A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equations

1968 ◽  
Vol 11 (1) ◽  
pp. 13-37 ◽  
Author(s):  
G. T. McAllister
Keyword(s):  
Difference Equations ◽  
A Priori ◽  
A Priori Bounds ◽  
Elliptic Difference ◽  
Difference Quotients

Some convergence results in elliptic difference equations

1971 ◽  
Vol 323 (1553) ◽  
pp. 191-199 ◽  
Keyword(s):  
Difference Equations ◽  
Convergence Theory ◽  
Discrete Problem ◽  
Positive Type ◽  
Norm Estimates ◽  
Elliptic Difference ◽  
Difference Approximations ◽  
Elliptic Differential Equations ◽  
Convergence Results ◽  
Difference Quotients

The purpose of the paper is to present a survey of some recent results in convergence theory for finite-difference approximations to Dirichlet’s problem for second-order linear elliptic differential equations. The basic approach is to first derive a priori inequalities for the discrete problem and then to use these to deduce corresponding convergence estimates. First difference approximations of positive type and related maximum-norm estimates are considered. Then some schemes of non-positive type are described and analysed by L 2 methods. Finally, interior estimates and in some cases estimates up to plane portions of the boundary are derived for difference quotients of solutions of elliptic difference equations.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

Sign in / Sign up

Export Citation Format

Share Document