Partial Differential Equations Research Articles

Preservation of global solvability and estimation of solutions of some controlled nonlinear partial differential equations of the second order

Let $U$ be the set of admissible controls, $T>0$, and let $W[0;\tau]$, $\tau\in(0;T]$, be a scale of Banach spaces such that the set of restrictions of functions from $W=W[0;T]$ to $[0;\tau]$ coincides with $W[0;\tau]$; let $F[.;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. Earlier, for the operator equation $x=F[x;u]$, $x\in W$, the author introduced a comparison system in the form of a functional integral equation in the space $\mathbf{C}[0;T]$. It was established that to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation, it is sufficient to preserve the global solvability of the specified comparison system, and the corresponding sufficient conditions were established. In this paper, further examples of application of this theory are considered: nonlinear wave equation, strongly nonlinear wave equation, nonlinear heat equation, strongly nonlinear parabolic equation.

Numerical investigation of Williamson fluid in presence of chemical reaction towards a parabolic surface

ABSTRACT Our focus here is on examining the behavior of a boundary layer that regulates the movement of a non-Newtonian fluid undergoing a chemical reaction on a radiative paraboloid surface. This includes studying the fluid’s movement, temperature, and mass transfer. The fluid being studied pertains to the Williamson fluid model, which is described as a type of extended Newtonian fluid. We have considered the effects of Joule heating and dissipation through viscous phenomena based on the rules of heat and mass movement. The equations in the mathematical structure of partial differential equations (PDEs) governing the fluid continue to flow model in the problem. A subsequent evolution of these equations into ODEs by incorporating similarity variables. The velocity field’s schematic behavior is attributable to the magnetic parameter, thickness parameter, and the Weissenberg number, which exhibit velocity profile decline. In this research, both thermal radiation and Eckert number are highlighted, and it is concluded that both of these variables substantially raise the temperature field. We continue to track the concentration-reducing chemical reaction characteristic that remains tracked along with concentration transfer. Furthermore, the study includes a comprehensive presentation of comparative analyses with previously published scholarly works.

Mixed convective heat transfer to flexible viscosity nanofluid over a radially elongated convective exterior with temperature dependent heat source and Arrhenius activation energy

Purpose This study aims to explore the collective influence of several factors, namely, thermal radiation, Brownian motion, magnetic field and variable viscosity parameter, on the boundary layer flow, heat and mass transfer of an electrically steering nanofluid over a radially stretching exterior subjected to convective heating. In addition, the impacts of thermal and solutal buoyancy forces and activation energy are taken into account. The enlarging velocity is assumed to vary linearly with radial distance. Design/methodology/approach Through the similarity transformation technique, the governing highly nonlinear partial differential equations are transformed into a set of nonlinear ordinary differential equations, which are then numerically solved using the Runge–Kutta–Fehlberg method with a shooting technique. Findings Graphical depictions are provided to analyze the velocity, temperature and nanoparticle concentration fields under the influence of various pertinent parameters. Furthermore, local skin friction, local Nusselt and Sherwood numbers are quantitatively presented and discussed. A comparison with previous results demonstrates good agreement. Originality/value This study uniquely integrates multiple factors influencing boundary layer flow in electrically conducting nanofluids, offering a nuanced understanding of heat and mass transfer over radially stretching surfaces. By using advanced numerical methods, it provides valuable insights and quantitative data that can inform practical applications in engineering and materials science.

A Solution-Structure B-Spline-Based Framework for Hybrid Boundary Problems on Implicit Domains

Solving partial differential equations (PDEs) on complex domains with hybrid boundary conditions presents significant challenges in numerical analysis. In this paper, we introduce a solution-structure-based framework that transforms non-homogeneous hybrid boundary problems into homogeneous ones, allowing exact conformity to the boundary conditions. By leveraging B-splines within the R-function method structure and adopting the stability principles of the WEB method, we construct a well-conditioned basis for numerical analysis. The framework is validated through a number of numerical examples of Poisson equations with hybrid boundary conditions on different implicit domains in two and three dimensions. The results reflect that the approach can achieve the optimal approximation order in solving hybrid problems.

Additive operator splitting scheme for a general mean curvature flow and application in edges enhancement

Many models that use non-linear partial differential equations (PDEs) have been extensively applied for different tasks in image processing. Among these PDE-based approaches, the mean curvature flow filtering has impressive results, for which feature directions in the image are important. In this paper, we explore a general model of mean curvature flow, as proposed in [4, 5]. The modelcan be re-arranged to a reaction-diffusion form, facilitating the creation of an unconditionally stable semi-implicit scheme for image filtering. The method employs the Additive Operator Split (AOS) technique. Experiments demonstrated that the modified general model of mean curvature flow is highly effective for reducing noise and has a superior job of preserving edges.

Strong solution of stochastic differential equations with discontinuous and unbounded coefficients

In this paper we study the existence and uniqueness of strong solution of the following d d -dimensional stochastic differential equation (SDE) driven by Brownian motion: d X t = b ( t , X t ) d t + σ ( t , X t ) d B t , X 0 = x , \begin{equation*} dX_t=b(t,X_t)dt+\sigma (t,X_t)dB_t, X_0=x, \end{equation*} where B B is a d d -dimensional standard Brownian motion; the diffusion coefficient σ \sigma is a Hölder continuous and uniformly nondegenerate d × d d\times d matrix-valued function and the drift coefficient b b may be discontinuous and unbounded, not necessarily in L p q \mathbb {L}_p^q , extending the previous works to discontinuous and unbounded drift coefficient situation. The idea is to combine the Zvonkin’s transformation with the Lyapunov function approach. Zvonkin’s transformation is a one-to-one (and quasi-isometric) transformation of a phase space that allows us to pass from a diffusion process with nonzero drift coefficient to a process without drift. To this end, we need to establish a local version of the connection between the solutions of the SDE up to the exit time of a bounded connected open set D D and the associated partial differential equation on this domain. As an interesting by-product, we establish a localized version of the Krylov estimates and a localized version of the stability result of the SDEs of discontinuous coefficients.

Lie symmetries for the shallow water magnetohydrodynamics equations in a rotating reference frame

Abstract We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition ∇ h B ≠ 0 for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential g and the Coriolis term f 0, related to the constant rotation of the reference frame. For four different cases, namely g = 0, f 0 = 0; g ≠ 0 , f 0 = 0; g = 0, f 0 ≠ 0; and g ≠ 0, f 0 ≠ 0 the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the L 10 = A 3 , 3 ⋊ A 2 , 1 ⨂ s A 5 , 34 a ; L 8 = A 2,1 ⋊ A 6,22; L 7 = A 3 , 5 ⋊ A 2 , 1 ⋊ A 2 , 1 ; and L 6 = A 3,5 ⋊ A 3,3respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where f 0 g ≠ 0, we derive all the invariants for the Adjoint action of the Lie algebra L 6 and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system.

Effects of Newtonian Heating on MHD Jeffrey Hybrid Nanofluid Flow via Porous Medium

In recent years, hybrid nanoparticles have gained significant attention for their ability to enhance thermal conductivity in various fluid systems, making them effective heat transport catalysts. Despite advancements in thermal fluid technology, a gap remains in understanding how hybrid nanoparticles interact within non-Newtonian Jeffrey fluid systems, particularly under complex boundary conditions like Newtonian heating. The present study aims to shed light on the effect of hybrid nanoparticles (alumina and copper) incorporated into a Jeffrey fluid model on flow and heat transport, considering them as heat transport catalyst and subject to Newtonian heating to optimize thermal efficiency. An exponentially accelerated plate is used to induce the fluid flow, taking into account the effects of porosity, MHD, and thermal radiation. The examined fluid exhibits an unsteady one-dimensional flow, formulated by deriving partial differential equations, which are subsequently transformed into ordinary differential equations using suitable non-dimensional variables and the Laplace transformation. This research distinguishes itself by presenting a novel mathematical model for MHD Jeffrey hybrid nanofluid, accounting for porosity and Newtonian heating effects. The inverse of Laplace is used to generate the exact solutions for velocity and temperature profiles, which is not explored in existing literature. Graphical representations are generated using Mathcad, depicting the velocity and temperature distributions. A comparison with prior study from the literature demonstrates strong agreement between our findings and theirs. The findings indicate that the velocity and temperature profiles of the hybrid nanofluid are higher with Newtonian heating than without it. Additionally, an increase in the Grashof number, radiation, acceleration, and porosity parameters also leads to an enhanced velocity profile.

Influence of MHD Flow on Shrinking Sheet with Partial Slip and Heat Generation at Stagnation Point

Fluid dynamics encompasses the fundamental principles of continuity, momentum, and energy conservation, which are applied through mathematical models like the Navier-Stokes equations. These equations are essential for describing how fluid properties like velocity, pressure, and density change in response to forces and environmental conditions. Thus, this study attempted to explore the characteristics of flow and heat transfer of a shrinking sheet in magnetohydrodynamics (MHD), along with the effect of partial slip and heat generation on the system. We employ a similarity transformation technique for turning the governing partial differential equations into ordinary differential equations. These equations are solved numerically through shooting method in Maple, and the results are compared to the previous research. The analysis shows that the suction parameter and velocity slip parameter have an increasing effect on both the skin friction rate and the heat transfer rate. In the meantime, the heat transfer rate decreases as the parameter increases for the heat generation, magnetic parameter, Eckert number and thermal slip parameter. The bvp4c solver in MATLAB is implemented to conduct a stability analysis and determine the physically feasible solution. According to our research, the stability of the solution occurs only in the first solution.

Computing the Set of RBF-FD Weights Within the Integrals of a Kernel-Based Function and Its Applications

This paper offers an approach to computing Radial Basis Function–Finite Difference (RBF-FD) weights by integrating a kernel-based function. We derive new weight sets that effectively approximate both the first and second differentiations of a function, demonstrating their utility in interpolation and the resolution of Partial Differential Equations (PDEs). Particularly, the paper evaluates the theoretical weights in interpolation tasks, highlighting the observed numerical orders, and further applies these weights to solve two distinct time-dependent PDE problems.

Numerical Simulation of Lake Pollution Model via Crank-Nicolson Finite Difference Scheme

Lake pollution has become a serious challenge considering the number of infectious diseases caused by contaminated water. To understand more about the concentration of pollutant released in a lake, this study formulates a partial differential equation (PDE) model of parabolic type, discretized via the Crank-Nicolson scheme. The accuracy of the formulated scheme was ascertained through the Von Neumann analysis. Our numerical simulation reveals that the concentration of pollutants spreads exponentially at each time steps

Investigating Salt‐Finger Convection Under Time‐Dependent Gravity Modulation in Micropolar Liquids

ABSTRACTThis paper investigates how gravity modulation affects salt‐finger convection in a micropolar liquid layer confined between two parallel, infinitely long plates separated by a thin gap. The system is heated and has solute added from above. The study uses linear stability analysis to examine when and how salt‐finger convection, driven by the salt‐finger process, begins. To analyze this, the partial differential equations governing the system are solved numerically using normal mode analysis. The Venezian approach is applied to find the critical Rayleigh number and the solutal Rayleigh number, which are key to understanding the onset of convection. Also, the paper explores how different micropolar fluid parameters—such as the coupling parameter, micropolar heat conduction parameter, couple stress parameter, and inertia parameter—affect the system when gravity modulation is present. It is found that gravity modulation can either stabilize or destabilize convection, depending on its frequency. At very high frequencies (approaching infinity), the effect of gravity modulation becomes minimal, having little impact on the convection process. The paper also examines the relationship between the critical Rayleigh number and the solutal Rayleigh number, which are related to heat and solute concentration, respectively.

Highly accurate simplification of physics-based electrochemical model of all-solid-state battery for online state of charge estimation

Abstract All-solid-state lithium batteries offer superior energy density and safety features, making them highly attractive for electric vehicles and wearable devices. Original physics-based electrochemical model can effectively simulate the internal electrochemical reactions, but they are difficult to be applied to embedded battery management systems. To facilitate the development of real-time applications based on physical models, this paper proposes a SOC prediction method based on a simplified electrochemical model (SEM). First, the transcendental transfer function is converted to third-order transfer functions using the Padé approximation. The electric field in the mass transfer overpotential is then solved for using average numerical integration method. Then, the proposed SEM method under variable load conditions is verified by comparing to the results from original partial differential equations. The results show that the developed SEM strikes a balance between high fidelity and computational efficiency. By taking into account the concentration of Li+ ions in the solid electrolyte, the estimation accuracy of SOC in the range of 0.1 to 0.3 is significantly improved, compared with prior studies. Strong support is provided for the advanced control design of smart management systems of ASSBs.

Numerical Solutions for Radiative MHD Casson Nanofluid Flow with Slip, Heat Source, and Chemical Reactions over a Stretching Surface

MHD Casson nanofluid flow in the presence of viscous dissipation, thermal radiation and chemical reaction past a linear stretching surface is investigated. The problem is subjected to slip boundary conditions. The analysis of heat and mass transfer in the presence of Brownian motion and the thermophoretic diffusion effects are incorporated into heat and mass transfer. Nonlinear partial differential equations model governing the problems are transformed into the dimensionless nonlinear ordinary differential equations via similarity variables, then shooting method with sixth-order Runge- Kutta numerical scheme is used to solve the reduce system of Ordinary Differential Equations (ODE). Maple software is used to perform the simulation. The results are presented graphically and in tabular form and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters. Comparison between the obtained results and previous works are well in agreement. Numerical values of the local skin-friction, Nusselt number and nanoparticle Sherwood number are computed and analyzed. It is noted that the skin-friction coefficient decreases for the larger values of velocity ratio parameter, slip parameter, and increases with an increasing value of Casson parameter. It is also found that enhancing the chemical reaction parameter leads to decrease in concentration profile

Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics

Abstract The present research explores nematicons in liquid crystals (LCs) with quadruple power law nonlinearity utilizing the modified extended Fan sub-equation technique as an analytical tool to investigate the optical spatial soliton solutions. For the inaugural time, a novel version of nonlinearity is investigated in relation to LCs. There are distinct applications for the several wave solutions that have been created in optical handling data. The aforementioned modified extended Fan subequation approach offers novel, comprehensive solutions that are relatively easy to deploy in comparison to earlier, regular methodologies. This approach translates a coupled non-linear partial differential equation into a coupled ordinary differential equation through implementing a traveling wave conversion. This approach indicates that a large variety of traveling and solitary solutions that rely upon five parameters can be incorporated by the nematicons in LCs. In addition, the investigation yields solutions of the single and mixed non-degenerate Jacobi elliptic function form. Novel solutions, such as the periodic pattern, kink and anti-kink patterns, N-pattern, W-pattern, anti-Z-pattern, M-pattern, V-pattern, complexion pattern and anti-bell pattern, or dark soliton solutions of nematic LCs, have been constructed by means of modified extended Fan subequation technique through granting suitable values for the parameters. The computer software Mathematica 14 is used to illustrate several modulus, real and imaginary solutions visually in the form of contour, 2D, and 3D visualizations that help understand the concrete importance of the nematicons in LCs. This research additionally offers a physical comprehension of the obtained solutions and applications of model. The imposed approach is ultimately thought to be more potent and effective than alternative approaches, and the solutions found in this work could be beneficial in our understanding of soliton structures in LCs.

Wavelet Total Variation Fusion and Fuzzy Logic Decision-Based Haze Density and Sky Region-Aware De-Hazing

In this study, we present a fuzzy logic-based classifier with a decision-based partial differential equation (PDE) for single image de-hazing. This enhanced partitioning is achieved via adaptive segmentation and morphological region-growing processes. Also, the first stage fusion of enhanced inverted log illumination and log reflectance components leads to minimization or elimination of edge artifacts. Additionally, improved scene discrimination between sky and non-sky regions due to the fuzzy logic decision-based approach is achieved. This results in effective de-hazing without sky region over-enhancement. Furthermore, a modified wavelet multi-level decomposition-based de-hazing algorithm is presented. It improves details and local contrast in de-hazed images, especially for images with thick haze. Ultimately, these algorithms are fused to obtain the best and most consistent results for all images. Experimental comparisons with the state-of-the-art algorithms show improved sky discrimination and improved clarity.

Comparison of the combinations of Fourier pseudospectral method, finite volume method, Euler method and fourth order Runge-Kutta method used to solve Burgers equation

The Burgers equation is a mathematical model frequently used in Computational Fluid Dynamics. It is often employed to test and calibrate numerical methods, as it is one of the few nonlinear transport equations with an exact analytical solution. In this paper, numerical solutions are obtained using the Finite Difference Method (FDM) and the Fourier Pseudospectral Method (FPSM) for spatial discretization, combined with the Euler Method and the Fourth-Order Runge-Kutta Method (FRKM) for time discretization. The results are compared with the exact analytical solution in terms of accuracy, convergence rate, and computational cost. The findings indicate that the combination of the FDM and Euler Method achieves excellent computational efficiency when compared to the other approaches. Meanwhile, the combination of FPSM and FRKM demonstrates superior accuracy (achieving round-off errors) and a high order of convergence (spectral convergence order). Thus, combining methods with similar convergence rates and accuracy is the optimal strategy for obtaining efficient numerical solutions of partial differential equations (PDEs).

Multi‐indice B$B$‐series

AbstractWe propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi‐indice. The main idea is to replace rooted trees in Butcher's ‐series by multi‐indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi‐indices ‐series uniquely characterize the Taylor expansion of 1‐dimensional local and affine equivariant maps.

GROUP ANALYSES, REDUCTIONS AND EXACT SOLUTIONS OF MONGE–AMPERE EQUATION OF MAGNETIC HYDRODYNAMICS

We study the Monge–Amp`ere equation with three independent variables which occurs in electron magnetohydrodynamics. A group analysis of this strongly nonlinear partial derivative equation is carried out. An eleven-parameter transformation preserving the form of the equation is found. A formula is obtained that makes it possible to construct multiparametric families of solutions based on simpler solutions. Two-dimensional reductions leading to simpler partial differential equations with two independent variables. One-dimensional reductions are described, which make it possible to obtain self-similar and other invariant solutions that satisfy ordinary differential equations. Exact solutions with additive, multiplicative and generalized separation of variables are constructed, many of which admit representation in elementary functions. The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by strongly nonlinear partial differential equations.

ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A NONLINEAR EQUATION

For a nonlinear partial differential equation generalizing a damped sixth-order Boussinesq equation with double dispersion and the equation of transverse oscillations of a viscoelastic Voigt–Kelvin beam under the action of external and internal friction and whose deformation is considered taking into account the correction for the inertia of section rotation, sufficient conditions for the existence and exponential decay of a global solution of the Cauchy problem are found.