Linear Partial Differential Equations Research Articles

Quantum simulation for partial differential equations with physical boundary or interface conditions

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions.

Application of Adomian Decomposition Method for Solving Helmholtz Equation

The Adomian decomposition method (ADM) is effective for solving linear and nonlinear ordinary and partial differential equations, integral equations, algebraic equations and integro-differential equations and it leads to an analytical solution in the form of an infinite power series. We will solve the Helmholtz equation in two dimensional by the Adomian decomposition method.
 Key words: Application of Adomian Decomposition Method, Helmholtz Equation, Adomian Decomposition Method. 

Problem of Determining the Density of Sources in a Multidimensional Heat Equation with the Caputo Time Fractional Derivative

In this paper, we propose a new formula for representing the solution of the third initial-boundary value problem for multidimensional fractional heat equation with the Caputo derivative. This formula is obtained by the continuation method used in the theory of partial differential equations with integer derivatives. The Green’s function of the problem is also constructed in terms of the Fox H- function. Involving the results of solving a direct problem and the overdetermination condition, a uniqueness theorem for the definition of the spatial part of the multidimensional source function is proved.

Sparse successive approximation for nonlinear [formula omitted] and [formula omitted] optimal control problems under residual errors

Successive approximation techniques are effective approaches to solve the Hamilton–Jacobi–Bellman (HJB)/Hamilton–Jacobi–Isaacs (HJI) equations in nonlinear H2 and H∞ optimal control problems (OCPs), but residual errors in the solving process may destroy its convergence property, and related numerical methods also pose computational burden and difficulties. In this paper, the HJB/HJI partial differential equations (PDEs) for infinite-horizon nonlinear H2 and H∞ OCPs are handled in a unified formulation, and a sparse successive approximation method is proposed. Taking advantage of successive approximation techniques, the nonlinear HJB/HJI PDEs are transformed into sequences of easily solvable linear PDEs, to which the solutions can be computed point-wise by handling simple initial value problems. Extra constraints are also incorporated in the solving process to guarantee the convergence under residual errors. The sparse grid based collocation points and basis functions are then employed to enable efficient numerical implementation. The performance of the proposed method is also numerically demonstrated in simulations.

The Combined Effect of Coriolis Force and Double Diffusive for the Convention in a Rectangular Isotropic and Anisotropic Porous Channel

The combined effects of Coriolis force, heat, and mass diffusion for the convection in a horizontal rectangular channel filled with fluid in both isotropic and anisotropic porous channels is studied in the three-dimensional case for which the governing equations with the boundary conditions are derived with the help of the following assumptions: the fluid is incompressible; the flow is axisymmetric about the y-axis; the fluid is steady; thermal diffusion is larger than the viscous diffusion. With the help of the stream function, the resultant equations are made dimensionless, and non-dimension parameters like Rac, Rs along with anisotropic aspect ratio of permeabilities and thermal diffusivities are introduced. Later, the resulting linear partial differential equations are solved by the method of Fourier series analysis. The effect of the non-dimensional numbers plays a vital role in controlling parameters like velocity, temperature, and concentration for both isotropic and anisotropic cases. The expression for critical Rayleigh number has been established for both isotropic and anisotropic cases. Various computations are carried out in order to study the effects of velocity, temperature, concentration, and Critical Rayleigh number for different values of non-dimensional parameters. The obtained results are in good agreement with previous literary works and also have a wide application in the real world which can be seen in different industries.

Approximation of curve-based sleeve functions in high dimensions

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function—the profile—is recovered, and second, the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximates the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

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.

Models and numerical methods for XVA pricing under mean reversion spreads in a multicurrency framework

In this article we make some new relevant contributions to the computation of total valuation adjustments (XVA) for financial derivatives involving several currencies. From the modelling point of view, for the credit spreads we consider the more realistic exponential Vasicek and CIR positive mean reversion processes. Moreover, the derivative is partially collateralized in cash in a foreign currency and the collateral value is a percentage of the derivative prices. Under this modelling assumptions and using appropriate dynamic hedging methodologies, we obtain formulations in terms of linear and nonlinear partial differential equations, which are solved with Lagrange-Galerkin methods in low dimension. For higher dimensions, we use the Monte Carlo techniques for the equivalent formulations in terms of expectations. These techniques include a multilevel Picard iteration method for the nonlinear case. Finally, the methodologies are applied to several European options with different payoffs and the numerical results are discussed.

On some structural properties of generalized Lyapunov eigenproblems and application to operator preconditioning

We are interested in generalized matrix eigenvalue problems of the type AX+XA=λHXH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A X + X A = \\lambda {H} X {H}$$\\end{document} and AX+XA=λ(HX+XH)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A X + X A = \\lambda ({H} X + X {H})$$\\end{document} with A and H both symmetric and positive definite, and in their tensor counterparts. We collect several structural properties, some of which are known, together with some new spectral results. We also analyze in detail the case where the second problem stems from the discretization of linear elliptic partial differential equations by finite differences. In particular, we derive spectral properties that can be used in the numerical solution of the resulting algebraic linear system.

Toward null-state equations in d > 2

In two-dimensional CFTs with a large central charge, the level-two BPZ equation governs the heavy-light scalar four-point correlator when the light probe scalar has dimension h = −12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ -\\frac{1}{2} $$\\end{document} ; the corresponding linear ordinary differential equation can be recast into a schematic form x2uxx + u = 0. In this paper, we make an observation that in a class of four-dimensional CFTs with a large central charge, the heavy-light scalar correlator in the near-lightcone limit obeys a similar equation, x3uxxxy + u = 0, when the light scalar has dimension ∆ = –1. We focus on the multi-stress tensor sector of the theory and also discuss the corresponding equations for the cases with ∆ = –2, –3. The solutions to these linear partial differential equations in higher dimensions are shown, after a suitable change of variables, to reproduce the near-lightcone correlators previously obtained via holography and the conformal bootstrap.

Ensemble controllability of parabolic type equations

In this paper, we extend the concept of ensemble controllability introduced in 2009 by Jr-Shin Li and Navin Khaneja to a class of linear partial differential equation. More precisely, we consider some abstract parabolic equation, where the system depends on some unknown parameter which is assumed to belong to a compact interval. We investigate the possibility of approximatively reaching (in L2-norm) any target state from any initial state, with an open loop control. Here the initial and target states might depend on the unknown parameter, but the control is assumed to be parameter independent.

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker–Planck equation in high dimensions and the Kuramoto–Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.

Reproducing kernel Banach space defined by the minimal norm property and applications to partial differential equation theory

Abstract It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any z ∈ U {z\in U} , in the set V z := { f ∈ V ∣ f ⁢ ( z ) = 1 } {V_{z}:=\{f\in V\mid f(z)=1\}} , if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality K ⁢ ( z , w ) = K ⁢ ( w , z ) ¯ {K(z,w)=\overline{K(w,z)}} does not have to be true without assumptions that K ⁢ ( z , w ) ≠ 0 , K ⁢ ( w , z ) ≠ 0 {K(z,w)\neq 0,K(w,z)\neq 0} . We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory.

An interwoven composite tailored finite point method for two dimensional unsteady Burgers' equation

This paper centers on constructing a lucid and utilitarian approach to tackle linear and non-linear two-dimensional partial differential equations. To test the applicability of the proposed algorithm a variant of the classical two-dimensional unsteady Burgers' equation is set up as a testing ground. The method in a nutshell reduces to solving one-dimensional Burgers' equations resulting from the application of appropriate operator splitting techniques in the temporal direction. In solving these one-dimensional Burgers' equations a refined tailored finite point method in conjunction with an apposite linearization to the purpose is employed. The conditional stability, consistency, and convergence of the method are established theoretically and the method is found to be first-order convergent in time and second-order convergent in space. To illustrate the accuracy of the scheme, divers examples have been solved and the results obtained prove that this method is top-notch in terms of cost-cutting and time efficiency through the sufficiency of coarse meshes.

First-order linear Marcus SPDEs

In this paper, we solve a Lévy driven linear stochastic first-order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita [First order stochastic partial differential equations, in Stochastic Analysis. Proc. Taniguchi Int. Symp. Stochastic Analysis, Katata and Kyoto, 1982, North-Holland Mathematical Library, Vol. 32, ed. K. Itô (North-Holland Publishing Company, 1984), pp. 249–269; Stochastic Flows and Stochastic Differential Equations, Vol. 24 (Cambridge University Press, 1997)].

Reinforcement learning for continuous-time mean-variance portfolio selection in a regime-switching market

We propose a reinforcement learning (RL) approach to solve the continuous-time mean-variance portfolio selection problem in a regime-switching market, where the market regime is unobservable. To encourage exploration for learning, we formulate an exploratory stochastic control problem with an entropy-regularized mean-variance objective. We obtain semi-analytical representations of the optimal value function and optimal policy, which involve unknown solutions to two linear parabolic partial differential equations (PDEs). We utilize these representations to parametrize the value function and policy for learning with the unknown solutions to the PDEs approximated based on polynomials. We develop an actor-critic RL algorithm to learn the optimal policy through interactions with the market environment. The algorithm carries out filtering to obtain the belief probability of the market regime and performs policy evaluation and policy gradient updates alternately. Empirical results demonstrate the advantages of our RL algorithm in relatively long-term investment problems over the classical control approach and an RL algorithm developed for the continuous-time mean-variance problem without considering regime switches.

Some remarks on the solution of the cell growth equation

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al . 2015 Solutions to an advanced functional partial differential equation of the pantograph type ( Proc. R. Soc. A 471 , 20140947 ( doi:10.1098/rspa.2014.0947 )). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups.

Actuator Anomaly Detection in Linear Parabolic Distributed Parameter Cyber-Physical Systems

Distributed parameter cyber-physical systems (DPCPSs) are characterized by two features: 1) they exhibit spatio-temporal (distributed parameter) dynamics and 2) there exist cyber components in the form of communication channels between the physical plant and control system. DPCPSs are vulnerable to anomalies such as nonmalicious physical faults and malicious cyber-attacks. Although the safety and security aspects of CPSs modeled by ordinary differential equations (ODEs) have been extensively explored during the past decade, security of DPCPSs has not received its due attention despite its safety-critical nature. In this work, we explore anomaly diagnostics in DPCPSs from a system theoretic viewpoint. Specifically, we focus on DPCPSs modeled by linear parabolic partial differential equations (PDEs) subject to physical faults and cyber-attacks in the actuation channel. First, we explore the detectability of malicious anomalies (i.e., cyber-attacks) and derive conditions for stealthy attacks in the context of output-feedback-based monitoring. Such stealthy attacks essentially illustrate the fundamental limitations of feedback-based monitoring algorithms given perfect adversarial knowledge. Subsequently, focusing on the anomalies that are detectable, we develop a design framework for anomaly detection algorithms based on output injection observers. Such detection algorithm explicitly considers the conflicting objectives of robustness to uncertainties and sensitivity to anomalies in their design. Finally, theoretical analysis and simulation studies are performed to illustrate the effectiveness of the proposed approach.

An Approach for Approximating Analytical Solutions of the Navier-Stokes Time-Fractional Equation Using the Homotopy Perturbation Sumudu Transform’s Strategy

In this study, we utilize the properties of the Sumudu transform (SuT) and combine it with the homotopy perturbation method to address the time fractional Navier-Stokes equation. We introduce a new technique called the homotopy perturbation Sumudu transform Strategy (HPSuTS), which combines the SuT with the homotopy perturbation method using He’s polynomials. This approach proves to be powerful and practical for solving various linear and nonlinear fractional partial differential equations (FPDEs) in scientific and engineering fields. We demonstrate the efficiency and simplicity of this method through examples, showcasing its ability to approximate solutions for FPDEs. Additionally, we compare the numerical results obtained using this technique for different values of alpha, showing that as the value moves from a fractional order to an integer order, the solution becomes increasingly similar to the exact solution. Furthermore, we provide the tabular representations of the solution for each example.