Separable Partial Differential Equations Research Articles

A Nonlinear Coupled Diffusion System for Image Despeckling and Application to Ultrasound Images

Despite extensive availability of filters for denoising, speckle noise suppression remains a challenging task. Speckle noise also hinders tasks such as efficient extraction of features, recognition, analysis, detection of edges, etc. Therefore, in this paper, motivated by the impressive performance of time delay regularization in additive noise removal, we develop a class of nonlinear diffusion-based coupled partial differential equation models for multiplicative noise removal. This denoising framework considers separate partial differential equations to handle diffusion function as well as fidelity term. By using a maximum a posteriori regularization approach, we can derive an energy functional and study the associated evolution problem which corresponds to the denoised image we want to recover. We then evaluate the effectiveness of our model with several standard test images and real ultrasound images. Qualitative and quantitative studies confirm that the proposed model is robust in comparison with state-of-the-art approaches. The denoised images have appealing visual characteristics with different levels of noise and textures while preserving the important details of the original image.

On bivariate classical orthogonal polynomials

We deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi-definite moment functional, and we revise old properties for these polynomials. More precisely, new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal Pearson-type equation are proved: they are solutions of two separate partial differential equations one for every partial derivative, their partial derivatives are again orthogonal, and every vector polynomial can be expressed in terms of its partial derivatives by means of a linear relation involving only three terms of consecutive total degree. Moreover, we study general solutions of the matrix second order partial differential equation satisfied by classical orthogonal polynomials, and we deduce the explicit expressions for the matrix coefficients of the structure relation. Finally, some illustrative examples are given.

Mesh Spacing Estimates and Efficiency Considerations for Moving Mesh Systems

AbstractAdaptive numerical methods for solving partial differential equations (PDEs) that control the movement of grid points are called moving mesh methods. In this paper, these methods are examined in the case where a separate PDE, that depends on a monitor function, controls the behavior of the mesh. This results in a system of PDEs: one controlling the mesh and another solving the physical problem that is of interest. For a class of monitor functions resembling the arc length monitor, a trade off between computational efficiency in solving the moving mesh system and the accuracy level of the solution to the physical PDE is demonstrated. This accuracy is measured in the density of mesh points in the desired portion of the domain where the function has steep gradient. The balance of computational efficiency versus accuracy is illustrated numerically with both the arc length monitor and a monitor that minimizes certain interpolation errors. Physical solutions with steep gradients in small portions of their domain are considered for both the analysis and the computations.

Analytical solution of the asymmetric transient wave in a transversely isotropic half-space due to both buried and surface impulses

With the aid of a complete set of two scalar potential functions, the problem of transient wave propagation in transversely isotropic half-space, subjected to time dependent tractions applied on a finite patch at an arbitrary depth below the free surface of the half-space is investigated. With the use of the displacement–potential function relationships in a cylindrical coordinate system, the coupled equations of motion are uncoupled; resulting in two separate partial differential equations one of which is second order and the other is fourth order. These two partial differential equations are solved with the aid of both Fourier series expansion and joint Hankel–Laplace integral transforms. The solutions are also investigated in details for tractions varying with time as Heaviside step function, which may be used as a kernel in any integral based method for more complicated elastodynamic initial-boundary value problems. Moreover, some displacement Green׳s functions are numerically evaluated for a synthetic transversely isotropic material to graphically demonstrate the transient motion of the free surface of the half-space.

Efficient segmentation based on Eikonal and diffusion equations

Segmentation of regions of interest in an image has important applications in medical image analysis, particularly in computer aided diagnosis. Segmentation can enable further quantitative analysis of anatomical structures. We present efficient image segmentation schemes based on the solution of distinct partial differential equations (PDEs). For each known image region, a PDE is solved, the solution of which locally represents the weighted distance from a region known to have a certain segmentation label. To achieve this goal, we propose the use of two separate PDEs, the Eikonal equation and a diffusion equation. In each method, the segmentation labels are obtained by a competition criterion between the solutions to the PDEs corresponding to each region. We discuss how each method applies the concept of information propagation from the labelled image regions to the unknown image regions. Experimental results are presented on magnetic resonance, computed tomography, and ultrasound images and for both two-region and multi-region segmentation problems. These results demonstrate the high level of efficiency as well as the accuracy of the proposed methods.

Godunov Mixed Methods on Triangular Grids for Advection–Dispersion Equations

A time-splitting approach for advection–dispersion equations is considered. The dispersive and advective fluxes are split into two separate partial differential equations (PDEs), one containing the dispersive term and the other one the advective term. On triangular elements a triangle-based high resolution Finite Volume (FV) scheme for advection is combined with a Mixed Hybrid Finite Element (MHFE) technique to solve dispersion. This approach introduces an error proportional to the time step and the overall scheme is only first order accurate if special care is not taken in the definition of the numerical flux approximation for advection. By incorporating the diffusive effects into the definition of this numerical flux, near second order accuracy (up to a log h factor) can be proved theoretically and validated by numerical experiments in both one- and two-dimensional cases.

An Expansion Method for Computing Axisymmetric Sunspot Oscillations

A method is proposed to solve for the linear axisymmetric oscillations of a general, axisymmetric, potential, magnetostatic sunspot equilibrium. The basic approach is to express the solution as a series of terms that are products of a prescribed radial planform times an unknown function of the vertical spatial coordinate and time. If the potential sunspot magnetic field is strictly uniform and aligned with the prevailing gravitational stratification, then a single term in the proposed series solution suffices, and the familiar problem first considered by Ferraro & Plumpton is readily recovered. For less trivial magnetic fields, which possess both vertical and radial gradients, the proposed series solution does not truncate after a finite number of terms, but the equations that determine the unknown functions of the vertical coordinate and time enjoy the advantage of being separable partial differential equations, which can be attacked through the solution of subordinate ordinary differential equations by the method of separation of variables. It is also demonstrated that the proposed series solution encompasses the thin flux tube expansion. Consequently, a rigorous mathematical basis is provided for this popular method employed to describe the dynamics of slender magnetic flux tubes, which proves useful in understanding the intrinsic astrophysical limitations of the approach. Whether this proposed method of solution is also a practical and efficient means to calculate the oscillation modes of axisymmetric sunspot equilibria is not answered here but will be addressed in a forthcoming companion paper.

Stable and efficient numerical method for solving the Schr�dinger equation to determine the response of tunneling electrons to a laser pulse

The form Ψ(x, t)=[F0(x)+F1(x, t)e−iωt+F−1(x, t)eiωt]e−iEt/ℏ is used for the wave function in the transient solutions. This expression is similar to the three dominant terms in the steady-state solution from the Floquet theory, except that now F1 and F−1 depend on t as well as x. The function F0 is the static solution, and separate partial differential equations are given for F1 and F−1. Polynomial extrapolation is used to satisfy boundary conditions at the ends of the grid. The numerical solutions are shown to converge and to be numerically stable even for simulated times exceeding 2000 cycles of the radiation field. The examples show delays corresponding to the semiclassical tunneling transit time, the classical time for traversing the inverted barrier. A resonance is seen when electrons promoted above the barrier by absorbing quanta from the radiation field have the closed line integral of momentum between the turning points equal to an integral multiple of Planck's constant. A second resonance occurs when the period of oscillation for the radiation equals the semiclassical tunneling transit time for electrons that absorb one photon from the radiation but are still below the barrier. This resonance decays at a rate corresponding to the tunneling dwell time, and, thus, it is not present in the steady state. These observations suggest a semiclassical picture of the tunneling process. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 703–710, 1998

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N2log2N) andO(N2log2log2N) arithmetic operations, respectively.

Families of separable partial differential equations

In this paper the authors investigate classes of partial differential equations (mostly in two independent variables, of order at most 4) which can be solved by the well‐known technique of separation of variables. In particular, several classical ordinary differential equations, some of which have been generalized (by means of certain parameters), are used to assist in the construction of such partial differential equations. The process gives rise to (potentially) many families of equations which appear to be difficult to solve. The article not only illustrates the technique used in the determination of solutions to such equations, but also contrasts the effect of the nature of solutions as certain parameters effect them. In addition, the paper indicates how other classes of solvable equations might be obtained; equations with more than two variables and of any order. Only analytical methods are discussed; however it would not be difficult to use computing machinery as the number of independent variables ...

Factorisation of separable partial differential equations

The author studies the factorisation of partial differential equations which admit a separation of variables. As a particular application the author studies the algebraic structure of the resulting raising and lowering operators for the spherical harmonics in three and four dimensions.

2-D MULTIPLE REFLECTIONS

Starting with a 1-D subsurface model, a method is developed for modeling and inverting the class of multiple reflections involving the near‐perfect reflector at the free surface. A solution to the practical problem of estimating the source waveform is discussed, and application of the 1-D algorithm to field data illustrates the successful elimination of seafloor and peg‐leg multiples. Extending the analysis to waves in two dimensions, we make the approximation that the subsurface behaves as an acoustic medium. Based on several numerical and theoretical considerations, the scalar wave equation is split into two separate partial differential equations: one governing propagation of upcoming waves and a second describing downgoing waves. The result is a pair of propagation equations which are coupled where reflectors exist. Finite difference approximations to the initial boundary value problem are developed to integrate numerically the surface reflection seismogram. Use of the 2‐D algorithm for modeling free‐surface multiple reflections is illustrated by several reflector models. The 2-D inverse problem of simultaneously migrating primary reflections and inverting diffracted multiples consists of reversing the forward calculation with the data as boundary conditions. Causal directions of propagation are related to downward continuation of surface data. Reflector mapping principles are used to develop a general reflection coefficient estimator. The inverse algorithm is illustrated using the results of the 2-D forward calculation as the boundary conditions.

On the modal control of distributed systems with distributed feedback

This paper presents a solution to the problem of the control of a class of linear distributed systems. The system is described by a linear operator acting on functions of time and distance. It is shown that if the operator separates in time and distance and the distance operator has a real discrete spectrum (self-adjoint and completely continuous), the operator can be represented by an infinite diagonal matrix in which the entries are functions of the Laplace transform variable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> . In particular, if the problem stems from separable partial differential equations, the entries are rational ratios of polynomials of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> . The system can be compensated with a series of discrete conventional filters using techniques of conventional lumped, single loop control system design. To implement the control system, the assumption is made that the distance dependent part of the output and forcing functions have negligible eigenfunction content beyond the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> th one. If this assumption holds, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> sensors, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> filters, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> manipulators, plus 2 matrix multipliers and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> subtractors provide a synthesis of the feedback control system. An illustrative numerical example is given.

Direct methods for the solution of finite-difference approximations to separable partial differential equations

Direct methods for the solution of finite-difference approximations to separable partial differential equations

Iterative procedures for solving finite-difference approximations to separable partial differential equations

Iterative procedures for solving finite-difference approximations to separable partial differential equations