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A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Neumann’s condition

100%
Sapa L.
Commentationes Mathematicae
|
2009
|
tom 49
|
nr 1
EN
Classical solutions of nonlinear second-order partial differential functional equations of parabolic type with Neumann’s condition are approximated in the paper by solutions of associated explicit difference functional equations. The functional dependence is of the Volterra type. Nonlinear estimates of the generalized Perron type for given functions are assumed. The convergence and stability results are proved with the use of the comparison technique. These theorems in particular cover quasi-linear equations, but such equations are also treated separately. The known results on similar difference methods can be obtained as particular cases of our simple result.
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A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Dirichlet's condition

100%
Sapa L.
Annales Polonici Mathematici
|
2008
|
tom 93
|
nr 2
113-133
EN
We deal with a finite difference method for a wide class of nonlinear, in particular strongly nonlinear or quasi-linear, second-order partial differential functional equations of parabolic type with Dirichlet's condition. The functional dependence is of the Volterra type and the right-hand sides of the equations satisfy nonlinear estimates of the generalized Perron type with respect to the functional variable. Under the assumptions adopted, quasi-linear equations are a special case of nonlinear equations. Quasi-linear equations are also treated separately. It is proved that our numerical methods are consistent, convergent and stable. Error estimates are given. The proofs are based on the comparison technique. Examples of physical applications and numerical experiments are presented.
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