Academic literature on the topic 'Nonlinear PDE`s'

Under the assumptionλ>0andfverifyingf(x,y,0)=0inD,2F(x,y,u)−uf(x,y,u)≥0,u≠0, and ifΩ=R×D, we show the convexity of functionE(t)=∬D|u(t,x,y)|2dxdy, whereu:Ω→ℝis a solution of problemλ(∂2u/∂t2)−(∂/∂x)(p(x,y)(∂u/∂x))−(∂/∂y)(q(x,y)(∂u/∂y))+f(x,y,u)=0 in Ω,u+ε(∂u/∂n)=0 on ∂Ω, considered inH2(Ω)∩L∞(Ω),p,q:D¯→ℝare two nonnull functions onD,εis a positive real number, andD=]a1,b1[×]a2,b2[,(F(x,y,s)=∫0sf(x,y,t)dt).

AbstractWe study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s-th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter s are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.

This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding Ito^’s formula, the B-S type PDE is derived by a self-financing strategy. Furthermore, the explicit pricing formula for barrier options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the barrier price, the Hurst index, the jump intensity and the volatility on the value of barrier option respectively.

We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.

Burger’s equations, an extension of fluid dynamics equations, are typically solved by several numerical methods. In this article, the laplace-Somé Blaise Abbo method is used to solve nonlinear Burger equations. This method is based on the combination of the laplace transform and the SBA method. After reminders of the laplace transform, the basic principles of the SBA method are described. The process of calculating the Laplace-SBA algorithm for determining the exact solution of a linear or nonlinear partial derivative equation is shown. Thus, three examplesof PDE are solved by this method, which all lead to exact solutions. Our results suggest that this method can be extended to other more complex PDEs.

AbstractWe consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form$\frac{d}{dt}\rho(u_{t})+Au_{tt}+\gamma A^{\theta}u_{t}+Au-\int_{0}^{t}g(s)Au(t% -s)=0,$where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ${\rho(s)}$ is a continuous, monotone increasing function, and the relaxation kernel ${g(s)}$ is a continuous, decreasing function in ${L_{1}(\mathbb{R}_{+})}$ with ${g(0)>0}$. Of particular interest is the case when ${A=-\Delta}$ with appropriate boundary conditions and ${\rho(s)=|s|^{\rho}s}$. This model arises in the context of solid mechanics accounting for variable density of the material. While finite energy solutions of the underlying PDE solutions exhibit exponential decay rates when strong damping is active (${\gamma>0,\theta=1}$), this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say, ${\theta=0}$ and ${g=0}$. In the absence of mechanical damping (${\gamma=0}$), the linearized version of the model reduces to a Volterra equation generated by bounded generators and, hence, it is exponentially stable for exponentially decaying kernels. The aim of the paper is to study intrinsic decays for the energy of the nonlinear model accounting for large classes of relaxation kernels described by the inequality ${g^{\prime}+H(g)\leq 0}$ with H convex and subject to the assumptions specified in (1.13) (a general framework introduced first in [1] in the context of linear second-order evolution equations with memory). In the context of frictional damping, such a framework was introduced earlier in [15], where it was shown that the decay rates of second-order evolution equations with frictional damping can be described by solutions of an ODE driven by a suitable convex function H which captures the behavior at the origin of the dissipation. The present paper extends this analysis to nonlinear equations with viscoelasticity. It is shown that the decay rates of the energy are intrinsically described by the solution of the dissipative ODE${S_{t}+c_{1}H(c_{2}S)=0}$with given intrinsic constants ${c_{1},c_{2}>0}$. The results obtained are sharp and they improve (by introducing a novel methodology) previous results in the literature (see [20, 19, 21, 6]) with respect to (i) the criticality of the nonlinear exponent ρ and (ii) the generality of the relaxation kernel.

The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

<abstract><p>The deployment of multi-agent systems (MASs) is widely used in the fields of unmanned agricultural machineries, unmanned aerial vehicles, intelligent transportation, etc. To make up for the defect that the existing PDE-based results are overly idealistic in terms of system models and control strategies, we study the PDE-based deployment of clustered nonlinear first-order and second-order MASs over a finite-time interval (FTI). By designing special communication protocols, the collective dynamics of numerous agents are modeled by simple fist-order and second-order PDEs. Two practical factors, external disturbance and Markov switching topology, are considered in this paper to better match actual situations. Besides, T–S fuzzy technology is used to approximate the unknown nonlinearity of MASs. Then, by using boundary control scheme with collocated measurements, two theorems are obtained to ensure the finite-time $ H_\infty $ deployment of first-order and second-order agents, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the proposed approaches.</p></abstract>

AbstractIn this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent$$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < $\begin{array}{} \displaystyle 2_s^* \end{array}$, where $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by$$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).