Smooth Radial Functions Research Articles

A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation

We propose and analyze a meshless method of lines by considering some hybrid radial kernels. These hybrid kernels are constructed by linearly combining infinite smooth radial functions to piecewise smooth radial functions; which are then used for spatial approximation on trial spaces spanned by translates of positive definite radial functions. After spatial approximation, a high-order ODE solver is invoked for efficient and stable time-integration of the resultant semi-discrete system of ordinary differential equations (ODEs). Unlike the mesh-based method of lines, the proposed method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. The proposed method is tested on one-, two- and three-dimensional reaction–diffusion Fisher equation for its numerical stability, accuracy, and efficiency against the contemporary meshless and mesh-based methods. The economical computational cost, improved accuracy, eigenvalues stability, and well-conditioning of system matrices are observed against RBF collocation and RBF-PS methods.

Limiting profile of the blow-up solutions for the fourth-order nonlinear Emden-Fowler equation with a singular source

We consider the anisotropic Emden-Fowler equation with exponential nonlinearity in fourth-dimensional with a singular source, given by \begin{document}$ \begin{equation*} \left\{\begin{array}{rclll}\Delta(a(x)\Delta u)-V(x)\,div(a(x)\nabla\,u) & = & \rho^{4} a(x)\,e^{u} -32 \pi^2\;\alpha\,a(x)\,\delta_{p}&\mbox{in} \; \; \Omega\subset \mathbb R^{4}\\\\u = \Delta u & = &0 &\mbox{on} \; \; \partial\Omega. \end{array} \right. \end{equation*} $\end{document} The leading part $ \Delta $ is, usually, called Laplacian operator. $ V = V(|x|) $ is a smooth bounded radial potential and $ a = a(|x|) $ is a given smooth radial function over $ \bar{\Omega} $, called the Schrödinger wave function. Namely, we extend the result of [8] by allowing the presence in the equation a singular source given by Dirac measure with pole at point $ p $ and a generalization in higher dimensional, of the interest problem treated in [11]. We construct a family of solutions $ u_{\rho} $ that concentrate at $ p $ as $ \rho $ tends to $ 0 $.

SPHERICAL NEWTON DISTANCE FOR OSCILLATORY INTEGRALS WITH HOMOGENEOUS PHASE FUNCTIONS

Abstract. In this paper we study oscillatory integrals with ana-lytic homogeneous phase functions for smooth radial functions. Wegive their sharp asymptotic behavior in terms of spherical Newtondistance. 1. IntroductionIn this paper we consider oscillatory integrals I(;’) de ned byI(;’) =Z R k e if(x) ’(x)dx;where is a large positive real parameter. Here f is a homogeneouspolynomial and ’is a smooth cut-o function. In this chapter we con-sider the cases k= 2;3. Main purpose of this chapter is to reduce thedimension by one to treat the higher dimensional problems in lower di-mension. For the case k= 3 we introduce \spherical Newton distance in\superadapted coordinates. A superadapted coordinate system was in-troduced by Greenblatt in [8] to describe the oscillatory index of smoothphase function in R 2 and it turned out to be a stronger notion of adaptedcoordinate system in [9].In view of the work of Varchenko [9], it is not surprising that theasymptotic behavior of I(;’) bears a close connection to the blow-upproperties of integrals of the formZ

Nodal solutions for an elliptic problem involving large nonlinearities

Let us consider the problem (0.1) { − Δ u + a ( | x | ) u = | u | p − 1 u in B 1 , u = 0 on ∂ B 1 , where B 1 is the unit ball in R N , N ⩾ 3 , and a ( | x | ) ⩾ 0 is a smooth radial function. Under some suitable assumptions on the regular part of the Green function of the operator − u ″ − N − 1 r u + a ( r ) u we prove the existence of a changing sign solution to (0.1) for p large enough.

Radial solutions for the Brezis–Nirenberg problem involving large nonlinearities

Let us consider the problem (0.1) { − Δ u + a ( | x | ) u = u p in B 1 , u > 0 in B 1 , u = 0 on ∂ B 1 , where B 1 is the unit ball in R N , N ⩾ 3 , and a ( | x | ) ⩾ 0 is a smooth radial function. Under some suitable assumptions on the regular part of the Green function of the operator − u ″ − N − 1 r u + a ( r ) u , we prove the existence of a radial solution to (0.1) for p large enough.

Existence of radial solutions for an elliptic problem involving exponential nonlinearities

Let us consider the problem $-\Delta u+a(|x|)u=\lambda e^u$in$ B_1,$ (0.1) $u=0$ on$ \partial B_1.$ where $B_1$ is the unit ball in $R^N$, $N\ge2$, $\lambda>0$ and $a(|x|)\ge0$ is a smooth radial function. Under some suitable assumptions on the regular part of the Green function of the operator $-u''- \frac{N-1}{r}u+a(r)u$, we prove the existence of a radial solution to (0.1) for $\lambda$ small enough.

A new class of radial basis functions with compact support

Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature.