Differential Equations Of Evolution Type Research Articles

Singular Perturbation of Nonlinear Systems with Regular Singularity

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type.

Humeanism and Exceptions in the Fundamental Laws of Physics

It has been argued that the fundamental laws of physics do not face a ‘problem of provisos’ equivalent to that found in other scientific disciplines (Earman, Roberts and Smith 2002) and there is only the appearance of exceptions to physical laws if they are confused with differential equations of evolution type (Smith 2002). In this paper I argue that even if this is true, fundamental laws in physics still pose a major challenge to standard Humean approaches to lawhood, as they are not in any obvious sense about regularities in behaviour. A Humean approach to physical laws with exceptions is possible, however, if we adopt a view of laws that takes them to be the algorithms in the algorithmic compressions of empirical data. When this is supplemented with a distinction between lossy and lossless compression, we can explain exceptions in terms of compression artefacts present in the application of the lossy laws.

On the relaxation gap for PDE mixed‐integer optimal control problems

AbstractMixed‐integer optimal control problems require taking discrete and continuous control decisions for the optimization of a dynamical system. We consider dynamics governed by partial differential equations of evolution type and assess the problem by relaxation and rounding strategies. For this solution approach, we present a priori estimates for semilinear evolutions on Banach spaces concerning the optimality gap. The theoretical results show that the gap can be made arbitrary small. We demonstrate the numerical performance of the approach on benchmark problems of parabolic type motivated from thermal manufacturing and of hyperbolic type motivated from traffic flow control. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On SPDEs with Variable Coefficients in One Space Dimension

Stochastic second-order partial differential equations of evolution type are considered in one space dimension. Existence and uniqueness results are given in weighted Sobolev spaces of functions given on an interval in the space variable.

Operational methods, fractional operators and special polynomials

We use methods of operational nature to deal with families of partial differential equations of evolution type to treat problems involving fractional differential operators. We also discuss the properties of families of special polynomials or special functions (like the Riemann ζ function), naturally associated with the proposed formalism.

Generalized solutions to partial differential equations of evolution type

This paper deals with a new solution concept for partial differential equations in algebras of generalized functions. Introducing regularized derivatives for generalized functions, we show that the Cauchy problem is wellposed backward and forward in time for every system of linear partial differential equations of evolution type in this sense. We obtain existence and uniqueness of generalized solutions in situations where there is no distributional solution or where even smooth solutions are nonunique. In the case of symmetric hyperbolic systems, the generalized solution has the classical weak solution as ‘macroscopic aspect’. Two extensions to nonlinear systems are given: global solutions to quasilinear evolution equations with bounded nonlinearities and local solutions to quasilinear symmetric hyperbolic systems.

AXIOMATIC DERIVATION OF IMAGE PROCESSING MODELS

We briefly review the derivation due to Alvarez, Guichard, Morel and the author of mathematical models in Image Processing. We deduce from classical axions in Computer Vision some nonlinear partial differential equations of evolution type that correspond to general multi-scale analysis (scale-space). We also obtain specific nonlinear models that satisfy additional invariances which are relevant for the analysis of images.

Perturbation of m-accretive operators in banach spaces

THERE ARE not many results on multiplicative perturbation theory dealing with development of criteria for an m-accretive operator to remain m-accretive after being composed with other operators. In this connection we mention the work of Gustafson and Lumer [l 11, who gave a criterion for m-accretivity of the composition KA on a Banach space X, where K is assumed to be an everywhere defined strongly accretive bounded linear operator and KA is accretive. The main purpose of this note is to weaken the requirement of strong accretivity of K to simple accretivity to conclude that the closure of densely defined operator KA is m-accretive. An example is given to illustrate applicability of this result even though the Gustafson and Lumer result is not applicable. Further a nonlinear analogue of Lumer’s result is proved. We suggest two unsolved problems in this direction. It is well known that the class of m-accretive operators arise in initial boundary value problems of differential equations of evolution type. Multiplicative perturbations of such operators have been studied previously in [l, 5, 6, 7, 8, 9, 10, 11, 131. For a description of m-accretive operators and their properties one can refer to [2, 3, 6, 12, 151. We state the above mentioned result of Gustafson and Lumer [l l] in the following form which can be compared more easily with our results in this work.

Unconditional stability of difference formulas

The paper concerns the solution of partial differential equations of evolution type by the finite difference method. The author discusses the general assumptions on the original equation as well as its discretization, which guarantee that the difference scheme is unconditionally stable, i.e. stable without any stability condition for the time-step. A new notion of the $A_n$-acceptability of the integration formula is introduced and examples of such formulas are given. The results can be applied to ordinary differential equations as well.

Implicit schemes for differential equations

Implicit high-order accurate temporal-integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only. The ordinary differential equations are solved using various accurate integration methods. The schemes are free from extraneous boundary and/or initial conditions and are stable for large Courant number.