The Dirichlet Problem for Some Semilinear Elliptic Differential Equations of Arbitrary Order

Analysis ◽  
1991 ◽  
Vol 11 (1) ◽  
Author(s):  
Hans-Christoph Granau
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Arbitrary Order ◽  
Semilinear Elliptic ◽  
Elliptic Differential Equations

The problem of dirichlet for quasilinear elliptic differential equations with many independent variables

1969 ◽  
Vol 264 (1153) ◽  
pp. 413-496 ◽  
Keyword(s):  
Dirichlet Problem ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Central Position ◽  
Second Primary ◽  
Elliptic Differential Equations ◽  
High Level ◽  
Quasilinear Elliptic

This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of necessary conditions and sufficient conditions for this problem to be solvable in a given domain with arbitrarily assigned smooth boundary data. A central position in the discussion is played by the concept of global barrier functions and by certain fundamental invariants of the equation. With the help of these invariants we are able to distinguish an important class of ‘ regularly elliptic5 equations which, as far as the Dirichlet problem is concerned, behave comparably to uniformly elliptic equations. For equations which are not regularly elliptic it is necessary to impose significant restrictions on the curvatures of the boundaries of the underlying domains in order for the Dirichlet problem to be generally solvable; the determination of the precise form of these restrictions constitutes a second primary aim of the paper. By maintaining a high level of generality throughout, we are able to treat as special examples the minimal surface equation, the equation for surfaces having prescribed mean curvature, and a number of other non-uniformly elliptic equations of classical interest.

scholarly journals TWO-GRID WEAK GALERKIN METHOD FOR SEMILINEAR ELLIPTIC DIFFERENTIAL EQUATIONS

2022 ◽  
Author(s):  
luoping chen ◽  
fanyun wu ◽  
guoyan zeng
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Mesh Size ◽  
Initial Guess ◽  
Coarse Mesh ◽  
Weak Galerkin ◽  
Fine Mesh ◽  
Semilinear Elliptic ◽  
Elliptic Differential Equations ◽  
Linearized System

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Małgorzata Klimek ◽  
Marek Błasik
Keyword(s):  
Differential Equations ◽  
Arbitrary Order ◽  
Continuous Solution ◽  
Initial Value ◽  
Uniqueness Results ◽  
Banach Theorem ◽  
Arbitrary Partition ◽  
Particular Solution ◽  
Stationary Function ◽  
Fractional Differential

AbstractTwo-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.

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