Partial Differential Equations Research Articles

Numerical Identification of Boundary Condition for Reaction-Advection-Diffusion Partial Differential Equation

In this study, we consider a reaction-advection-diffusion partial differential equations (PDEs) in a plane domain with missed boundary data. We applied both the KMF algorithm and the conjugate gradient method to reconstruct the missed data by using the spectral element method. Several numerical examples were given illustrating the convergence of the used algorithms.

Influence of Higher Order Chemical Reaction on Rotational and Permeable Radiative Flow with Interruption of Micro-organism

Objective: The concentration on the function of the non-Newtonian radiative Casson Nanofluid model via a permeable and stretchable cone has been investigated. The innovation of the problem is Casson, taken as base fluid, and Manganese Ferrite is dropped in the base fluid under the action of a higher-order chemical reaction. This study examines the convective magneto-hydrodynamic (MHD) flow of a conducting micro-organism fluid confined within a horizontal channel via stimulus of a transverse magnetic field (MHD) and thermal radiation parameter. Methods: The foremost partial differential equations (PDE) are changed into an arrangement of ordinary differential equations by applying adequate corresponding transform. The outputs of these equations are performed numerically employing the RK method and computed by well-established BVP4C MATLAB software Special features such as radiation, heat generation/absorption, Joule heating, and electrically conductive MHD are examined. Skin friction and Nusselt number are incorporated into the flow and results are tabulated and outcomes are discussed graphically. Findings: The system incorporates the effects of micro-organism, pressure gradient and the movement of the moving cone. The generated heat is significant enough to cause radiation from both the fluid and the dust particles. Solutions for velocity and temperature distributions of the fluid, as well as the velocity and energy of the dust particles, are obtained using a perturbation scheme. The analytical impact of numerous geometrical parameters on both fluid and dust particle profiles has been analyzed and illustrated graphically. The impact of nonlinear partial derivative formulations is developed subjected to boundary layer theory. Novelty: As labelled in the results, heterogeneous is the examination of thermal radiation in science and technology such as liquid metal fluids different momentum devices for missiles, and turbines of gas effects on thermal radiation are exhibited and distinguished in various investigations and mathematical applications uses of the view in which radiant production be dependent on temperature are point out as a new value to the prevailing literature. Keywords: Thermal radiation, Brownian motion, Buoyancy flow, Chemical reaction, Heat and Mass transfer, Rotating cone

Dynamic coefficient of flexural motion of beam experiencing simple support under successive moving loads

AbstractAnalytic expressions of the dynamic coefficient (DC) factor and vibrational behavior of a uniformly elastic isotropic beam with a simple boundary condition caused by accelerating masses with varying velocities are analyzed. The motion of this problem is described by a fourth‐order partial differential equation, which governs its behavior. The weighted residual method converts the governing equation into a sequence of linked second‐order differential equations to facilitate the analysis. A rewritten version of Struble's asymptotic method further simplifies the transformed governing equation. This modification aids reduction in the complexity of the equation. The closed‐form response is contrasted across three force motions: acceleration, deceleration, and uniform motion. The study thoroughly examines how different velocities and frequencies of the moving force affect the dynamic behavior of the beam. The study also examines the influence of load velocity on the DC of the beam subjected to pinned–pinned boundary conditions.

A proof of convergence and equivalence for 1D finite element methods and ReLU neural networks

This paper investigates the convergence and equivalence properties of the Finite Element Method (FEM) and Rectified Linear Unit Neural Networks (ReLU NNs) in solving differential equations. We provide a detailed comparison of the two approaches, highlighting their mutual capabilities in function space approximation. Our analysis proves the subset and superset inclusions that establish the equivalence between FEM and ReLU NNs for approximating piecewise linear functions. Furthermore, a comprehensive numerical evaluation is presented, demonstrating the error convergence behavior of ReLU NNs as the number of neurons per basis function varies. Our results show that while increasing the number of neurons improves approximation accuracy, this benefit diminishes beyond a certain threshold. The maximum observed error between FEM and ReLU NNs is 10-4, reflecting excellent accuracy in solving partial differential equations (PDEs). These findings lay the groundwork for integrating FEM and ReLU NNs, with important implications for computational mathematics and engineering applications.

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Euler–Lagrange equation for gradient-type Lagrangian and related conservation laws

AbstractVariational calculus with gradient-type variations has often been neglected, although it proves to be suitable for certain concrete problems governed by several evolution variables. These kinds of variations lead to Euler–Lagrange partial differential equations controlled by the right-hand member. In this context, we also introduce anti-trace Euler–Lagrange partial differential equations that are suitable for some innovative ideas. Also, some applications are provided for the theoretical results derived in the paper.

A scientific report on heat transfer analysis due to generated and absorbed heat over an oscillatory stretching sheet: a finite difference-based study.

A non-Newtonian liquid motion across a stretchable surface is relevant to various industrial applications, including extruding plastic sheets and stretching plastic films. In connection with this, the impact of endothermic and exothermic chemical reactions on the flow of rate-type liquid via an oscillatory stretching sheet in the presence of permeable media with the Maxwell liquid model is elucidated in the present study. Scientists and engineers may enhance the efficiency of chemical reactions or heat transmission by optimising system flow and investigating the effect of reactions on flow dynamics. The present study's governing partial differential equations (PDEs) are transformed into their non-dimensional form using similarity variables. The resulting equations are numerically solved using the finite difference method (FDM). Outcomes disclose that the temperature profile declines as the activation energy and unsteadiness parameters increase. The influence of the Maxwell and unsteadiness parameters on the fluid's velocity with respect to time is represented. The rise in the values of chemical reaction parameter upsurges the thermal profile. As the activation energy and unsteadiness parameters upsurge, the thermal profile declines. As the chemical reaction parameter and the ratio of oscillation frequency to stretching rate values rise, the concentration profile falls.

On Sawada-Kotera and Kaup-Kuperschmidt integrable systems

To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods that are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the M2 -extension of integrable such scalar equations. For illustration, we give Korteweg–de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations, we obtain some new integrable systems with recursion operators. We also give the soliton solutions of the systems and integrable standard nonlocal and shifted nonlocal reductions of these systems.

Analytical simulation of Darcy–Forchheimer nanofluid flow over a curved expanding permeable surface

Abstract This research paper presents an analytical simulation of Darcy–Forchheimer flow over a porous curve stretching surface. In fluid dynamics, the Darcy–Forchheimer model combines Forchheimer adjustment and high-velocity effects with Darcy’s formula for porous media flow: two nanofluid particles, molybdenum disulphide, and graphene oxide, form nanofluid with the base fluid blood. The governing partial differential equations for momentum and energy are converted into a nonlinear ordinary differential equations system by applying the appropriate similarity transformations. The homotopy analysis method is used to solve the transform equations analytically. The impact of essential factors includes the Forchheimer parameter, porosity parameter, slip parameter, Eckert number, nanoparticle volume friction, magnetic field parameter, and curvature parameter. The results have applications in the design of sophisticated cooling systems, where effective thermal control is essential.

Optimization of heat transfer in a double lid-driven cavity with isoperimetric heated blocks using GFEM.

This article is concerned with the examination of flow dynamics and heat transfer characteristics in a 1:4 double lid driven cavity in presence of isoperimetric heated blocks of various shapes. The focus is to identify the optimal shape that enhances the heat transfer in a tall cavity. The parametric settings are chosen in such a way that all the convection regimes including natural, forced and mixed convection could be generated. This cavity has lids positioned at the top and bottom, moving in opposite directions along the x-axis. The physical system is represented as a set of coupled partial differential equations incorporating the rheological properties of the power-law fluids (PL). The governing equations in conjunction with various non-dimensional physical parameters are simulated via Galerkin's Finite Element Method (GFEM) on a very fine hybrid grid. The study includes the computation of the Kinetic Energy and Average Nusselt number to determine the optimal shape. It is concluded that the circular block is superior to the other two in terms of heat transmission efficiency.

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Filtered partial differential equations: a robust surrogate constraint in physics-informed deep learning framework

Embedding physical knowledge into neural network (NN) training has been a hot topic. However, when facing the complex real world, most of the existing methods still strongly rely on the quantity and quality of observation data. Furthermore, the NNs often struggle to converge when the solution to the real equation is very complex. Inspired by large eddy simulation in computational fluid dynamics, we propose an improved method based on filtering. We analysed the causes of the difficulties in physics-informed machine learning, and proposed a surrogate constraint (filtered partial differential equation, FPDE) of the original physical equations to reduce the influence of noisy and sparse observation data. In the noise and sparsity experiment, the proposed FPDE models (which are optimized by FPDE constraints) have better robustness than the conventional PDE models. Experiments demonstrate that the FPDE model can obtain the same quality solution with 100 % higher noise and 12 % quantity of observation data of the baseline. Besides, two groups of real measurement data are used to show the FPDE improvements in real cases. The final results show that the FPDE still gives more physically reasonable solutions when facing the incomplete equation problem and the extremely sparse and high-noise conditions. The proposed FPDE constraint is helpful for merging real-world experimental data into physics-informed training, and it works effectively in two real-world experiments: simulating cell movement in scratches and blood velocity in vessels.

PICL: Physics informed contrastive learning for partial differential equations

Neural operators have recently grown in popularity as Partial Differential Equation (PDE) surrogate models. Learning solution functionals, rather than functions, has proven to be a powerful approach to calculate fast, accurate solutions to complex PDEs. While much work has been performed evaluating neural operator performance on a wide variety of surrogate modeling tasks, these works normally evaluate performance on a single equation at a time. In this work, we develop a novel contrastive pretraining framework utilizing generalized contrastive loss that improves neural operator generalization across multiple governing equations simultaneously. Governing equation coefficients are used to measure ground-truth similarity between systems. A combination of physics-informed system evolution and latent-space model output is anchored to input data and used in our distance function. We find that physics-informed contrastive pretraining improves accuracy for the Fourier neural operator in fixed-future and autoregressive rollout tasks for the 1D and 2D heat, Burgers’, and linear advection equations.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

A novel linetype recognition algorithm based on track geometry detection data

Abstract The detection and recognition of track geometry are crucial for ensuring railway transportation safety and maintenance efficiency. Traditional linetype recognition algorithms struggle to effectively handle the complexities of variable track geometry and noise interference. To address these challenges, a novel linetype recognition algorithm (NLRA) is proposed, which thoroughly analyzes the characteristics of track curvature across straight lines, transition curves, and circular segments. By employing partial differential equations for data preprocessing, NLRA effectively filters out noise. Utilizing curve design parameters and integrating line anti-noise recognition with multi-segment line fitting techniques, the algorithm accurately identifies each line segment and incorporates a single-point recognition method for data gaps. Experimental results demonstrate that NLRA outperforms traditional algorithms in terms of real-time performance, recognition accuracy, anti-interference capability, and noise handling. Notably, NLRA enhances accuracy by at least 35.1% and recall by at least 17.4% on selected typical routes. This work underscores the algorithm's potential to provide efficient and accurate tools for track maintenance, facilitate advancements in track detection software, and significantly contribute to railway transportation safety.

Analytical solutions of the Caudrey-Dodd-Gibbon equation using Khater II and variational iteration methods.

This study focuses on solving the Caudrey-Dodd-Gibbon ( ) equation using the Khater II ( ) method and the Variational Iteration ( ) method. The equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the and methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation's behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics.

Cattaneo‐Christov model for cross nanofluid with Soret and Dufour effects under endothermic/exothermic reactions: a modified Buongiorno tetra‐hybrid nanofluid approach

AbstractThe previous researchers used a slightly different version of the Buongiorno hybrid nanofluid (BHNF) concept. In the existing research, there is not a single effort that was undertaken to explore the effect that tetra hybrid nanoparticles (tethnf) implanted with the BHNF model have on liquid movement that is then exposed to an elastic sheet. This endeavor focuses on researching the implication of a modified Buongiorno tetra hybrid cross nanofluid concept on magnetized radiative cross nanofluid transport, along with impacts such as Cattaneo–Christov heat flux and Cattaneo–Christov mass flux, variable thermal conductivity, diffusion‐thermo and thermo‐diffusion effects, and endothermic/exothermic type chain reactions. This approach integrates both Buongiorno and tethnf. By inserting similar variables, the controlling model partial differential equations have been transformed into dimensionless Ordinary differential equations. These equations are then treated computationally with the help of the Lobatto III A technique. The MATLAB computer program is used to produce simulation solutions, as well as the findings, which are displayed in the form of graphs and tables. Based on the results that have been gathered, it was discovered that the speed of heat delivery increases when there is an increase in the speed of an endothermic reaction. When the Dufour and thermal relaxation factors are increased, the speed at which heat is transmitted also increases. The fluid viscosity becomes more pronounced as a result of an increase in the viscosity index n, the Weissenberg number We, and the magnetic field M, all of which contribute to a reduction in the velocity distribution. A progressive change in the radiation parameters Rd, Dufour effect, and thermal conductivity which magnifies Nusselt number causes an increase in the heat delivery rate, which in turn amplifies Nusselt number.

Kolmogorov-Arnold Representation Based Informed Orchestration in Solving Partial Differential Equations for Carbon Balance Sustainability

This paper introduces the Kolmogorov-Arnold Credit Informed Network Orchestration (KACINO), a novel framework proposed to comprehensively structure and analyze the carbon dynamic system. By synthesizing dynamic processes such as carbon emission and sequestration, a credit information of carbon dynamics is built accumulation, KACINO provides a holistic approach to model the complexities inherent in carbon dynamics. Through systematic analysis and scenario simulations, KACINO facilitates informed policy recommendations aimed at optimizing carbon credit utilization and achieving sustainable environmental outcomes. This paper outlines the theoretical foundation, methodology, and practical applications of KACINO, highlighting its potential to support transformative strategies in climate change mitigation and sustainable development.

Uncertainty analysis of MHD oscillatory flow of ternary nanofluids through a diverging channel: a comparative study of nanofluid composites

Purpose This study aims to investigate the heat and mass transfer characteristics of a temperature-sensitive ternary nanofluid in a porous medium with magnetic field and the Soret–Dufour effect through a tapered asymmetric channel. The ternary nanofluid consists of Boron Nitride Nanotubes (BNNT), silver (Ag) and copper (Cu) nanoparticles, with a focus on understanding the thermal behaviour and performance across mono, hybrid and tri-hybrid nanofluids. This paper also examines the thermal behaviour of MHD oscillatory nanofluid flow and carries out an uncertainty analysis of the model using the Taguchi method. Design/methodology/approach The governing equations for this system are transformed into coupled linear partial differential equations using non-similarity transformations and solved numerically with the Crank–Nicolson scheme. The impact of temperature sensitivity at three distinct temperatures (5°C, 20°C and 60°C) is incorporated to analyse variations in viscosity and Prandtl number. The study also examines the combined effects of Soret–Dufour numbers and thermal radiation on heat and mass transfer within the nanofluid. Findings The results demonstrate that the inclusion of BNNT, Ag and Cu nanoparticles significantly enhances heat and mass transfer rate, with copper nanoparticles showing superior performance in terms of skin friction and heat transfer rates. The Soret and Dufour effects play critical roles in modulating heat and mass diffusion within tri-hybrid nanofluids. The study reveals that temperature sensitivity alters heat and mass transfer characteristics depending on the temperature range, with pronounced variations at elevated temperatures. The influence of thermal radiation and the Peclet number is found to significantly impact temperature distribution and overall heat transfer performance within the asymmetric channel. Originality/value To the best of the authors’ knowledge, this study is the first to analyse the heat and mass diffusion in a ternary nanofluid composed of BNNT, Ag and Cu nanoparticles, considering porous media, oscillatory flow and thermal radiation within a tapered asymmetric channel. The research extends to a novel examination of temperature sensitivity in mono, hybrid and tri-hybrid nanofluids at varying temperature gradients. Furthermore, a comparative analysis of skin friction and heat transfer rates between copper, alumina and ferro composites is presented for optimising the nanofluid performance.

Calibration of the Ueno’s Shadow Rate Model of Interest Rates

Shadow rate models of interest rates are based on the assumption that the interest rates are determined by an unobservable shadow rate. This idea dates back to Fischer Black, who understood the interest rate as an option that cannot become negative. Its possible zero values are consequences of negative values of the shadow rate. In recent years, however, the negative interest rates have become a reality. To capture this behavior, shadow rate models need to be adjusted. In this paper, we study Ueno’s model, which uses the Vasicek process for the shadow rate and adjusts its negative values when constructing the short rate. We derive the probability properties of the short rate in this model and apply the maximum likelihood estimation method to obtain the parameters from the real data. The other interest rates are—after a specification of the market price of risk—solutions to a parabolic partial differential equation. We solve the equation numerically and use the long-term rates to fit the market price of risk.