Partial Differential Equations Research Articles

Nanofluid dissipative Reiner-Philippoff model with thermal radiation: numerical investigation using the modified Adomian decomposition method associated with Mohand transforms

Abstract The flow of nanofluids over a stretched sheet situated within a porous medium is the main subject of this work. The Reiner-Philippoff model, which includes a magnetic field, chemical reaction, thermal radiation, viscous dissipation, and variable thermal conductivity, is examined. The study investigates how these complex processes affect the system’s heat transfer characteristics and flow dynamics. A system of partial differential equations describes the physical model. We arrive at a system of ordinary differential equations that, due to its highly nonlinear nature, requires numerical treatment by employing the proper similarity transformations. The governing equations are solved numerically, namely by combining the Mohand transform and the Adomian decomposition method. For computer-based solutions, complicated equations are simplified using the sophisticated Modified Decomposition Method (MDM). To guarantee convergence, it combines the Mohand transform with Adomian decomposition methods, yielding a series solution that almost matches the precise solution to the issue.

$s$-stability for $W^{s,n/s}$-harmonic maps in homotopy groups

We study s -dependence for minimizing W^{s,n/s} -harmonic maps u\colon \mathbb{S}^{n} \to \mathbb{S}^{\ell} in homotopy classes. Sacks–Uhlenbeck theory shows that, for each s , minimizers exist in a generating subset of \pi_{n}(\mathbb{S}^{\ell}) . We show that this generating subset can be chosen locally constant in s . We also show that as s varies, the minimal W^{s,n/s} -energy in each homotopy class changes continuously. In particular, we provide progress on a question raised by Mironescu [in: Perspectives in nonlinear partial differential equations (2007), 413–436] and Brezis–Mironescu [Sobolev maps to the circle (2021)].

Unveiling the biological side of PET-derived biomarkers: a simulation-based approach applied to PDAC assessment.

Radiomics has revolutionized clinical research by enabling objective measurements of imaging-derived biomarkers. However, the true potential of radiomics necessitates a comprehensive understanding of the biological basis of extracted features to serve as a clinical decision support. In this work, we propose an end-to-end framework for the in silico simulation of [18F]FLT PET imaging process in Pancreatic Ductal Adenocarcinoma, accounting for the biological characterization of tissues (including perfusion and fibrosis) on tracer delivery. We thus establish a direct association between radiomics features and the underlying biological properties of tissues. We considered 4 immunohistochemically stained Whole Slide Images of pancreatic tissue of one healthy control and three patients with PDAC and/or precursor lesions. From marker-specific images, tissue-depending diffusivity properties were estimated and computational domains were built to simulate the [18F]FLT spatial-temporal uptake exploiting Partial Differential Equations and Finite Elements Method. Consequently, we simulated the imaging process obtaining surrogated PET images for the considered patients, and we performed image-derived features extraction from PET images to be mapped with biological properties via correlation estimation. The framework captured the phenotypic differences and generated Time Activity Curves reflecting the underlying tissue composition. Image-derived biomarkers were ranked in view of their association with biological characteristics of the tissue, unveiling their molecular correlative. Moreover, we showed that the proposed pipeline could serve as a digital phantom to optimize the image acquisition for lesion detection. This innovative framework holds the potential to enhance interpretability and reliability of radiomics, fostering the adoption in personalized nuclear medicine and patient care.

Exploring chaos and sensitivity in the Ivancevic option pricing model through perturbation analysis.

This study explores the Ivancevic Option Pricing Model, a nonlinear wave-based alternative to the Black-Scholes model, using adaptive nonlinear Schrödingerr equations to describe the option-pricing wave function influenced by stock price and time. Our focus is on a comprehensive analysis of this equation from multiple perspectives, including the study of soliton dynamics, chaotic patterns, wave structures, Poincaré maps, bifurcation diagrams, multistability, Lyapunov exponents, and an in-depth evaluation of the model's sensitivity. To begin, a wave transformation is applied to convert the partial differential equation into an ordinary differential equation, from which soliton solutions are derived using the [Formula: see text] method. We explore various forms of the option price function at different time points, including singular-kink, periodic, hyperbolic, trigonometric, exponential, and complex solutions. Furthermore, we simulate 3D surface plots and 2D graphs for the real, imaginary, and modulus components of some of the obtained solutions, assigning specific parameter values to enhance visualization. These graphical representations offer valuable insights into the dynamics and patterns of the solutions, providing a clearer understanding of the model's behavior and potential applications. Additionally, we analyze the system's dynamic behavior when a perturbing force is introduced, identifying chaotic patterns using the Lyapunov exponent, Sensitivity, multistability analysis, RK4 method, wave structures, bifurcation diagrams, and Poincaré maps.

Randomized radial basis function neural network for solving multiscale elliptic equations

Abstract Ordinary deep neural network-based methods frequently encounter difficulties when tackling multiscale and high-frequency partial differential equations. To overcome these obstacles and improve computational accuracy and efficiency, this paper presents the Randomized Radial Basis Function Neural Network (RRNN), an innovative approach explicitly crafted for solving multiscale elliptic equations. The RRNN method commences by decomposing the computational domain into non-overlapping subdomains. Within each subdomain, the solution to the localized subproblem is approximated by a randomized radial basis function neural network with a Gaussian kernel. This network is distinguished by the random assignment of width and center coefficients for its activation functions, thereby rendering the training process focused solely on determining the weight coefficients of the output layer. For each subproblem, similar to the Petrov-Galerkin finite element method, a linear system will be formulated on the foundation of a weak formulation. Subsequently, a selection of collocation points is stochastically sampled at the boundaries of the subdomain, ensuring the satisfaction of $C^0$ and $C^1$ continuity and boundary conditions to couple these localized solutions. The network is ultimately trained using the least squares method to ascertain the output layer weights. To validate the RRNN method's effectiveness, an extensive array of numerical experiments has been executed. 
The comparative analysis demonstrates the RRNN's superior performance with respect to computational accuracy and training time compare to deep neural network methods based on gradient descent method, and can attain enhanced accuracy at a comparable computational cost compare to Local Extreme Learning Machine method and the classical finite element method.

A Physics‐Driven GraphSAGE Method for Physical Field Simulations Described by Partial Differential Equations

AbstractPhysics‐informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and oscillations, trained solutions usually suffer low accuracy. In addition, most current works only offer the trained solution for predetermined input parameters. If any change occurs in input parameters, transfer learning or retraining is required, and traditional numerical techniques also need recomputation. In this work, a physics‐driven GraphSAGE approach (PD‐GraphSAGE) based on the Galerkin method and piecewise polynomial nodal basis functions is presented to solve computational problems governed by irregular PDEs and to develop parametric PDE surrogate models. This approach employs graph representations of physical domains, thereby reducing the demands for evaluated points due to local refinement. A distance‐related edge feature and a feature mapping strategy are devised to help training and convergence for singularity and oscillation situations, respectively. The merits of the proposed method are demonstrated through a couple of cases. Moreover, the robust PDE surrogate model for heat conduction problems parameterized by the Gaussian random field source is successfully established, which not only provides the solution accurately but is several times faster than the finite element method in the experiments.

Modeling the structure and relaxation in glycerol-silica nanocomposites.

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

Advanced Dynamic Vibration Control Algorithms of Materials Terfenol-D Si3N4 and SUS304 Plates/Cylindrical Shells with Velocity Feedback Control Law

A numerical, generalized differential quadrature (GDQ) method is presented on applied heat vibration for a thick-thickness magnetostrictive functionally graded material (FGM) plate coupled with a cylindrical shell. A nonlinear c1 term in the z axis direction of a third-order shear deformation theory (TSDT) displacement model is applied into an advanced shear factor and equation of motions, respectively. The equilibrium partial differential equation used for the thick-thickness magnetostrictive FGM layer plate coupled with the cylindrical shell under thermal and magnetostrictive loads can be implemented into the dynamic GDQ discrete equations. Parametric effects including nonlinear term coefficient of TSDT displacement field, advanced nonlinear varied shear coefficient, environment temperature, index of FGM power law and control gain on displacement, and stress of the thick magnetostrictive FGM plate coupled with cylindrical shell are studied. The vibrations of displacement and stress can be controlled by the control gain algorithms in velocity feedback control law.

Analytical Study of Magnetohydrodynamic Casson Fluid Flow over an Inclined Non-Linear Stretching Surface with Chemical Reaction in a Forchheimer Porous Medium

This study investigates the steady, two-dimensional boundary layer flow of a Casson fluid over an inclined nonlinear stretching surface embedded within a Forchheimer porous medium. The governing partial differential equations are transformed into a set of ordinary differential equations through similarity transformations. The analysis incorporates the effects of an external uniform magnetic field, gravitational forces, thermal radiation modeled by the Rosseland approximation, and first-order homogeneous chemical reactions. We consider several dimensionless parameters, including the Casson fluid parameter, magnetic parameter, Darcy and Forchheimer numbers, Prandtl and Schmidt numbers, and the Eckert number to characterize the flow, heat, and mass transfer phenomena. Analytical solutions for the velocity, temperature, and concentration profiles are derived under simplifying assumptions, and expressions for critical physical quantities such as the skin friction coefficient, Nusselt number, and Sherwood number are obtained.

Analytical solutions and molecule states of the (3+1)-dimensional variable coecient Date-Jimbo-Kashiwara-Miwa equation

Analytical solutions and molecule states of the (3+1)-dimensional variable coecient Date-Jimbo-Kashiwara-Miwa equation

Viscous Marangoni migration of an inviscid bubble by surfactant spreading: an exactly solvable model

A model is formulated of a two-dimensional migrating, or swimming, inviscid bubble in a viscous fluid whose unsteady displacement is caused by the spreading over its surface of an initial distribution of insoluble surfactant. Assuming small capillary and Reynolds numbers, and a linear equation of state giving the surface tension as a function of surfactant concentration, the quasi-steady Stokes flow around the bubble is found analytically and explicit formulas are determined for the time-dependent bubble speed and its final overall displacement. At infinite surface Péclet number this is done using a complex version of the method of characteristics to solve a complex partial differential equation of Burgers type. For a finite non-zero surface Péclet number, the problem is shown to be linearizable by a complex variant of the classical Cole–Hopf transformation. The formulation allows general statements to be made on the bubble speed and its total net displacement in terms of the initial surfactant distribution. A weak finite-time singularity in the surface activity associated with an isolated clean point on the bubble surface is also identified and studied in detail.

Time fractional (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation: its Lie symmetries, exact solutions and conservation laws

Abstract In this paper, Lie symmetry analysis method is applied to (2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation. We obtain all the Lie symmetries admitted by the governing equation and reduce the corresponding (2+1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1+1)-dimensional counterparts with Erd'{e}lyi-Kober fractional derivative. Then we obtain the power series solutions of the reduced equations, prove their convergence and analyze their dynamic behavior graphically. In addition, the conservation laws for all the obtained Lie symmetries are constructed by the new conservation theorem and the generalization of Noether operators.

Numerical analysis of bioconvective heat transport through Casson nanofluid over a thin needle.

Bioconvective flows over a thin needle hold significant importance in various fields, particularly in biomedical engineering, microfluidics, and environmental science. This paper examines the bioconvective flow properties of a copper and blood-based Casson nanofluid over a thin needle, accounting for gyrotactic microorganisms in the presence of a magnetic field. The two-phase nanofluid model is applied to formulate the flow problem. The system of non-dimensional ordinary differential equations is obtained by reducing the governing partial differential equations with the help of similarity variables. Further, the ODEs are numerically solved using the 4th-order Runge-Kutta method based Shooting technique. The similar solutions of the non-dimensional ODEs are represented graphically and the blood-based nanofluid's velocity, temperature, concentration, and presence of microorganisms are examined with reference to the accompanying diagrams. A detailed analysis is provided for skin friction, Nusselt number, and microorganism density number. The primary outcomes reveal that the augmentation of the mixed convection parameter and buoyancy ratio parameter enhance the rate of heat transfer.

A thresholding algorithm for Willmore-type flows via fourth-order linear parabolic equation

We propose a thresholding algorithm for Willmore-type flows in \mathbb{R}^{N} . This algorithm is constructed based on the asymptotic expansion of the solution to the initial value problem for a fourth-order linear parabolic partial differential equation whose initial data is the indicator function on the compact set \Omega_{0} . The main results of this paper demonstrate that the boundary \partial\Omega(t) of the new set \Omega(t) , generated by our algorithm, is included in O(t) -neighborhood of \partial\Omega_{0} for small t>0 and that the normal velocity from \partial\Omega_{0} to \partial\Omega(t) is nearly equal to the L^{2} -gradient of Willmore-type energy for small t>0 . Finally, numerical examples of planar curves governed by the Willmore flow are provided by using our thresholding algorithm.

Finite-Time Stability Analysis of a Discrete-Time Generalized Reaction–Diffusion System

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab’s LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems.

Adaptive Bounded Bilinear Control of a Parallel‐Flow Heat Exchanger

ABSTRACTThis work investigates the adaptive constrained control design for a parallel‐flow heat exchanger represented by a system of two coupled linear first‐order hyperbolic partial differential equations (PDEs). This system incorporates structured uncertainty involving unknown in‐domain parameters that characterize neglected dynamics in the heat exchanger model. These parameters may encompass unmodeled heat transfer phenomena, variations in fluid properties, and modeling simplifications. The objective is to regulate the internal fluid outlet temperature to track a desired reference trajectory by adjusting the external fluid velocity. Due to inherent physical constraints, this manipulated variable is upper and lower‐bounded. Accordingly, the control problem is bounded and bilinear. Using the set‐invariance principle and an energy‐like framework, we first develop a bounded state‐feedback controller. Then, since the measurements are considered only at the boundaries, we propose an adaptive boundary observer using a swapping scheme and a recursive least squares identifier. The proposed adaptive observer provides online estimates of the distributed state and the unknown parameters. Next, the state‐feedback controller is associated with the boundary observer and parameter identifier, and the exponential stability of the closed‐loop system is guaranteed using Lyapunov's stability theory. Finally, we provide numerical simulations to demonstrate the efficiency of the proposed control scheme.

Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients

In this work, we obtain non-Gaussian (NG) stochastic solutions to χ-Wick-type stochastic (χ-Wk-TS) Burgers’ equations with variable coefficients. An Exp-function method, the connection between white noise theory and hypercomplex systems (HCSs), the χ-Wick product (χ-Wk-product) and an χ-Hermite transform (χ-Hr-transform) are proposed. We provide a new set of non-Gaussian solitary wave solutions (NG-SWSs) to Burgers’ equations with variable coefficients. NG white noise functional solutions (NG-WNFSs) to χ-Wk-TS Burgers’ equations with variable coefficients are shown. The symmetry coefficients of partial differential equations and the symmetrical properties of SPDEs are critical in determining the best solution.

On the adjoint state method for the gradient computation in full waveform inversion: a complete mathematical derivation for the (visco-)elastodynamics approximation

Summary High resolution seismic imaging at all scales using full waveform inversion is now routinely used in the industry and in the academy. One key element for the success of this approach is a numerical method, named adjoint state method, originally designed for optimization problems constrained by partial differential equations, a category to which full waveform inversion belongs. This method provides an efficient way to compute the gradient of the full waveform inversion misfit function, which is the most computationally demanding task in the implementation of full waveform inversion. While well known, the complete and rigorous mathematical derivation of the adjoint state method for full waveform inversion remains missing in the scientific bibliography. The aim of this study is to remedy this lack. The derivation is performed in general settings, that is in the elastodynamics approximation, with and without considering viscosity. Through the calculus, the mechanism of the adjoint state strategy makes clear the connection between the incident and adjoint fields, especially regarding their initial and boundary conditions. The impact of introducing the viscosity is carefully analyzed. The resulting gradient formulas are analyzed and shown to be consistent with already published ones. The generic approach which is adopted also makes it possible to derive misfit function gradients with respect to other quantities than the subsurface mechanical parameters, for instance with respect to the initial or the boundary conditions, which could be of interest for specific applications where the reconstructed parameters are not only volumetric mechanical parameters but boundary parameters or initial field values.

Starshaped Compact Hypersurfaces in Warped Product Manifolds II: A Class of Hessian Type Equations

In this note, we prove the existence of one particular class of starshaped compact hypersurfaces, by deriving global curvature estimates for such hypersurfaces; this generalizes a recent result on this problem from the Euclidean space to Riemannian warped products. Moreover, we show that interior second order a priori estimates for admissible solutions to the associated fully nonlinear elliptic partial differential equations can be readily established by similar arguments.

Novel analytical perspectives on nonlinear instabilities of viscoelastic Bingham fluids in MHD flow fields

The nonlinear stability of a plane interface separating two Bingham fluids and fully saturated in porous media is inspected in the existing work. The two fluids are compressed by a normal magnetic field. The two fluids have diverse viscoelasticity, densities, magnetic, and porosity medium, with the existence of surface tension at the interface. The motivation of applied physics and engineering relations has encouraged the discussion of the current paper. Because the mathematical behavior is rather complex, the viscoelasticity involvement is reproduced only at the surface of separation, which is well-known as the viscous potential theory. Thereby, the equations of movement are scrutinized in a linear form, whereas a set of nonlinear boundary conditions are supposed. This procedure produces a nonlinear expressive nonlinear partial differential equation of the interface displacement. The non-perturbative approach which is based on the He’s frequency formula is employed to transform the nonlinear distinguishing ordinary differential equation with complex coefficients into a linear one. A novel process relying on the non-perturbative approach is utilized to examine the nonlinear stability and scrutinize the interface presentation. A non-dimensional analysis produces several dimensionless physical numerals. To validate the new approach, a comparison between the non-perturbative approach and its corresponding linear ordinary differential equation via the Mathematica Software is described and interpreted through a set of diagrams. Additionally, the Polar graphs have been elucidated. It is found that the mechanism of the stability does not change in the cases of real and complex coefficients.