Search results for "partial differential equation"
showing 10 items of 326 documents
A linearization technique and error estimates for distributed parameter identification in quasilinear problems
1996
The identification problem of a nonlinear functional coefficient in elliptic and parabolic quasilinear equations is considered. A distributed observation of the solution of the corresponding equation is assumed to be known a priori. An identification method is introduced, which needs only a linear equation to be solved in each iteration step of the optimization. Estimates of the rate of convergence for the proposed approach are proved, when the equation is discretized with the finite element method with respect to space variables. Some numerical results are given.
Singular Double Phase Problems with Convection
2020
We consider a nonlinear Dirichlet problem driven by the sum of a $p$ -Laplacian and of a $q$ -Laplacian (double phase equation). In the reaction we have the combined effects of a singular term and of a gradient dependent term (convection) which is locally defined. Using a mixture of variational and topological methods, together with suitable truncation and comparison techniques, we prove the existence of a positive smooth solution.
Linearly implicit-explicit schemes for the equilibrium dispersive model of chromatography
2018
Abstract Numerical schemes for the nonlinear equilibrium dispersive (ED) model for chromatographic processes with adsorption isotherms of Langmuir type are proposed. This model consists of a system of nonlinear, convection-dominated partial differential equations. The nonlinear convection gives rise to sharp moving transitions between concentrations of different solute components. This property calls for numerical methods with shock capturing capabilities. Based on results by Donat, Guerrero and Mulet (Appl. Numer. Math. 123 (2018) 22–42), conservative shock capturing numerical schemes can be designed for this chromatography model. Since explicit schemes for diffusion problems can pose seve…
The body-mind problem from a personality relativity theory approach: (Relativity of personality)
2012
The body-mind problem is here posed fromthe General Factor of Personality Theory and mathematically presented as a relativity theory: 1. A two-level-invariant model; 2. The transformation equation between both levels. On a hand, the investigation of personality dynamicsas a consequenceof a stimulant drug provides a system of two coupled differential equations, which adapts to both biological (body) and psychological (mind)levels ofdescription. On the other hand, a system of two coupled partial differential equations is here deduced as the transformation model between the biological and psychological levels. Both models are presented and validated in this paper. The experimental design to va…
Spatial seismic point pattern analysis with Integrated Nested Laplace Approximation
2020
This paper proposes the use of Integrated Nested Laplace Approximation (Rue et al., 2009) to describe the spatial displacement of earthquake data. Specifying a hiechical structure of the data and parameters, an inhomogeneuos Log-Gaussian Cox Processes model is applied for describing seismic events occurred in Greece, an area of seismic hazard. In this way, the dependence of the spatial point process on external covariates can be taken into account, as well as the interaction among points, through the estimation of the parameters of the covariance of the Gaussian Random Field, with a computationally efficient approach.
Generalized finite difference schemes with higher order Whitney forms
2021
Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first ord…
On critical behaviour in generalized Kadomtsev-Petviashvili equations
2016
International audience; An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the disp…
Minimizing total variation flow
2000
We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in $L^1$. We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as $t \to \infty$. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts.
Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data
2018
We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$ - {\rm{div}}(a(x,\nabla u)) = h(x,u) + \mu ,\;\;\;\;\;{\rm{in}}\;{\rm{\Omega }} \subset \mathbb{R}^{N},$$ where the left-hand side is a Leray-Lions operator from $$- {W}^{1,p}_0(\Omega)$$ into W−1,p′(Ω) with 1 < p < N, h(x,s) is a Caratheodory function which grows like ∣s∣p−1 and μ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Holder-continuous far from the support of μ.
Optimal shape design and unilateral boundary value problems: Part II
2007
In the first part we give a general existence theorem and a regularization method for an optimal control problem where the control is a domain in R″ and where the system is governed by a state relation which includes differential equations as well as inequalities. In the second part applications for optimal shape design problems governed by the Dirichlet-Signorini boundary value problem are presented. Several numerical examples are included.