principles

We report on a novel method for solving one-dimensional moving boundary problems based on wave digital principles. Here, we exploit multidimensional wave digital algorithms to derive an efficient and robust algorithm for the solution of the considered problem. Our method lets the wave digital model, on which the algorithm is based, expand according to the size of the solution domain. The expanding model introduces new dynamical elements, which must be properly initialized to obtain a calculable algorithm. To deal with this problem, we make use of linear multistep methods to extrapolate the missing values. Our results show the proposed method to indeed be capable of correctly solving a one-dimensional partial differential equation describing a growing biological axon.

The Large Deviation of Semi-linear Stochastic Partial Differential Equation Driven by Brownian Sheet

We prove the the large deviation principle(LDP) for the law of the one-dimensional semilinear stochastic partial differential equations driven by nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of global solution. Then the large deviation principle is obtained via weak convergence method.

Damping-induced dispersion in simple waveguides

The goal is to gain a non-dimensional formulation of the complex wavenumber knowing the onedimensional partial differential equation for wave propagation and the harmonic wave approach. This can be disassembled into its components, namely the imaginary part, showing the decay of a wave, and the real part, showing its spatial propagation. Having an eye on damping in waveguides including internal and external damping, we discuss the resulting consequences, which are frequency-dependent phase and group velocities (dispersion), as well as the energy transport and dissipation. This bears relevance for many domains of physics such as seismic and electromagnetic waves.

Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

There are various linear and nonlinear one-dimensional partial differential equations that are the focus of this research. There are a large number of these equations that cannot be solved analytically or precisely. The evaluation of nonlinear partial differential equations, even if analytical solutions exist, may be problematic. Therefore, it may be necessary to use approximate analytical methodologies to solve these issues. As a result, a more effective and accurate approach must be investigated and analyzed. It is shown in this study that the Lagrange multiplier may be used to get an ideal value for parameters in a functional form and then used to construct an iterative series solution. Linear and nonlinear partial differential equations may both be solved using the variational iteration method (VIM) method, thanks to its high computing power and high efficiency. Decoding and analyzing possible Korteweg-De-Vries, Benjamin, and Airy equations demonstrates the method’s ability. With just a few iterations, the produced findings are very effective, precise, and convergent to the exact answer. As a result, solving nonlinear equations using VIM is regarded as a viable option.

Solutions of Some Wave Mechanics Models

We consider one-dimensional second order partial differential equations describing waves in inhomogeneous and nonlinear media. Contact transformations and Euler differential substitution are used to construct general solutions. General and partial solutions of some nonstationary continuum mechanics models are found.

Fun with Finite Difference

We summarize most of what you need to know about finite difference solution for one-dimensional partial differential equations in finance. We demonstrate the theoretical results by numerical experiments. The reader is invited and encouraged to replicate the experiments using the C++ code and spreadsheets in our GitHub repository.

A Dynamic Model of CO2 Adsorption by Zeolite 13X Pellets

Novel techniques to capture carbon dioxide (CO2) from gas mixtures include physical adsorption on pelleted solid substrates. In this paper, the behaviour of spherical zeolite pellets is modelled, by resorting to one-dimensional partial differential equations, along the pellet radial direction, to express CO2 mass and energy conservation inside the porous material. One of the innovative contributions is that a dynamic description is adopted also for adsorption/desorption kinetics. The uncertain parameters in the model can be identified with a two-step procedure thanks to real data collected in two different sets of experiments. First, static isothermal tests carried out on few pellets by means of a gas adsorption analyzer allow to identify the available sorbent active sites (saturation capacities) and the ratio of the desorption and adsorption kinetic coefficients. Then, dynamic tests carried out with the same analyzer with small progressive CO2 injections allow to estimate desorption and adsorption kinetic coefficients separately. The pellet dynamic model thus tuned can be adopted in future work as a core element to mathematically describe a whole adsorption column filled with many pellets.

Fingero-imbibition phenomenon with magnetic field effect in two-phase flow through porous medium

This paper discusses the problem of fingero-imbibition phenomenon with magnetic field effect. The problem arises in two-phase flow through homogeneous porous medium during secondary oil recovery process. The mathematical formulation gives us one-dimensional nonlinear partial differential equation. Homotopy analysis method is adopted to solve the governing equation. The solution is discussed numerically and graphically with the help of Mathematica BVPh package.

Fuzzy solution of nonlinear Boussinesq equation

Abstract In this paper, the solution of the one-dimensional second-order unsteady nonlinear fuzzy partial differential Boussinesq equation is examined. The physical problem concerns unsteady flow in a semi-infinite unconfined aquifer bordering a lake. In the examined problem, there is a sudden rise and subsequent stabilization of the lake's water level, thus the aquifer is recharging from the lake. The aquifer boundary conditions are considered fuzzy and, therefore, ambiguities are created to the solution of the overall physical problem. Then, the fuzzy problem is translated to a system of crisp boundary value problems. By using a Boltzmann transformation, the crisp problem is transformed into an integro-differential equation and solved with the help of a special numerical method. This method has a simple iterative procedure, which converges rapidly and is proven very accurate in comparison with other analytical methods. Additionally, the algebraic equation estimates very close to the storage coefficient, the flux at the stream-aquifer origin, and the water stored volume with respect to other exact solutions. This method is very useful for practical cases (artificial recharge problems, irrigation, and drainage projects), giving the possibility to decision-makers to take the right decision.

Comparison of barrel-shaped fixed bed drying for different ventilation methods by mathematic simulation

Unsteady one-dimensional partial differential equations for heat and mass transfer in peanut drying are used in mathematical simulations to compare the drying performance of a barrel-shaped bed in three ventilation modes: bottom up, inside out and outside in. Results obtained using the finite difference scheme show that the mathematical simulation reliably predicts the drying process of peanuts, and that the drying delay of bottom-up ventilation is the most significant at the beginning of drying. This is because the drying enhancement afforded by the high average velocity of air along the ventilation direction is insufficient to offset the drying delay caused by the large ventilation thickness compared with that afforded by the inside-out and outside-in ventilation. Furthermore, as the air volume flow increases, the time consumption and moisture content difference decrease, whereas the productivity, energy consumption, and uniformity increase for all three ventilation modes, consistent with the classical ventilation drying law. Under the same air volume flow, the time and energy consumptions for the bottom-up ventilation are the lowest, followed by those for the outside-in and inside-out ventilation. The productivity afforded by the bottom-up ventilation is the highest, followed by that by the outside-in and inside-out ventilation. The outside-in ventilation indicates the least moisture content difference and the best drying uniformity. Meanwhile, the inside-out ventilation indicates the most significant moisture content difference and the worst drying uniformity.