Class Of Piecewise Smooth Functions Research Articles

The flow of a viscous fluid with a predetermined pressure gradient through periodic structures

In the Stokes approximation, the problem of viscous fluid flow through two-dimensional and three-dimensional periodic structures is solved. A system of thin plates of a finite width is considered as a two-dimensional structure, and a system of thin rods of finite length is considered as a three-dimensional structure. Plates and rods are periodically located in space with certain translation steps along mutually perpendicular axes. On the basis of the procedure developed earlier, the authors constructed an approximate solution of the equations for fluid flow with an arbitrary orientation of structures relative to a given vector of pressure gradient. The solution is sought in a finite region (cells) around inclusions in the class of piecewise smooth functions that are infinitely differentiable in the cell, and at the cell boundaries they satisfy the continuity conditions for velocity, normal and tangential stresses. Since the boundary value problem for the Laplace equation is solved, it is assumed that the solution found is unique. The type of functions allows us to separate the variables and to reduce the problem's solution to the solution of ordinary differential equations. It is found that the change in the flow rate of a fluid through a characteristic cross section is determined mainly by the geometric dimensions of the cells of the free liquid in such structures and is practically independent of the size of the plates or rods.

Convex multiclass segmentation with shearlet regularization

Segmentation plays an important role in many preprocessing stages in image processing. Recently, convex relaxation methods for image multi-labelling were proposed in the literature. Often these models involve the total variation (TV) semi-norm as regularizing term. However, it is well known that the TV-functional is not optimal for the segmentation of textured regions. In recent years, directional representation systems were proposed to cope with curved singularities in images. In particular, curvelets and shearlets provide an optimally sparse approximation in the class of piecewise smooth functions with C 2 singularity boundaries. In this paper, we demonstrate that the discrete shearlet transform is suited as regularizer for the segmentation of curved structures. Neither the shearlet nor the curvelet transform where used as regularizer in a segmentation model so far. To this end, we have implemented a translation invariant finite discrete shearlet transform based on the fast Fourier transform. We describe how the shearlet transform can be incorporated within the multi-label segmentation model and show how to find a minimizer of the corresponding functional by applying an alternating direction method of multipliers. Here, the Parseval frame property of our shearlets comes into play. We demonstrate by numerical examples that the shearlet-regularized model can better segment curved textures than the TV-regularized one and that the method can also cope with regularizers obtained from non-local means.

Stability and convergence of the difference schemes for equations of isentropic gas dynamics in Lagrangian coordinates

For the initial-boundary value problem (IBVP) if isentropic gas dynamics written in Lagrangian coordinates written in terms of Riemann invariants we show how to obtain necessary conditions for existence of global smooth solution using the Lax technique. Under these conditions we formulate the existence theorem in the class of piecewise-smooth functions. A priori estimates with respect to the input data for the difference scheme approximating this problem are obtained. The estimates of stability are proved using only restrictions on the initial and boundary conditions corresponding the differential problem. In the general case the estimates have been obtained only for the finite instant of time t < t0. The monotonicity has been proved in the both cases. The uniqueness and convergence of the difference solution are also considered. The results of the numerical experiment illustrating theoretical statements are given.

Gibbs phenomenon removal by adding Heaviside functions

We define a kind of spectral series to filter off completely the Gibbs phenomenon without overshooting and distortional approximation near a point of discontinuity. The construction of this series is based on the method of adding the Fourier coefficients of a Heaviside function to the given Fourier partial sums. More precisely, we prove the uniform convergence of the proposed series on the class of piecewise smooth functions. Also, we attach two numerical examples which illustrate the uniform convergence of the suggested series in comparison with the Fourier partial sums.

Singularities of the radon transform of a class of piecewise smooth functions in R 2

A class of piecewise smooth functions in R2 is considered. The propagation law of the Radon transform of the function is derived. The singularities inversion formula of the Radon transform is derived from the propagation law. The examples of singularities and singularities inversion of the Radon transform are given.

Solitary Waves of the Regularized Short Pulse and Ostrovsky Equations

We derive a model for the propagation of short pulses in nonlinear media. The model is a higher-order regularization of the short-pulse equation (SPE). The regularization term arises as the next term in the expansion of the susceptibility in derivation of the SPE. Without the regularization term there do not exist traveling pulses in the class of piecewise smooth functions with one discontinuity. However, when the regularization term is added, we show, for a particular parameter regime, that the equation supports smooth traveling waves which have structure similar to solitary waves of the modified Korteweg–deVries equation. The existence of such traveling pulses is proved via the Fenichel theory for singularly perturbed systems and a Melnikov-type transversality calculation. Corresponding statements for the Ostrovsky equations are also included.

Solution of the nerve cable equation using Chebyshev approximations

The propagation of excitation along the dendrites and the axon of a neurone is described by a partial differential equation which is nonlinear when voltage-gated conductances are present. In this case, numerical methods are employed to obtain a solution: the evolution of the membrane potential in space and time. Even when the membrane is passive (linear), numerical methods might still be preferred to analytical ones that are often too cumbersome to obtain. In this paper, we present the Chebyshev pseudospectral or collocation method as an alternative to the hitherto commonly used finite difference schemes (compartmental models) that are based on sufficiently fine equidistant subdivisions of the spatial structure (dendrites or axon). In the Chebyshev method, solutions are approximated by finite Chebyshev series. The solutions have uniform, usually high, numerical accuracy at any spatial point, not only at the original collocation points. Often, truncation errors become negligible, hence, the total error is essentially the rounding error of the computations. Furthermore, quantities involving spatial derivatives, and in particular the axial current, can be computed exactly from the solution, i.e. the membrane potential. Space-dependent parameter distributions (channel densities, non-uniform dendritic geometries), as well as mixed linear boundary conditions can easily be implemented, and can be chosen from the large class of piecewise smooth functions.

Shock Waves for a Model System of the Radiating Gas

This paper is concerned with the existence and the asymptotic stability of traveling waves for a model system derived from approximating the one-dimensional system of the radiating gas. We show the existence of smooth or discontinuous traveling waves and also prove the uniqueness of these traveling waves under the entropy condition, in the class of piecewise smooth functions with the first kind discontinuities. Furthermore, we show that the C3 -smooth traveling waves areasymptotically stable and that the rate of convergence toward these waves is t-1/4 , which looks optimal. The proof of stability is given by applying the standard energy method to the integrated equation of the original one.

Pointwise fourier inversion and related eigenfunction expansions

AbstractNecessary and sufficient conditions are found for the convergence at a pre‐assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.

Canonical representation: from piecewise-linear function to piecewise-smooth functions

The canonical representation of piecewise-linear (PWL) functions provides a global compact formulation of continuous PWL functions, which has significant advantages in the research and applications concerning nonlinear systems. This work studies the generalization of the canonical representation from PWL functions to piecewise-smooth (PWS) functions. First a class of PWS functions, called the regular PWS functions, is defined as a generalization of the continuous PWL functions. An important example of the regular PWS functions is the continuous piecewise-polynomial function. The continuous PWL function with a PWL partition is also covered by the regular PWS function. Then the canonical representation of the PWS function is defined and the existence conditions are discussed. The PWS generalization of the canonical representation is significant in applications where a PWS scheme can improve the performance of a PWL scheme in the approximation of a nonlinear function, i.e., in approximating the input/output (I/O) relation of a nonlinear system or a mapping neural network or in nonlinear signal processing. >

Convergence analysis of a computational method for optimal control of non-linear differential-algebraic systems

ABSTRACT The convergence analysis of a computational method for optimal control problems of non-linear differential-algebraic systems is considered. The class of admissible controls is taken to be the class of piecewise smooth functions. A control parametrizution technique is used to approximate the optimal control problem into a sequence of optimal parameter selection problems. The solution of each of these approximate problems gives rise to a suboptimal solution to the original optimal control problem in an obvious way. The gradients of the cost functional with respect to parameters are derived. Furthermore, the error bounds between the suboptimal costs and the true optimal cost are derived.

Convergence analysis of a computational method for time-lag optimal control problems

Abstract We consider a class of non-linear time-lag optimal control problems. The class of admissible controls are taken to be the class of piecewise smooth functions. A control parameterization technique is used to approximate the optimal control problem by a sequence of optimal parameter selection problems. The solution of each of these approximate problems gives rise to a sub-optimal solution to the true optimal control problem in an obvious way. The error bound is derived for the sub-optimal costs and the true optimal cost.

DISCONTINUOUS SOLUTIONS OF HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS

CONTENTS Introduction § 1. Systems of quasilinear equations § 2. Some properties of the solutions of quasilinear equations § 3. The concept of a generalized solution of a system of quasilinear equations § 4. Conditions for stability of a generalized solution § 5. Generalized solutions of the Cauchy problem for systems of quasilinear equations, and the irreversibility of the process described by the equations § 6. Conservativeness of systems of quasilinear equations § 7. Example of a non-conservative system of quasilinear equations § 8. The discontinuity of solutions of a conservative system of quasilinear equations § 9. The concept of the potential of a generalized solution of a system of quasilinear equations § 10. Uniqueness of the generalized solution of the Cauchy problem for a single quasilinear equation § 11. Construction of a generalized solution of the Cauchy problem for a single equation by the method of the potential § 12. Uniqueness of the generalized solution of the Cauchy problem for systems of quasilinear equations § 13. Theorem of uniqueness for the generalized solution of the Cauchy problem for systems of equations, in the class of piece-wise smooth functions § 14. Some remarks on hyperbolic systems of linear equations § 15. Centred solutions of systems of quasilinear equations § 16. Uniqueness of the generalized solution of the problem of decay of a discontinuity § 17. The asymptotic behaviour of solutions of quasilinear equations as § 18. The “viscosity” method for quasilinear equations § 19. The system of equations of gas dynamics § 20. A comparison of the properties of solutions of linear and quasilinear equations References