On spherically symmetric solutions of the relativistic Euler equation

This work investigates the spherical symmetric solutions of more realistic equation of states. We generalize the method of Hsu et al. (Methods Appl. Anal. 8 (2001) 159) to show the existence of spherical symmetric weak solution of the relativistic Euler equation with initial data containing the vacuum state.     

Barenblatt solution and spherically symmetric solutions of the porous media equation

We consider the Cauchy problem of the porous media equation. We show that it is spherically symmetric solution has the same property as Barenblatt solution, with respct to some regularity property.     

A self-similar solution to the equation of gas filtration in a spherically symmetric porous medium

We consider the Cauchy problem for the Boussinesq equation which describes filtration of a gas in a spherically symmetric porous medium. For the self-similar solution to this problem we construct a formal in the neighborhood of the point r → ∞ expansion and a convergent near r = 0 one.     

On the topology of spherically symmetric space-times

Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent results concerning the global geometry of spherically symmetric space-times [16] is presented below.     

On the numerical solutions for the fractional diffusion equation

Fractional differential equations have recently been applied in various area of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional diffusion equation (FDE) is considered. The fractional derivative is described in the Caputo sense. The method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FDE to a system of ordinary differential equations, which solved by the finite difference method. Numerical simulation of FDE is presented and the results are compared with the exact solution and other methods.     

On the regularity of spherically symmetric wave maps

Wave maps are critical points U: M → N of the Lagrangian ??[U] = ∞_M ‖dU‖², where M is an Einsteinian manifold and N a Riemannian one. For the case M = ?^2,1 and U a spherically symmetric map, it is shown that the solution to the Cauchy problem for U with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (1) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded; and (2) there are two constants c and C such that the smallest eigenvalue λ and the largest eigenvalue λ of the second fundamental form k_AB of any geodesic sphere Σ(p, s) of radius s centered at p ? N satisfy sλ ≧ c and s A ≦ C(1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. © 1993 John Wiley & Sons, Inc.     

Numerical treatment of the spherically symmetric solutions of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov equation

In the present work, the connection of the generalized Fisher-KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher-KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher-KPP equation, the Heaviside equation, and the generalized Fisher-KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher-KPP problems.     

Identifying a spherically symmetric conductivity in a nonlinear parabolic equation

In this article we determine, i.e. we prove existence and uniqueness of a radially symmetric operator conductivity in a nonlinear heat equation of the form (1.1) related to a spherical corona Ω ? R ³ under a suitable additional information.     

On the unimodality of spherically symmetric stable distribution functions

Several theorems are obtained concerning the unimodality of spherically symmetric distribution functions. These theorems are used to show that a class of spherically symmetric infinitely divisible distribution functions that contains the class of spherically symmetric stable distribution functions is unimodal.     

Stabilizing the nonlinear spherically symmetric backward heat equation via two-parameter regularization method

In this paper, we consider a time-inverse problem for a nonlinear spherically symmetric backward heat equation which is a severely ill-posed problem. Using a modified integral equation method with two regularization parameters: one related to the error in a measurement process and the other is related to the regularity of solution, we regularize this problem and obtain the Hölder-type estimation error for the whole time interval. Numerical results are presented to illustrate the accuracy and efficiency of the method.     

On the numerical solution of the Klein‐Gordon equation

A predictor–corrector (P–C) scheme based on the use of rational approximants of second‐order to the matrix‐exponential term in a three‐time level reccurence relation is applied to the nonlinear Klein‐Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double‐soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Identifying an unknown source term in a spherically symmetric parabolic equation

The aim of this work is to solve the inverse problem of determining an unknown source term in a spherically symmetric parabolic equation. The problem is ill-posed: the solution (if it exists) does not depend continuously on the final data. A spectral method is applied to formulate a regularized solution, and a Hölder type estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter.     

Chebyshev pseudospectral method for computing numerical solution of convection–diffusion equation

A method for computing highly accurate numerical solutions of 1D convection–diffusion equations is proposed. In this method, the equation is first discretized with respect to the spatial variable, transforming the original problem into a set of ordinary differential equations, and then the resulting system is integrated in time by the fourth-order Runge–Kutta method. Spatial discretization is done by using the Chebyshev pseudospectral collocation method. Before describing the method, we review a finite difference-based method by Salkuyeh [D. Khojasteh Salkuyeh, On the finite difference approximation to the convection–diffusion equation, Appl. Math. Comput. 179 (2006) 79–86], and, contrary to the proposal of the author, we show that this method is not suitable for problems involving time dependent boundary conditions, which calls for revision. Stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out. Superiority of the proposed method over a revised version of Salkuyeh’s method is verified by numerical examples.     

On the numerical solution of the generalized Prandtl equation using variation-diminishing splines

Convergence results are proved for Cauchy principal value integrals of the Schoenberg variation-diminishing splines and its first derivative. The use of such splines in the numerical solution of the Prandtl and generalized Prandtl integral equations is proposed. A Nyström-type method and a modified Nyström method are used and compared computationally.     

On a numerical algorithm for the solution of the radial Loewner equation

The Loewner partial differential equation provides a one‐parametric family of conformal maps on the unit disk. The images describe a flow of an expanding simply‐connected domain, called the Loewner flow, on the complex plane. In this paper, we present a numerical algorithm for solving the radial Loewner partial differential equation. The algorithm is applied to visualization of Loewner flows with the performance and precision. From the theoretical point of view, our algorithm is based on a recursive formula for determining coefficients of polynomial approximations. We prove that each coefficient converges to true values with reasonable regularity.     

A numerical solution for advection‐diffusion equation using dual reciprocity method

This article describes a numerical method based on the boundary integral equation and dual reciprocity method(DRM) for solving the one‐dimensional advection‐diffusion equations. The concept of DRM is used to convert the domain integral to the boundary that leads to an integration free method. The time derivative is approximated by the time‐stepping method. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of the new approach. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013