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Some properties of positive radial solutions for some semilinear elliptic equations. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Positive radial solutions for some quasilinear elliptic systems in exterior domains. American Institute of Mathematical Sciences. We consider the structure and the stability of positive radial solutions of a semilinear inhomogeneous elliptic equation with multiple growth terms. Equations similar to this kind have come up both in geometry and in physics, and have been a subject of extensive studies.

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Our result shows that any positive radial solution is stable or weakly asymptotically stable with respect to certain norm. Keywords: Cauchy problem , positive radial solutions , stability , sub- and super-solutions , Matukuma-type equation. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. References: [1] D. Google Scholar [2] S. Google Scholar [3] S. Google Scholar [4] S. Google Scholar [5] S.

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## Local and global existence theorems for the Einstein equations

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## The Cauchy problem for solutions of elliptic equations

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Note that Eq. We can now consider using Eq. The finite difference form of Eq. We must also specify two boundary conditions BCs for Eq. Equations 49 , 51 , 53 and 55 constitute the full system of equations for the calculation of the numerical solution to Eq. Note that we have replaced the original PDE, Eq.

Also, an analytical solution to Eq. Representative output from this program that compares the numerical solution from Eqs. Additional parameters follow from the values in Table 2. We can note two additional points about these values:. Finally, some descriptive comments about the details of the program in Appendix 1 are given immediately after the program listing. We again have an analytical solution to evaluate the numerical solution.

Since Eq.

An important difference between the parabolic problem of Eq. Equations 66 , 67 , 68 , 69 , 70 , 71 constitute the complete finite difference approximation of Eqs. A small MATLAB program for this approximation is given in Appendix 2 , including some descriptive comments immediately after the program.

The general analytical solution to Eq. The plotted output from the program is given in Figure 1 and includes both the numerical solution of Eqs. We can note the following points about Figure 1 :. Additional parameters follow from the values in Table 3.

This is a stringent test of the numerical solution since the curvature of the solution is greatest at these peaks. Of course, the peak analytical values given by Eq. This explicit finite difference numerical solution also has a stability limit like the preceding parabolic problem. In other words, the parameters of Table 3 were chosen primarily for accuracy and not stability.

Rather, we will convert Eq. The idea then is to integrate Eq. The analytical solution to Eqs. The analytical solution of Eq. To develop a numerical solution to Eq. The output from this program is listed in Table 5. The convergence of the solution of Eq. Additional parameters follow from the values in Table 6.

The stability constraint for the 2D problem of Eq. The actual path that the parabolic problem takes to the solution of the elliptic problem is not relevant so long as the parabolic solution converges to the elliptic solution. Some other trial values indicated that this initial value is not critical but it should be as close to the final value, for example 2.

This parametrization is an example of continuation in which the solution is continued from the given assumed starting value of Eq. The concept of continuation has been applied in many forms and not just through the addition of a derivative as in Eq. In general, the errors in the numerical solution of PDEs can result from the limited accuracy of all of the approximations used in the calculation.

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For example, the 0. In formulating a numerical method or algorithm for the solution of a PDE problem, it is necessary to balance the discretization errors so that one source of error does not dominate, and generally degrade, the numerical solution. Thus, control of approximation errors is central to the calculation of a numerical solution of acceptable accuracy.