Class Of Partial Differential Equations Research Articles

A Direct Method to Approximate Solution of the Space-fractional Diffusion Equation

A class of partial differential equations with fractional derivatives in the spatial variables are called space-fractional diffusion equations. They can be applied to simulate anomalous diffusion, in which the classical diffusion equation does not accurately describe how a plume of particles disperses. Analytically solving fractional diffusion equations can be problematic due to the typically complex structures of fractional derivative models. Hence, this study proposes the utilisation of a satisfier function in combination with the Ritz method to effectively address fractional diffusion equations in the Caputo sense. By employing this approach, the equations are transformed into an algebraic system, so facilitating their solution and providing a numerical result. This method can achieve a high level of accuracy in solving the Caputo fractional diffusion equations by utilising only a small number of terms from the shifted Legendre polynomials in two variables. The precision and effectiveness of our approach may be evaluated, as it yielded dependable approximations of the solutions.

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

A hidden convexity in continuum mechanics, with application to classical, continuous-time, rate-(in)dependent plasticity

A methodology for defining variational principles for a class of PDE (partial differential equations) models from continuum mechanics is demonstrated, and some of its features are explored. The scheme is applied to quasi-static and dynamic models of rate-independent and rate-dependent, single-crystal plasticity at finite deformation.

A finite-dimensional approximation for partial differential equations on Wasserstein space

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme (Barles and Souganidis, 1991) to our context, as well as on a key precompactness result for semimartingale measures. We illustrate our convergence result with the example of the Hamilton–Jacobi–Bellman and Bellman–Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.

Gradient-Based Monte Carlo Methods for Relaxation Approximations of Hyperbolic Conservation Laws

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier–Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the ’90s and have several interesting features, such as being grid-free, automatically adapting to the solution by concentrating elements where the gradient is large, and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

Performance of GRKM-method for solving classes of ordinary and partial differential equations of sixth-orders

Abstract A general quasilinear sixth-order ordinary differential equation (ODE) is an important class of ODEs. The primary objective of this study is to establish a numerical method for solving a general class of quasilinear sixth-order partial differential equations (PDEs) and ODEs. However, the Runge–Kutta method (RKM) approach for solving special classes of ODEs has been generalized as an effort to solve the general class of ODEs. Nonlinear algebraic order condition (OCs) equations have been obtained up to the tenth order using the Taylor-series expansion methodology which is used to derive the novel generalized Runge–Kutta method (GRKM). In this study, a GRKM integrator has been derived for solving a general class of quasilinear sixth-order ODEs and then this method is modified subsequently to solve a class of PDEs. Accordingly, the proposed GRKM is modified to solve a quasilinear sixth-order PDE by converting it to a system of sixth-ODEs using the method of lines. Nine problems have been implemented to prove the efficiency and accuracy of the proposed method. Simulation results of these problems showed that the proposed numerical GRKM is an accurate and efficient method. In contrast, by comparing the proposed GRKM numerical approach with the classical RK method, the numerical results demonstrate that the direct integrator outperforms the indirect classical RK method in terms of algorithm complexity and function evaluations, proving that the numerical GRKM is efficient.

Investigating invariance and conservation laws for the classes of nonlinear parabolic and wave systems

We study various classes of the nonlinear dynamics of some high order parabolic equations like the Oskolkov–Benjamin–Bona–Mahony–Burgers and Benjamin–Bona–Mahony–Peregrine–Burger equations that arise in the study of some wave phenomena. Also, a broader class of partial differential equations are used in modelling ocean waves that originate from the Ostrovsky equation. We study the invariance properties via the Lie invariance method for the nonlinear systems and further establish classes of conservation laws for models arises in this study. We show how the relationship leads to double reductions of the systems. This relationship is determined by a recent result involving multipliers that lead to a total divergence or closed form of the differential equation under investigation.

Equivariant neural operators for gradient-consistent topology optimization

Abstract Most traditional methods for solving partial differential equations (PDEs) require the costly solving of large linear systems. Neural operators (NOs) offer remarkable speed-ups over classical numerical PDE solvers. Here, we conduct the first exploration and comparison of NOs for three-dimensional topology optimization. Specifically, we propose replacing the PDE solver within the popular Solid Isotropic Material with Penalization (SIMP) algorithm, which is its main computational bottleneck. For this, the NO not only needs to solve the PDE with sufficient accuracy but also has the additional challenge of providing accurate gradients which are necessary for SIMP’s density updates. To realize this, we do three things: (i) We introduce a novel loss term to promote gradient-consistency. (ii) We guarantee equivariance in our NOs to increase the physical correctness of predictions. (iii) We introduce a novel NO architecture called U-Net Fourier neural operator (U-Net FNO), which combines the multi-resolution properties of U-Nets with the Fourier neural operator (FNO)’s focus on local features in frequency space. In our experiments we demonstrate that the inclusion of the novel gradient loss term is necessary to obtain good results. Furthermore, enforcing group equivariance greatly improves the quality of predictions, especially on small training datasets. Finally, we show that in our experiments the U-Net FNO outperforms both a standard U-Net, as well as other FNO methods.

Some New Families of Exact Solitary Wave Solutions for Pseudo-Parabolic Type Nonlinear Models

The objective of the current study is to provide a variety of families of soliton solutions to pseudo-parabolic equations that arise in nonsteady flows, hydrostatics, and seepage of fluid through fissured material. We investigate a class of such equations, including the one-dimensional Oskolkov (1D OSK), the Benjamin-Bona-Mahony (BBM), and the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation. The Exp (-ϕξ)-expansion method is used for new hyperbolic, trigonometric, rational, exponential, and polynomial function-based solutions. These solutions of the pseudo-parabolic class of partial differential equations (PDEs) studied here are new and novel and have not been reported in the literature. These solutions depict the hydrodynamics of various soliton shapes that can be utilized to study the nature of traveling wave solutions of other nonlinear PDE’s.

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0≤ ξ ≤ 1. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example.

C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\infty }$$\\end{document}-Regularization by Noise of Singular ODE’s

In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the “roughness” and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of “higher order averaging operators along highly fractal stochastic curves”. This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

A quantum algorithm for computing dispersal of submarine volcanic tephra

In this paper, we develop a quantum computing algorithm for solving the partial differential equation (PDE) for tephra dispersal through advection in the semi-infinite horizontal buoyant region of a submarine volcanic eruption. The concentration of pyroclastic particles in the fluid domain of a hydrothermal megaplume provides important information about the rate of volcanic energy release, mechanism of formation of the megaplume, and submarine depositional patterns. This work leveraging on previous works [F. Gaitan, NPJ Quantum Inf. 6, 61 (2020); F. Gaitan, Adv. Quantum Tech. 4, 2100055 (2021)] further opens up opportunities to solve wider classes of PDEs with different applications of interest. Some additional specific contributions of this work are transforming the semi-infinite spatial domain problem into a problem on a finite spatial domain for applying the quantum algorithm, and the investigation into the effect of spatial and temporal resolution on the solution of PDEs for the quantum algorithm. Furthermore, possible modification of the algorithm with different spatial discretization schemes has been presented and their influence and implications on the solution of the PDE have been discussed. Also, studies are conducted to examine the effect of regularity conditions in time and the presence of statistical noise in the spatial domain, on the solutions obtained using quantum algorithms. The study in this paper paves an important pathway to venture into other types of advection-diffusion problems.

Global dynamics of a diffusive SARS-CoV-2 model with antiviral treatment and fractional Laplacian operator

In this paper, we propose and investigate the global dynamics of a SARS-CoV-2 infection model with diffusion and antiviral treatment. The proposed model takes into account the two modes of transmission (virus-to-cell and cell-to-cell), the lytic and nonlytic immune responses. The diffusion into the model is formulated by the regional fractional Laplacian operator. Furthermore, the global asymptotic stability of equilibria is rigorously established by means of a new proposed method constructing Lyapunov functions for a class of partial differential equations (PDEs) with regional fractional Laplacian operator. The proposed method is applied to the classical reaction-diffusion equations with normal diffusion.

Time-space formulation of a conservative string subject to finite transformations

This paper investigates power-balanced descriptions of the conservative vibrating string. First, the hypotheses of continuum mechanics are recalled for the case of infinitesimal transformations. The string is described by classical partial differential equations (PDEs) and reformulated as a port Hamiltonian system (PHS). Second, the case of finite (possibly large) transformations is considered, for which the time variation of the elastic energy appears to be no longer the elastic power. To naturally solve this difficulty and ensure the invariance of the elastic power with respect to the superposition of rigid body motion, the problem is here addressed in the framework of a time-space formulation. Eulerian and Lagrangian conservative formulations are proposed in the context of non relativistic velocities. This work also yields perspectives for a time-space representation of a port-Hamiltonian vibrating string invariant to the change of observer in a relativistic context.

Numerical analysis of a chaotic coupled hyperbolic PDE system for secure image encryption

This paper focuses on the interest of using chaotic hyperbolic Partial Differential Equations (PDE) systems synchronization for secure image communication. Such systems offer a promising alternative to finite dimensional chaotic systems with enlarged key space, efficient computation, and more complex dynamics. Here, we modify the classical chaotic hyperbolic PDE system by adding integral boundary couplings and potential terms. The chaotic behaviour of the resulting system is observed in numerical simulations. Adding such couplings allows to enhance the security of the cryptosystem. The synchronization of receiver and transmitter chaotic systems is achieved by the use of a chaotic observer designed using the backstepping methodology. The initial image can then be encrypted into a chaotic signal on the transmitter side before being sent to the receiver side. Using observational data, the receiver can then reconstruct the initial image. Numerical illustrations complete the paper.

A unified hp-HDG framework for Friedrichs' PDE systems

This work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems.

A new treatment of boundary conditions in PDE solution with Galerkin methods via Partial Integral Equation framework

We present a new mathematical framework for solution of Partial Differential Equations (PDEs), which is based on an exact transformation of the underlying PDE that removes the boundary constraints from the solution state and moves them into the dynamics of the equivalent transformed equation. The framework is based on a Partial Integral Equation (PIE) representation of a PDE or a system of PDEs, where Partial Integral Equation does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized and consistent treatment of boundary conditions in constructing spectrally convergent solution approximations to a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries, among others. The significance of this result is that a solution to almost any linear PDE in a form of a function series approximation can now be systematically constructed, irrespective of the boundary conditions. Furthermore, in many cases, the resulting ODE system for the expansion coefficients can now be integrated analytically in time, which allows us to obtain solution approximations to a broad class of unsteady PDEs with unprecedented accuracy. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. We also present comparison of the PIE methods with some classical direct PDE solution methods, further demonstrating advantages and potential limitations of the PIE approach. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

Energy-dissipative evolutionary deep operator neural networks

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seek numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network consists of two sub-networks, the Branch net and the Trunk net. For an objective operator G, the Branch net encodes different input functions f at a fixed number of sensors xi,i=1,2,⋯,m, and the Trunk net evaluates the output function at any location. By minimizing the error between the evaluated output q and the expected output G(f)(y) at the test point y, DeepONet generates a good approximation of the operator G. A key distinction in our methodology is the utilization of DeepONet for the training of the initial state, which operates as a multi-parametric operator. Further, the evolution of parameters in our model is facilitated by the Scalar Auxiliary Variable (SAV) method, leading to a formulated iterative process for the parameter's progression over time. The SAV approach is adopted to preserve essential physical properties of PDEs, particularly the Energy Dissipation Law. It introduces a kind of modified energy and establishes unconditional energy dissipation law in the discrete level. By treating the parameters of the well-trained DeepONet as a representation of the initial operator and evolving them by a dynamic system, our network can predict the accurate solution at any further time, while the training data is only the initial state. In order to validate the accuracy and efficiency of our neural networks, we provide numerical simulations of several partial differential equations, including heat equations, parametric heat equations, Allen-Cahn equations and a reaction–diffusion equation in three dimensions.

A peridynamics approach to flexible multibody dynamics for fracture analysis of mechanical systems

The classical theory of continuum mechanics is formulated using partial differential equations (PDEs) that fail to describe structural discontinuities, such as cracks. This limitation motivated the development of peridynamics, reformulating the classical PDEs into integral-differential equations. In this theory, each material point interacts with its neighbours inside a characteristic length-scale through bond-interaction forces. However, while peridynamics can simulate complex multi-physics phenomena, its integration in the study of mechanical systems is still limited. This work presents a methodology that incorporates a peridynamics formulation into a planar multibody dynamics (MBD) formulation to allow the integration of flexible structures described by peridynamics into mechanical systems. A flexible body is described by a collection of point masses, in analogy with the meshless collocation scheme commonly used for peridynamics discretisations. Each point mass interacts with other point masses through nonlinear forces governed by a bond-based peridynamics (BBPD) formulation. The virtual bodies methodology enables the definition of kinematic joints connecting the flexible body with the neighbouring bodies. The implementation of the methodology proposed is illustrated using various mechanisms with different levels of complexity. Notched plates subjected to different loading conditions are compared with the results presented in the literature of the peridynamics field. The deformations of a flexible slider-crank mechanism compare well with the results obtained using a classical flexible MBD formulation. Additionally, three scenarios involving a rotating pendulum illustrate how the methodology proposed allows simulating impact scenarios. The results demonstrate how this methodology is capable to successfully simulate highly nonlinear phenomena, including crack propagation, in a multibody framework.