Narrow Results

Of concern is the nonlinear evolution equation

An earlier suggested parallel "ring" algorithm for solving the spatially one-dimensional initial-boundary-value problem (IBVP) for a parabolic equation using an explicit difference method is shortly described. Theoretical estimates for both of the speed-up function and the communication complexity of this parallel algorithm are studied. The speed-up function is determined as the ratio between times for realization of the algorithm in sequentional and parallel cases. Theoretical estimates of the speed-up function show the significant speed-up of the parallel algorithm in comparison with the serial one for a large number of values computed by one processor during one time level and it is shown that the the speed-up tends to the number of used processors. Communication complexity is determined as a ratio between the number of interchanges and the number of arithmetical operations. It has been proven that the coefficient of the communication complexity for spatially m-dimensional IBVP tends in general to ¾.

This paper considers the behavior of solutions from the neighborhood of an equilibrium state of nonlinear two-component parabolic problems with diffusion matrixes of one or two eigenvalues to zero. It has been shown that problems related to stability have infinite dimension. Reported here is the development of an algorithm that constructs universal families of nonlinear boundary-value problems which do not contain small parameters and whose nonlocal dynamics describes local dynamics of original boundary-value problems. In addition, an exhaustive set of universal systems for two-component parabolic equations is presented. It is concluded that a hyper multistability phenomenon is one characteristic of these systems.

An optimal harvesting problem for a parabolic partial differential system modeling two subpopulations of the same species is investigated. The two subpopulations are competing for resources. Under conditions on the smallness of the time interval and certain biological parameters, existence and uniqueness of an optimal control pair are established.

Despite their great importance in determining the dynamic evolution of solutions to mathematical models of tumor growth, equilibrium configurations within such models have remained largely unexplored. This was due, in part, to the complexity of the relevant free boundary problems, which is enhanced when the process deviates from radial symmetry. In this paper, we present the results of our investigation on the existence of non-spherical dormant states for a model of non-necrotic vascularized tumors. For the sake of clarity we perform the analysis on two-dimensional geometries, though our techniques are evidently applicable to the full three-dimensional problem. We rigorously show that there is, indeed, an abundance of steady states that are not radially symmetric. More precisely, we prove that at any radially symmetric stationary state with free boundary r=R₀ (which we first show to exist), there begin infinitely many branches of equilibria that bifurcate from and break the symmetry of that radial state. The free boundaries along the bifurcation branches are of the form , where ℓ=2,3,… and |ε|<ε₀; each choice of ℓ and ε determines a non-radial steady configuration.

In this paper we deal with a Cauchy–Dirichlet quasilinear parabolic problem containing a gradient lower order term; namely, u_t - Δu + |u|^{2 γ-2}u |∇u|² = |u|^p-2u. We prove that if p ≥ 1, γ ≥ ½ and p < 2 γ + 2, then there exists a global weak solution for all initial data in L¹ (Ω). We also see that there exists a non-negative solution if the initial datum is non-negative.

Tumor survival, growth and dissemination are associated with the formation of both new blood vessels (angiogenesis) and new lymph vessels (lymphangiogenesis). A mathematical model of tumor lymphangiogenesis was recently developed by the authors, based on experimental results established in the last five years. The model consists of a coupled system of eight parabolic equations. In this paper we prove existence and uniqueness of a solution for this system, for all t>0.

In this paper, we establish several results involving the minimum and maximum principles and the comparison principles for elliptic equations and parabolic equations on finite graphs. The results are then used to prove the monotonicity and asymptotic properties of solutions for parabolic equations whose initial values are given by the equation $\mathrm{\Delta }\psi +f(\psi )=0$ with Dirichlet boundary conditions. Finally, an illustration with numerical experiments is provided to demonstrate our main results.

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite difference schemes. A new compact scheme, generalizing the compact schemes of Rigal [29], is derived and proved to be unconditionally stable and non-oscillatory. The numerical results are compared to standard finite difference schemes. It turns out that the compact schemes have very satisfying stability and non-oscillatory properties and are generally more efficient than the considered classical schemes.

In the setting of a stochastic volatility model, we find a general pricing equation for the class of payoffs depending on the terminal value of a market asset and its final quadratic variation. This provides a pricing tool for European-style claims paying off at maturity a joint function of the underlying and its realized volatility or variance. We study the solution under various specific stochastic volatility models, give a formula for the computation of the delta and gamma of these claims, and introduce some new interesting payoffs that can be valued by means of the general pricing equation. Numerical results are given and compared to those from plain vanilla derivatives.

We are concerned with existence and uniqueness of the solutions for linear and nonlinear parabolic equations with time-dependent coefficients, in the class of bounded solutions satisfying appropriate conditions at infinity.

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

In this article, we continue the analysis of a class of singularly perturbed parabolic equations with alternating boundary layer type solutions. For such problems, the degenerate (reduced) equations obtained by setting a small parameter equal to zero correspond to algebraic equations that have several isolated roots. As time increases, solutions of these equations periodically go through two comparatively long lasting stages with fast transitions between these stages. During one of these stages, the solution outside the boundary layer (i.e. the regular part of the asymptotic solution) is close to one of the roots of the degenerate equation. During the other stage, the regular part of the asymptotic solution is close to the other root. Here we discuss some specific features of the solutions' behavior for such problems in certain two-dimensional spatial domains.

Parabolic partial differential equations possessing nonlocal initial and boundary specifications are used to model some real-life applications. This paper focuses on constructing fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on a double power series representation of the solution via ordinary polynomials. Consequently, a parabolic partial differential equation is reduced to a system involving algebraic equations. Exact solutions are obtained when the solutions are themselves polynomials. Some parabolic partial differential equations are treated by the technique to judge its validity and also to measure its accuracy as compared to the existing methods.

Let be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form . In particular, we establish results on the distribution kernels and mapping properties of e^-tL and (μ + L)^s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the W^s,p-Sobolev spaces on M and E. We also prove that L satisfies maximal L^p–L^q-regularity for 1 < p, q < ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂_tu + Lu = F(t, x, u, ∇ u).

This paper studies Γ-convergence for a sequence of parabolic functionals,

The authors consider a class of nonlinear parabolic problems where the lower order term is depending on a weighted integral of the solution, and address the issues of existence, uniqueness, stationary solutions and in some cases asymptotic behaviour.

A new efficient multigrid algorithm is proposed for solving parabolic equations. It is similar to implicit schemes by stability and accuracy, but the computational complexity is substantially reduced at each time step. Stability and accuracy of the proposed two-grid algorithm are analyzed theoretically for one- and two-dimensional heat diffusion equations. Good accuracy is demonstrated on model problems for one- and two-dimensional heat diffusion equations, including those with thermal conductivity defined as a discontinuous function of coordinates.

The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Car-leman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equations.