An Analog of the Tricomi Problem for a Mixed Type Equation with a Partial Fractional Derivative

Mathematics Subject Classification 2010: 35M10, 35R11, 26A33, 33C05, 33E12, 33C20.

A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Symbolic Solving of Partial Differential Equation Systems and Compatibility Conditions

An algorithm is produced for the symbolic solving of systems of partial differential equations by means of multivariate Laplace–Carson transform. A system of K equations with M as the greatest order of partial derivatives and right-hand parts of a special type is considered. Initial conditions are input. As a result of a Laplace–Carson transform of the system according to initial condition we obtain an algebraic system of equations. A method to obtain compatibility conditions is discussed.

Boundary identification in the domain of a parabolic partial differential equation

Bakgrunden och inspirationen till föreliggande studie är tidigare forskning i tillämpningar på randidentifiering i metallindustrin. Effektiv randidentifiering möjliggör mindre säkerhetsmarginaler och längre serviceintervall för apparaturen i industriella högtemperaturprocesser, utan ökad risk för materielhaverier. I idealfallet vore en metod för randidentifiering baserad på uppföljning av någon indirekt variabel som kan mätas rutinmässigt eller till en ringa kostnad. En dylik variabel för smältugnar är temperaturen i olika positioner i väggen. Denna kan utnyttjas som insignal till en randidentifieringsmetod för att övervaka ugnens väggtjocklek. Vi ger en bakgrund och motivering till valet av den geometriskt endimensionella dynamiska modellen för randidentifiering, som diskuteras i arbetets senare del, framom en flerdimensionell geometrisk beskrivning. I de aktuella industriella tillämpningarna är dynamiken samt fördelarna med en enkel modellstruktur viktigare än exakt geometrisk beskrivning. Lösningsmetoder för den s.k. sidledes värmeledningsekvationen har många saker gemensamt med randidentifiering. Därför studerar vi egenskaper hos lösningarna till denna ekvation, inverkan av mätfel och något som brukar kallas förorening av mätbrus, regularisering och allmännare följder av icke-välställdheten hos sidledes värmeledningsekvationen. Vi studerar en uppsättning av tre olika metoder för randidentifiering, av vilka de två första är utvecklade från en strikt matematisk och den tredje från en mera tillämpad utgångspunkt. Metoderna har olika egenskaper med specifika fördelar och nackdelar. De rent matematiskt baserade metoderna karakteriseras av god noggrannhet och låg numerisk kostnad, dock till priset av låg flexibilitet i formuleringen av den modellbeskrivande partiella differentialekvationen. Den tredje, mera tillämpade, metoden kännetecknas av en sämre noggrannhet förorsakad av en högre grad av icke-välställdhet hos den mera flexibla modellen. För denna gjordes även en ansats till feluppskattning, som senare kunde observeras överensstämma med praktiska beräkningar med metoden. Studien kan anses vara en god startpunkt och matematisk bas för utveckling av industriella tillämpningar av randidentifiering, speciellt mot hantering av olinjära och diskontinuerliga materialegenskaper och plötsliga förändringar orsakade av “nedfallande” väggmaterial. Med de behandlade metoderna förefaller det möjligt att uppnå en robust, snabb och tillräckligt noggrann metod av begränsad komplexitet för randidentifiering.

Edge detection and noise removal by use of a partial differential equation with automatic selection of parameters

This work deals with noise removal by the use of an edge preserving method whose parameters are automatically estimated, for any application, by simply providing information about the standard deviation noise level we wish to eliminate. The desired noiseless image u(x), in a Partial Differential Equation based model, can be viewed as the solution of an evolutionary differential equation u t(x) = F(u xx, u x, u, x, t) which means that the true solution will be reached when t ® ¥. In practical applications we should stop the time ''t'' at some moment during this evolutionary process. This work presents a sufficient condition, related to time t and to the standard deviation s of the noise we desire to remove, which gives a constant T such that u(x, T) is a good approximation of u(x). The approach here focused on edge preservation during the noise elimination process as its main characteristic. The balance between edge points and interior points is carried out by a function g which depends on the initial noisy image u(x, t0), the standard deviation of the noise we want to eliminate and a constant k. The k parameter estimation is also presented in this work therefore making, the proposed model automatic. The model's feasibility and the choice of the optimal time scale is evident through out the various experimental results.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

On a Stochastic Partial Differential Equation with a Noisy Term

2000 Mathematics Subject Classification: 60H15, 60H40

Darcy`s law and the differential equation of motion

This note addresses the relation between the differential equation of motion and Darcy`s law. It is shown that, in different flow conditions, three versions of Darcy`s law can be rigorously derived from the equation of motion.

Differential expression of AMPA-type glutamate receptor subunits during development of the chick optic tectum

Glutamate receptors have been often associated with developmental processes. We used immunohistochemical techniques to evaluate the expression of the AMPA-type glutamate receptor (GluR) subunits in the chick optic tectum (TeO). Chick embryos from the 5th through the 20th embryonic day (E5-E20) and one-day-old (P1) chicks were used. The three types of immunoreactivity evaluated (GluR1, GluR2/3, and GluR4) had different temporal and spatial expression patterns in the several layers of the TeO. The GluR1 subunit first appeared as moderate staining on E7 and then increased on E9. The mature GluR1 pattern included intense staining only in layer 5 of the TeO. The GluR2/3 subunits presented low expression on E5, which became intense on E7. The staining for GluR2/3 changed to very intense on E14 in tectal layer 13. Staining of layer 13 neurons is the most prominent feature of GluR immunoreactivity in the adult TeO. The GluR4 subunit generally presented the lowest expression starting on E7, which was similar to the adult pattern. Some instances of transient expression of GluR subunits were observed in specific cell populations from E9 through E20. These results demonstrate a differential expression of the GluR subunits in the embryonic TeO, adding information about their possible functions in the developmental processes of the visual system.

Differential recognition of the type I interferon receptor by interferons tau and alpha is responsible for their disparate cytotoxicities.

Interferon tau (IFN tau), originally identified as a pregnancy recognition hormone, is a type I interferon that is related to the various IFN alpha species (IFN alpha s). Ovine IFN tau has antiviral activity similar to that of human IFN alpha A on the Madin-Darby bovine kidney (MDBK) cell line and is equally effective in inhibiting cell proliferation. In this study, IFN tau was found to differ from IFN alpha A in that is was > 30-fold less toxic to MDBK cells at high concentrations. Excess IFN tau did not block the cytotoxicity of IFN alpha A on MDBK cells, suggesting that these two type I IFNs recognize the type I IFN receptor differently on these cells. In direct binding studies, 125I-IFN tau had a Kd of 3.90 x 10(-10) M for receptor on MDBK cells, whereas that of 125I-IFN alpha A was 4.45 x 10(-11) M. Consistent with the higher binding affinity, IFN alpha A was severalfold more effective than IFN tau in competitive binding against 125I-IFN tau to receptor on MDBK cells. Paradoxically, the two IFNs had similar specific antiviral activities on MDBK cells. However, maximal IFN antiviral activity required only fractional occupancy of receptors, whereas toxicity was associated with maximal receptor occupancy. Hence, IFN alpha A, with the higher binding affinity, was more toxic than IFN tau. The IFNs were similar in inducing the specific phosphorylation of the type I receptor-associated tyrosine kinase Tyk2, and the transcription factors Stat1 alpha and Stat2, suggesting that phosphorylation of these signal transduction proteins is not involved in the cellular toxicity associated with type I IFNs. Experiments using synthetic peptides suggest that differences in the interaction at the N terminal of IFN tau and IFN alpha with the type I receptor complex contribute significantly to differences in high-affinity equilibrium binding of these molecules. It is postulated that such a differential recognition of the receptor is responsible for the similar antiviral but different cytotoxic effects of these IFNs. Moreover, these data imply that receptors are "spare'' with respect to certain biological properties, and we speculate that IFNs may induce a concentration-dependent selective association of receptor subunits.

Differential involvement of N-type calcium channels in transmitter release from vasoconstrictor and vasodilator neurons

1 The effects of calcium channel blockers on co-transmission from different populations of autonomic vasomotor neurons were studied on isolated segments of uterine artery and vena cava from guinea-pigs. 2 Sympathetic, noradrenergic contractions of the uterine artery (produced by 200 pulses at 1 or 10 Hz; 600 pulses at 20 Hz) were abolished by the N-type calcium channel blocker omega-conotoxin (CTX) GVIA at 1-10 nM. 3 Biphasic sympathetic contractions of the vena cava (600 pulses at 20 Hz) mediated by noradrenaline and neuropeptide Y were abolished by 10 nM CTX GVIA. 4 Neurogenic relaxations of the uterine artery (200 pulses at 10 Hz) mediated by neuronal nitric oxide and neuropeptides were reduced < 50% by CTX GVIA 10-100 nM. 5 Capsaicin (3 muM) did not affect the CTX GVIA-sensitive or CTX GVIA-resistant neurogenic relaxations of the uterine artery. 6 The novel N-type blocker CTX CVID (100-300 nM), P/Q-type blockers agatoxin IVA (10-100 nM) or CTX CVIB (100 nM), the L-type blocker nifedipine (10 muM) or the 'R-type' blocker SNX-482 (100 nM), all failed to reduce CTX GVIA-resistant relaxations. The T-type channel blocker NiCl2 (100-300 muM) reduced but did not abolish the remaining neurogenic dilations. 7 Release of different neurotransmitters from the same autonomic vasomotor axon depends on similar subtypes of calcium channels. N-type channels are responsible for transmitter release from vasoconstrictor neurons innervating a muscular artery and capacitance vein, but only partly mediate release of nitric oxide and neuropeptides from pelvic vasodilator neurons.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Invariant discretizations of partial differential equations

Un algorithme permettant de discrétiser les équations aux dérivées partielles (EDP) tout en préservant leurs symétries de Lie est élaboré. Ceci est rendu possible grâce à l'utilisation de dérivées partielles discrètes se transformant comme les dérivées partielles continues sous l'action de groupes de Lie locaux. Dans les applications, beaucoup d'EDP sont invariantes sous l'action de transformations ponctuelles de Lie de dimension infinie qui font partie de ce que l'on désigne comme des pseudo-groupes de Lie. Afin d'étendre la méthode de discrétisation préservant les symétries à ces équations, une discrétisation des pseudo-groupes est proposée. Cette discrétisation a pour effet de transformer les symétries ponctuelles en symétries généralisées dans l'espace discret. Des schémas invariants sont ensuite créés pour un certain nombre d'EDP. Dans tous les cas, des tests numériques montrent que les schémas invariants approximent mieux leur équivalent continu que les différences finies standard.