Partial Differential Equations Research Articles

Level-set based shape optimization for plane elastic structures using radial basis functions and Hilbertian descent direction

Structural optimization problems are often associated with the so-called shape functionals depending on a shape through its geometry and the state being a solution of given partial differential equation. In such a framework it is convenient to work with the gradient-like method based on a concept of a shape derivative and level set method. The key idea of level set method is to represent the structural boundary with zero level set of given function (level set function—LSF). Now, changing the shape of a structure under optimization is equivalent to transport the LSF in such a direction that ensures decreasing the value of the objective functional. To this end, we make use of coercive bilinear form taken from the weak formulation of elasticity problem to obtain descent direction at each iteration. This descent direction is a solution of an additional variational problem, involving the bilinear form mentioned above and the volumetric expression of the shape derivative plays the role of a linear form. In this paper, we combine level set method with radial basis functions (RBFs) used to approximate LSF. We focus on the so-called multiquadric RBFs, but other classes of RBFs are also briefly considered. This eventually leads to transformation of partial differential equation (linear transport equation governing the evolution of shapes) to a system of linear ordinary differential equations which admits analytical formula for the solution. We apply our method to compliance minimization of a cantilever problem as well as to total potential energy minimization of a structure with kinematic loading. To run all the numerical experiments, we wrote our own code in Wolfram Mathematica environment.

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

Longitudinal wave propagation in FG rods under impact force

A well-known fact is that one-dimensional wave analysis is the theoretical basis of the famous Hopkinson bar dynamic testing technique. The current one-dimensional wave theory is mostly confined to the slender rods of isotropic materials. It is not easy to obtain an analytical solution to the wave equation of an anisotropic rod. In this work, rods with both elastic modulus and density graded in the length direction are presented and analyzed. The one-dimensional variable coefficient wave equation corresponding to the functionally graded rod is constructed and converted into a second-order variable coefficient partial differential equation using the Laplace approach. Then, the details of solving the partial differential equation of the second-order variable coefficients are given. It is worth noting that here we construct a variable coefficient equation that satisfies the form of Euler's equation. It is still difficult to obtain analytical solutions for other equations that do not satisfy this form. Subsequently, studies of the wave propagation characteristics of rods with different graded configurations are carried out. The theoretical results show that the wave propagation behavior and post-impact vibration of the rod are significantly influenced by the graded configuration. It is possible to adjust not only the impact response at the end but also the impact response in the middle of the rod by optimizing the design of the rod's graded configuration.

The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions

Abstract The purpose of this research is to implement the orthogonal polynomials associated with operational matrices to get the approximate solutions for solving two-dimensional elliptic partial differential equations (E-PDEs) with mixed boundary conditions. The orthogonal polynomials are based on the Standard polynomial (xi ), Legendre, Chebyshev, Bernoulli, Boubaker, and Genocchi polynomials. This study focuses on constructing quick and precise analytic approximations using a simple, elegant, and potent technique based on an orthogonal polynomial representation of the solution as a double power series. Consequently, a linear partial differential equation is transformed into a linear algebraic system which is solved by the Mathematica®12. Approximate solutions can be found if the answers are polynomials in and of itself. Three applications involving well-known linear problems Laplace, Poisson, and Helmholtz equations have been solved by using the proposed methods, and a comparison of the approaches has been provided. Furthermore, the computation of the error norm L∞ has been done to show the accuracy of the suggested approaches. The results clearly demonstrate how precise, efficient, and dependable the proposed methods are in obtaining rough solutions to the problem. Bernoulli was one of the best methods in most examples.

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

A cortical field theory – dynamics and symmetries

We characterise cortical dynamics using partial differential equations (PDEs), analysing various connectivity patterns within the cortical sheet. This exploration yields diverse dynamics, encompassing wave equations and limit cycle activity. We presume balanced equations between excitatory and inhibitory neuronal units, reflecting the ubiquitous oscillatory patterns observed in electrophysiological measurements. Our derived dynamics comprise lowest-order wave equations (i.e., the Klein-Gordon model), limit cycle waves, higher-order PDE formulations, and transitions between limit cycles and near-zero states. Furthermore, we delve into the symmetries of the models using the Lagrangian formalism, distinguishing between continuous and discontinuous symmetries. These symmetries allow for mathematical expediency in the analysis of the model and could also be useful in studying the effect of symmetrical input from distributed cortical regions. Overall, our ability to derive multiple constraints on the fields — and predictions of the model — stems largely from the underlying assumption that the brain operates at a critical state. This assumption, in turn, drives the dynamics towards oscillatory or semi-conservative behaviour. Within this critical state, we can leverage results from the physics literature, which serve as analogues for neural fields, and implicit construct validity. Comparisons between our model predictions and electrophysiological findings from the literature — such as spectral power distribution across frequencies, wave propagation speed, epileptic seizure generation, and pattern formation over the cortical surface — demonstrate a close match. This study underscores the importance of utilizing symmetry preserving PDE formulations for further mechanistic insights into cortical activity.

A novel post-processed finite element method and its convergence for partial differential equations

In this article, by combining high-order interpolation on coarse meshes and low-order finite element solutions on fine meshes, we propose a novel approach to improve the accuracy of the finite element method. The new method is in general suitable for most partial differential equations. For simplicity, we use the second-order elliptic problem as an example to show how the novel approach improves the accuracy of the finite element method. Numerical tests are also conducted to validate the main theoretical results.

Dynamics of stochastic differential equations with memory driven by colored noise.

In this paper, we will show two approaches to analyze the dynamics of a stochastic partial differential equation (PDE) with long time memory, which does not generate a random dynamical system and, consequently, the general theory of random attractors is not applicable. On the one hand, we first approximate the stochastic PDEs by a random one via replacing the white noise by a colored one. The resulting random equation does generate a random dynamical system which possesses a random attractor depending on the covariance parameter of the colored noise. On the other hand, we define a mean random dynamical system via the solution operator and prove the existence and uniqueness of weak pullback mean random attractors when the problem is driven by a more general white noise.

The data-driven solutions and inverse problems of some nonlinear diffusion convection-reaction equations based on Physics-Informed Neural Network

The Physics-Informed Neural Network (PINN) has achieved remarkable results in solving partial differential equations (PDEs). This paper aims to solve the forward and inverse problems of some specific nonlinear diffusion convection-reaction equations, thereby validating the practical efficacy and accuracy of data-driven approaches in tackling such equations. In the forward problems, four different solutions of the studied equations are reproduced effectively and the approximation errors can be reduced to 10−5. Experiments indicate that the PINNs method based on adaptive activation functions (PINN-AAF), outperforms the standard PINNs in dealing with inverse problems. The unknown parameters are estimated effectively and the approximation errors can lower to 10−4. Additionally, training rules for both PINN and PINN-AAF are summarized. The results of this study validate the exceptional performance of the data-driven approach in solving the complex nonlinear diffusion convection-reaction equation problems, and provide an effective mechanism for dealing with analogous, intricate nonlinear problems.

Theoretical analysis for dynamic model of U-shaped electrothermal microactuator based on Chebyshev spectral method

Abstract This paper presents a largely and accurate analysis for dynamic model of U-shaped actuator that considers the material parameters dependent on temperature. The electro-thermal analysis for temperature distribution and thermo-mechanical analysis for deflection is carried out. The electro-thermal dynamic behaviour is described by piecewise one-dimensional (1D) heat transfer partial differential equations (PDE) for each beam of U-shaped actuator. A combination of Chebyshev spectral method for space domain and Euler forward difference method for time domain is used to solve these equations. The convergence test is first done to ensure the result convinced.

A microstructural defect-orientation informed phase field model

Micro-cracks, micro-voids and other such defects, which typically coalesce to form macroscopic cracks, could be represented through incompatible, local rotations of material points, i.e. micro-regions. Specifically, based on a micropolar or Cosserat continuum-like hypothesis that requires attaching directors to material points, we track the evolving frame rotation and hence the microstructural orientation during quasi-brittle damage. We introduce a critical energy release rate incorporating the wryness tensor, which in turn is a function of the micro-rotation field and its gradient within the damaged region. The (pseudo) rotation field appears as additional degrees of freedom to describe a frame field, whose evolution is particularly significant within a diffused region of evolving damage as obtained through a phase-field formulation of brittle fracture. We emphasize that, unlike micropolar continua where it contributes to elastic energy, the wryness tensor appears only in the fracture energy in our approach. Thus, without damage, the solid conforms to the classical continuum. By making suitable modifications to the terms within the elastic energy and applying the principle of virtual work, we arrive at the governing partial differential equations (PDEs). For assessing the proposed framework, we choose a specific form of energy density and demonstrate, through numerical examples, the effect of the newly introduced parameters. The classical phase-field model is readily recovered by switching off the micro-rotation. Additionally, we explore a potential application of this model in representing and propagating initial defects.

Galerkin finite element analysis on radiative heat transfer in titanium dioxide-polyalphaolefin nanolubricant past a convergent/divergent channel with non-uniform heat source/sink effect

The current study inspects the radiative heat and mass transport of a polyalphaolefin (PAO) – based nanolubricant flow consisting of titanium dioxide (TiO2) nanoparticles in a convergent/divergent channel with thermophoresis, Brownian motion, and non-uniform heat source/sink effects. The mathematical modelling of the present problem results in a system of complex non-linear partial differential equations (PDEs). The Galerkin finite element method (GFEM) is adopted to solve complex equations and to obtain the solution for the field variables in convergent/divergent channels. The results revealed that as the values of Brownian motion get higher, the temperature dispersal in the channel quickens, whereas the concentration profile slows down. The rise in the value of the thermophoresis parameter increases the thermophoresis force, which causes liquid particles to be dispersed in the stream. The heat transmission rate increases with an increase in radiation parameter values.

Optimization of the pipe diameters and the dynamic operation of a district heating network using a robust initialization strategy

District heating networks (DHN) are systems offering a substantial solution for the decarbonization of heat production. As these systems involve significant costs, the development of digital tools for their optimization becomes a necessity. In the literature, there is a lack of studies optimizing the sizing of the DHN while considering a dynamic operation of the system with a detailed physics. In this work, we developed a Non-Linear Programming (NLP) optimization used in a case study of a DHN where the pipe sizing and the operation are optimized with a precise modeling of the distribution network. The method of Orthogonal Collocation on Finite Elements (OCFE) was employed to discretize the partial differential equation (PDE) governing the temperature evolution in the pipes. In addition, the pressure drops were modeled using the Darcy–Weisbach equation. Three optimization criteria were used: investment cost of the pipes, operational cost, and the global cost. The optimization provided the pipe diameters that best fit each optimization criterion. Furthermore, several initializations were tested to verify if the obtained optimum was a confidence optimum and if the resolution was robust. Finally, the optimization provided a coherent response to the variation in the cost of heat production.

Brownian motion in a magneto Thermo-diffusion fluid flow over a semi-circular stretching surface

The current study explores the mass and heat transport analysis of a Casson liquid stream past a curved surface. The current model considers the effect of magnetic strength brought on by the strength of the applied uniform magnetic field. The significance of thermophoresis and Brownian motion are also taken into account using the Buongiorno nano-liquid model. The study of liquid flow over stretching sheets frequently addresses practical issues that have garnered significant attention from researchers due to their importance in various domains, including microfluidics, fibreglass production, manufacturing, transportation, metal extrusion, thermal insulation, glass production, paper manufacturing, and acoustic blasting. The governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) using the similarity variables. These equations are numerically solved using the finite difference method (FDM). The concentration, temperature, and velocity graphs were produced by varying the different physical parameters. The upsurge in the magnetic parameter reduces the velocity profile. As the magnetic parameter increases, thermal and concentration profiles upsurge. The decrease in velocity profile can be seen as the Casson parameter rises. The intensification in values of thermophoretic parameter enhances the thermal and concentration profiles. The concentration and thermal profiles reduce as the curvature parameter upsurges.

Non-similar analysis of heat generation and thermal radiation on Eyring-Powell Hybrid nanofluid flow across a stretching surface

This paper portrays a two-dimensional mathematical model developed for the analysis of Eyring-Powell hybrid nanofluid flow, incorporating variable temperature and velocity profiles. This study investigates the flow through a porous media with a mixture of hybrid nanoparticles (copper dioxide, magnetite) integrated into ethylene glycol ([Formula: see text]) across a stretching sheet. The model takes magnetic effects, heat generation, and thermal radiation into account. Additionally, nonsimilar partial differential equations (PDEs) are transformed into ordinary differential equations using the local nonsimilarity approach. These equations are then numerically solved using the built-in Matlab function bvp4c. To examine the effects of adjusting parameters on heat, mass transfer, and skin friction, the results are displayed graphically. With an increase in the given values of magnetic field [Formula: see text], the velocity profile [Formula: see text] is seen to decay. Moreover, it is observed that the hybrid nanofluid density and dynamic viscosity increase as the volume fraction rises. It is estimated that the thermal conductivity and viscosity of the [Formula: see text] hybrid nanofluid are expected to increase to 4.1% and 12.35%, respectively, and to 71.55% and 78.01%, respectively, for volume concentrations ranging from 0.02% to 0.05%. It is also concluded that the local skin-friction coefficient has the opposite trend while the local Nusselt number rises monotonically with both the slip velocity parameter and the surface convection parameter. This research has broad applications In the glass and polymer industries, metallic plate cooling, plastic sheet contractions, etc.

Gyrotactic micro-organism dusty fluid flow over an extended surface

The conventional magnetohydrodynamics (MHD) equations are modeled for the dusty fluid flow past a stretching sheet embedded in a porous medium. These equations are coupled with motile microorganisms, temperature and concentration equations for both fluid and dust particles dynamical equations. The two-phase dusty fluid flow is subjected to magnetic effect, porosity factor, 2nd order chemical reaction, Brownian diffusion, thermophoretic effect and Arrhenius activation energy. An appropriate transformation is used to reset the system of partial differential equations (PDEs) into non-linear system of ODEs (ordinary differential equations). The flow equations are numerically solved by using the parametric continuation method (PCM). For validity of the applied methodology, the results are compared using numerical and graphical results of Matlab built-in package “BVP4c”. Both results show accuracy up to four decimal places. Furthermore, from graphical results, it can be noticed that the fluid particles interaction parameter causes an attractive force among fluid particles which prevents the dust particles phase flow. This technique can be used for biological separations, filters, fluid dust separators and crude oil purification.

A new approach to modeling transdermal ethanol kinetics.

Measurement of ethanol above the skin surface (supradermal) is used to monitor blood alcohol concentrations (BAC) in both legal and consumer settings. Previously, the relationship between supradermal alcohol concentration (SAC) and BAC was described using partial and ordinary differential equations (PDE model: J. Appl. Physiol. 100: 649-55, 2006). Using a range of BAC profiles by varying absorption times and peak concentrations, the PDE model accurately predicted experimental measures of SAC. Recently, other mathematical models have relied on the PDE model. This paper proposes a new approach to modeling transdermal ethanol kinetics using a mass transfer coefficient and only ordinary differential equations (ODE model). Using a range of BAC profiles, the ODE model performed very similarly to the PDE model. The ODE model had slightly slower washout rates and slightly slower times to peak SAC and to zero SAC. Similar to the PDE model, a sensitivity analysis on the ODE model showed changes in solubility and diffusivity within the stratum corneum, stratum corneum thickness, and the volume of gas above the skin affected model performance. This new model will streamline integration into larger physiologic models, reduce computation time, and decrease the time to transform skin alcohol measurements to blood alcohol concentrations.

Numerical Solution of the Forward Kolmogorov Equations in Population Genetics Using Eta Functions

This paper introduces a new numerical method for solving forward Kolmogorov equations in population genetics. Since there's no simple analytical expression for the distribution of allele frequencies (DAF), we use these equations to derive it. The accuracy of solving these equations depends on the choice of base functions, so we use Eta-based functions for better approximations. By employing the operational matrix of integral, we simplify the partial differential equation to an algebraic one. The method's error bounds, stability, and validity are demonstrated through numerical examples. Finally, we apply this method to analyze the behavior of forward Kolmogorov equations under various evolutionary forces.

Stochastic Burgers type equations with two reflecting walls: Existence, uniqueness and exponential ergodicity

In this paper, we study stochastic Burgers type equations with two reflecting walls driven by multiplicative noise. We first establish the existence and uniqueness of the solution by Picard iteration technique, the main trouble is to handle the complicated nonlinear term ∂g(u(x,t))∂x which involves the derivative of the solution u(x, t) and the singularities due to reflection. Furthermore, we prove the exponential ergodicity for the equations driven by possibly degenerate, multiplicative noise through asymptotic coupling. More generally, we also give a simplified criterion for exponential ergodicity of parabolic stochastic partial differential equations with reflections driven by degenerate multiplicative noise.

Numerical Heat Transfer Analysis of Carreau Nanofluid Flow over Curved Stretched Surface with Nonlinear Effects and Viscous Dissipation.

The article presents a comprehensive study that thoroughly assesses the effects of Carreau fluid flow over a curved stretch surface. The study meticulously considers various factors, including nonlinear thermal radiation, convective boundary conditions, viscous dissipation, thermophoresis, Brownian motion, and nonlinear mixed convection. By using similarity variables, the problem's governing equations are transformed from nonlinear partial differential equations to nonlinear ordinary differential equations, and then the shooting method is applied to obtain the results. The study also rigorously examines the effect of modifying flow factors on velocity, temperature, and concentration in shear-thickening situations through graphical assessment. Furthermore, according to the corresponding values, a table is provided with data on skin friction, Nusselt, and Sherwood numbers. This study offers new perspectives on the complex mechanisms of Carreau fluids flowing over curved surfaces, with potential applications in materials engineering and fluid mechanics.