Technique For Partial Differential Equations Research Articles

An order-N complexity meshless algorithm for transport-type PDEs, based on local Hermitian interpolation

This work describes a new numerical method utilising radial basis function interpolants. Based on local Hermitian interpolation of function values and boundary operators, and using an explicit time advancement formulation, the method is of order-N complexity. Computational cost to advance the solution in time is minimal, and is largely dependent on local system support size. The explicit time advancement formulation allows a novel solution technique for many nonlinear partial differential equations. The performance of the method is examined for a variety of linear convection–diffusion–reaction problems, featuring both steady and unsteady solutions. The method is also demonstrated with a nonlinear Richards’ equation model, solving an unsaturated flow in porous media problem. The technique is named the local Hermitian interpolation (LHI) method.

Reduced order modeling and dynamic optimization of multiscale PDE/kMC process systems

The problem of dynamic optimization for multiscale systems comprising of coupled continuum and discrete descriptions is considered. The solution of such problems is challenging owing to large computational requirements of the multiscale process model. This problem is addressed by developing a reduced multiscale model. This is achieved by combining order reduction techniques for dissipative partial-differential equations with adaptive tabulation of microscopic simulation data. The multiscale process optimization problem is subsequently solved using standard search algorithms. The proposed method is applied to two representative catalytic oxidation processes where optimal inlet concentration profiles are computed to guide the microscopic system from one stable stationary state to another stable stationary state.

The Credit Risk and Pricing of OTC Options

In the over-the-counter (OTC) markets, the options traded are always subject to credit risk. Therefore the counterparty’s credit risk is a striking factor when pricing options, whereas it is not considered in the classic Black-Scholes models. Based on the first passage time models, this paper develops the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism. The default probability and the pricing formulae of the OTC options are obtained by using partial differential equation (PDE) techniques, especially Green’s function.

Total Variation Wavelet Thresholding

We propose using Partial Differential Equation (PDE) techniques in wavelet based image processing to remove noise and reduce edge artifacts generated by wavelet thresholding. We employ a variational framework, in particular the minimization of total variation (TV), to select and modify the retained wavelet coefficients so that the reconstructed images have fewer oscillations near edges while noise is smoothed. Numerical experiments show that this approach improves the reconstructed image quality in wavelet compression and in denoising.

Free-form geometric modeling by integrating parametric and implicit PDEs.

Parametric PDE techniques, which use partial differential equations (PDEs) defined over a 2D or 3D parametric domain to model graphical objects and processes, can unify geometric attributes and functional constraints of the models. PDEs can also model implicit shapes defined by level sets of scalar intensity fields. In this paper, we present an approach that integrates parametric and implicit trivariate PDEs to define geometric solid models containing both geometric information and intensity distribution subject to flexible boundary conditions. The integrated formulation of second-order or fourth-order elliptic PDEs permits designers to manipulate PDE objects of complex geometry and/or arbitrary topology through direct sculpting and free-form modeling. We developed a PDE-based geometric modeling system for shape design and manipulation of PDE objects. The integration of implicit PDEs with parametric geometry offers more general and arbitrary shape blending and free-form modeling for objects with intensity attributes than pure geometric models.

Interval oscillation criteria for second order partial differential equations with delays

In this paper, we present new interval oscillation criteria related to integral averaging technique for second order partial differential equations with delays that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [ t 0 , ∞ ) , rather than on whole half-line. Our results are sharper than some previous results and handles the cases which are not covered by known criteria.

Implicit extension of taylor series method for initial value problems

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of these algorithms is based on the approximate calculation of higher derivatives using well-known technique for the partial differential equations. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval. This property offers different facility for adaptive error control. This paper describes several explicit Taylor series with implicit extension algorithms and examines its consistency and stability properties. The implicit extension based on a collocation term added to the explicit truncated Taylor series. This idea is different from the general collocation method construction, which conduces to the implicit R-K algorithms [12]. It demonstrates some numerical test results for stiff systems herewith we attempt to prove the efficiency of these new-old algorithms.

Analytical treatment of Schrödinger problems with geometrical perturbations

An analytical theory for calculating shifts in eigenvalues to Schrödinger problems due to geometrical perturbations is proposed. The possibility of growing practically any nano-heterostructure today makes it interesting to examine the influence of shape and size on, e.g., electronic, optical and magnetic properties. For that, detailed knowledge of shifts in eigenvalues induced by changes in geometry, either intentionally or unintentionally, is important. Several examples are presented so as (a) to verify the geometry-perturbative model against exact calculations and (b) to attack problems which cannot be solved exactly without resorting to numerical partial differential equation (PDE) techniques. The latter include the cases of a particle confined to a three-dimensional cylinder with a sinusoidally varying radius as a function of the axial coordinate in the presence (or absence) of a dc electric field along the cylinder axis.

Multiscale optimization using hybrid PDE/kMC process systems with application to thin film growth

The problem of optimal time-constant and time-varying operation for transport-reaction processes is considered, when the cost functional and/or equality constraints necessitate the consideration of phenomena that occur over disparate length scales. Multiscale process models are initially developed, linking continuum conservation laws with microscopic scale simulators. Subsequently, order reduction techniques for dissipative partial-differential equations are combined with adaptive tabulation of microscopic simulation data to reduce the computational requirements of the optimization problem, which is then solved using standard search algorithms. The method is applied to a conceptual thin film deposition process to compute optimal substrate-surface temperature profiles that simultaneously maximize film-deposition-rate uniformity (macroscale objective) and minimize surface roughness (microscale objective) across the film surface for a steady-state process operation. Subsequently, optimal time-varying policies of substrate temperature and precursor inlet concentrations are computed under the assumption of quasi-steady-state process operation.

OPTIMAL OPERATION OF THIN FILM GROWTH WITH MULTISCALE PROCESS OBJECTIVES

The issue of optimal time-varying operation for transport-reaction processes is considered, when the cost functional and/or equality constraints necessitate the consideration of phenomena that occur over disparate length scales. Multiscale process models are initially developed, linking continuum conservation laws with microscopic scale simulators. Subsequently, order reduction techniques for dissipative partial-differential equations are combined with adaptive tabulation methods for microscopic simulators to reduce the computational requirements of the optimization problem, which is then solved using standard search algorithms. The method is demonstrated on a thin film deposition process, where optimal surface temperature profiles and inlet switching times that simultaneously maximize thickness uniformity and minimize surface roughness across the film surface are computed.

Integral price formulas for lookback options

We derive an integral representation of the price formulas for European options whose terminal payoff involves path‐dependent lookback variable. The intricacies in the derivation procedures using the partial differential equation techniques stem from the degenerate nature of the pricing models, where the lookback state variables appear only in the auxiliary conditions but not in the governing differential equations. We also derive a parity relation between the price functions of the floating strike and fixed strike lookback options.

Sine transform based preconditioners for solving constant-coefficient first-order PDEs

In this paper, we study nonsymmetric and highly nondiagonally dominant linear systems that arise from discretizations of constant-coefficient first-order partial differential equations (PDEs). We apply the generalized minimal residual method [Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston] for solving the system with a preconditioner based on the fast sine transform. An analytic formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except some outliers. The outlier eigenvalues are bounded and well separated from the origin when the size of system increases. In numerical experiments, we compare our preconditioner with the semi-Toeplitz preconditioner proposed in [SIAM J. Sci. Comput. 17 (1996) 47]. We refer to [J. Numer. Linear Algebra Appl. 1 (1992) 77, Numer. Math. J. Chinese Univ. 2 (1993) 116, BIT 32 (1992) 650, Linear Algebra Appl. 293 (1999) 85] for the early works on preconditioning techniques for PDEs.

Computational Bayesian maximum entropy solution of a stochastic advection‐reaction equation in the light of site‐specific information

This work presents a computational formulation of the Bayesian maximum entropy (BME) approach to solve a stochastic partial differential equation (PDE) representing the advection‐reaction process across space and time. The solution approach provided by BME has some important features that distinguish it from most standard stochastic PDE techniques. In addition to the physical law, the BME solution can assimilate other sources of general and site‐specific knowledge, including multiple‐point nonlinear space/time statistics, hard measurements, and various forms of uncertain (soft) information. There is no need to explicitly solve the moment equations of the advection‐reaction law since BME allows the information contained in them to consolidate within the general knowledge base at the structural (prior) stage of the analysis. No restrictions are posed on the shape of the underlying probability distributions or the space/time pattern of the contaminant process. Solutions of nonlinear systems of equations are obtained in four space/time dimensions and efficient computational schemes are introduced to cope with complexity. The BME solution at the prior stage is in excellent agreement with the exact analytical solution obtained in a controlled environment for comparison purposes. The prior solution is further improved at the integration (posterior) BME stage by assimilating uncertain information at the data points as well as at the solution grid nodes themselves, thus leading to the final solution of the advection‐reaction law in the form of the probability distribution of possible concentration values at each space/time grid node. This is the most complete way of describing a stochastic solution and provides considerable flexibility concerning the choice of the concentration realization that is more representative of the physical situation. Numerical experiments demonstrated a high solution accuracy of the computational BME approach. The BME approach can benefit from the use of parallel processing (the relevant systems of equations can be processed simultaneously at each grid node and multiple integrals calculations can be accelerated significantly, etc.).

A new hybrid meshless-differential order reduction ( hM-DOR) method with applications to shape control of smart structures via distributed sensors/actuators

This paper presents the development of a new hybrid numerical method, the hM-DOR method, which is based on an order-reduction technique for partial differential equations, and combines the true-meshless collocation technique with a fixed reproducing kernel approximation. The proposed method is able to directly impose overlapping boundary conditions, a procedure difficult to accomplish with existing collocation-based meshless methods. The present method is applied to examine thin-plate bending for plates of various geometries and different boundary conditions. Its efficiency and accuracy are validated by numerical comparisons with finite element results. Furthermore, it is also used to simulate the deformation shape control of the uniformly loaded smart beams and plates integrated with piezoelectric sensors/actuators. The numerical results here again verify the efficiency of this newly developed method.

Solution techniques for elementary partial differential equations

Ordinary Differential Equations: Brief Revision First-Order Equations Homogeneous Linear Equations with Constant Coefficients Nonhomogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equations Functions and Operators Fourier Series The Full Fourier Series Fourier Sine Series Fourier Cosine Series Convergence and Differentiation Sturm-Liouville Problems Regular Sturm-Liouville Problems Other Problems Bessel Functions Legendre Polynomials Spherical Harmonics Some Fundamental Equations of Mathematical Physics The Heat Equation The Laplace Equation The Wave Equation Other Equations The Method of Separation of Variables The Heat Equation The Wave Equation The Laplace Equation Other Equations Equations with More than Two Variables Linear Nonhomogeneous Problems Equilibrium Solutions Nonhomogeneous Problems The Method of Eigenfunction Expansion The Heat Equation The Wave Equation The Laplace Equation Other Equations The Fourier Transformations The Full Fourier Transformation The Fourier Sine and Cosine Transformations Other Applications The Laplace Transformation Definition and Properties Applications The Method of Green's Functions The Heat Equation The Laplace Equation The Wave Equation General Second-Order Linear Partial Differential Equations with Two Independent Variables The Canonical Form Hyperbolic Equations Parabolic Equations Elliptic Equations The Method of Characteristics First-Order Linear Equations First-Order Quasilinear Equations The One-Dimensional Wave Equation Other Hyperbolic Equations Perturbation and Asymptotic Methods Asymptotic Series Regular Perturbation Problems Singular Perturbation Problems Complex Variable Methods Elliptic Equations Systems of Equations Answers to Odd-Numbered Exercises Appendix Bibliography Index Exercises appear at the end of each chapter.

Viscosity solutions of Hamilton-Jacobi equations, and asymptotics for Hamiltonian systems

In this paper we apply the theory of viscosity solutions of Hamilton-Jacobi equations to understand the structure of certain Hamiltonian flows. In particular, we describe the asymptotic behavior of minimizing orbits, and prove analogs of the classical Hamilton-Jacobi integrability theory that hold under very general conditions. Then, combining partial differential equations techniques with dynamical systems ideas (Mather measures, ergodicity) we study solutions of time-independent Hamilton-Jacobi equation, namely, uniform continuity, difference quotients and non-uniqueness.

Symmetric modified finite volume element methods for self-adjoint elliptic and parabolic problems

The finite volume element method is a discretization technique for partial differential equations, but in general case the coefficient matrix of its linear system is not symmetric, even for the self-adjoint continuous problem. In this paper we develop a kind of symmetric modified finite volume element methods both for general self-adjoint elliptic and for parabolic problems on general discretization, their coefficient matrix are symmetric. We give the optimal order energy norm error estimates. We also prove that the difference between the solutions of the finite volume element method and symmetric modified finite volume element method is a high order term.

New symmetry solution techniques for first-order non-linear PDEs

Using recent extensions of work of S. Lie (Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, Teubner, 1891) and É. Cartan (Leçons sur les invariants intégreaux, Hermann, Paris, 1922; Les systèms differentials extérieures et leurs applications géométrique, Hermann, Paris, 1945) for integrating Frobenius integrable vector field distributions via symmetry, we examine some symmetry techniques for finding local solutions of first-order non-linear partial differential equations (PDEs). In the language of exterior differential systems, we develop a technique for solving first-order quasilinear PDEs that is then applied to general, first-order non-linear partial differential equations. Our results are significant inasmuch as we give algorithms for solving first-order PDEs in the presence of symmetry that are essentially mechanical in nature and done in the original coordinate system.

Efficient image segmentation using partial differential equations and morphology

The goal of this paper is to investigate segmentation methods that combine fast preprocessing algorithms using partial differential equations (PDEs) with a watershed transformation with region merging. We consider two well-founded PDE methods: a nonlinear isotropic diffusion filter that permits edge enhancement, and a convex nonquadratic variational image restoration method which gives good denoising. For the diffusion filter, an efficient algorithm is applied using an additive operator splitting (AOS) that leads to recursive and separable filters. For the variational restoration method, a novel algorithm is developed that uses AOS schemes within a Gaussian pyramid decomposition. Examples demonstrate that preprocessing by these PDE techniques significantly improves the watershed segmentation, and that the resulting segmentation method gives better results than some traditional techniques. The algorithm has linear complexity and it can be used for arbitrary dimensional data sets. The typical CPU time for segmenting a 256 2 image on a modern PC is far below 1 s.

Discretization errors associated with reproducing kernel methods: one-dimensional domains

The reproducing kernel particle method (RKPM) is a discretization technique for partial differential equations that uses the method of weighted residuals, classical reproducing kernel theory and modified kernels to produce either “mesh-free” or “mesh-full” methods. Although RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quantified. In order to address the numerical performance of RKPM, von Neumann analysis is performed for semi-discretizations of three model one-dimensional PDEs. The von Neumann analyses results are used to examine the global and asymptotic behavior of the semi-discretizations. The model PDEs considered for this analysis include the parabolic and hyperbolic (first- and second-order wave) equations. Numerical diffusivity for the former and phase speed for the latter are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero. Group speed is also presented for the hyperbolic problems. Excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter. In contrast, the row-sum lumped mass matrix formulation severely degraded performance. The asymptotic analysis indicates that very good rates of convergence (O( x 6)–O( x 8)) are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter and quadrature rule.