In this paper we present the identification of parameters in the flux and diffusion functions for a quasilinear strongly degenerate parabolic equation which models the physical phenomenon of flocculated sedimentation. We formulate the identification problem as a minimization of a suitable cost function and we derive its formal gradient by means of an adjoint equation which is a backward linear degenerate parabolic equation with discontinuous coefficients. For the numerical approach, we start with the discrete Lagrangian formulation and assuming that the direct problem is discretized by the Engquist-Osher scheme obtain a discrete adjoint state associated with this scheme. The conjugate gradient method allows us to find numerical values of the physical parameters. It also allows us to identify the critical concentration level at which solid flocs begin to touch each other and determine the change of parabolic to hyperbolic behaviour in the model equation.
Abstract. We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definition of entropy solution of the model in the sense of Kružkov and prove convergence of the scheme to the unique BV entropy solution of the problem, up to satisfaction of one of the boundary conditions.
One of the main challenges in cancer modelling is to improve the knowledge of tumor progression in areas related to tumor growth, tumor-induced angiogenesis and targeted therapies efficacy. For this purpose, incorporate the expertise from applied mathematicians, biologists and physicians is highly desirable. Despite the existence of a very wide range of models, involving many stages in cancer progression, few models have been proposed to take into account all relevant processes in tumor progression, in particular the effect of systemic treatments and angiogenesis. Composite biological experiments, both in vitro and in vivo, in addition with mathematical modelling can provide a better understanding of theses aspects. In this work we proposed that a rational experimental design associated with mathematical modelling could provide new insights into cancer progression. To accomplish this task, we reviewed mathematical models and cancer biology literature, describing in detail the basic principles of mathematical modelling. We also analyze how experimental data regarding tumor cells proliferation and angiogenesis in vitro may fit with mathematical modelling in order to reconstruct in vivo tumor evolution. Additionally, we explained the mathematical methodology in a comprehensible way in order to facilitate its future use by the scientific community.
Artículo de publicación ISIThis paper deals with a systematic study of the convolution operator Kf = f (*) k defined on weighted pseudo almost periodic functions space PAP(X, rho) and with k is an element of L-1(R). Upon making several different assumptions on k, f and rho, we get five main results. The first two main results establish sufficient conditions on k and rho such that the weighted ergodic space PAP(0)(X, rho) is invariant under the operator kappa. The third result specifies a sufficient condition on all functions (k, f and rho) such that the kappa f is an element of PAP(0)(X, rho). The fourth result is a sufficient condition on the weight function p such that PAP(0)(X, rho) is invariant under kappa. The hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of PAP(0)(X, rho). As a consequence of the fifth result, we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of PAP(X, rho) with the supremum norm. In addition, the results on convolution are applied to general abstract integral and differential equations.Universidad del Bio-Bio, Chile
DIUBB GI 153209/C
DIUBB GI 152920/EF
FONDECYT
1120709
Universidad Central de Chile
CIR 141
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