Stability of the second order partial differential equations

Abstract

Abstract We say that a functional equation (ξ) is stable if any function g satisfying the functional equation (ξ) approximately is near to a true solution of (ξ). In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms: y x ( x , t ) = f ( x , t , y ( x , t ) ) , a y x ( x , t ) + b y t ( x , t ) = f ( x , t , y ( x , t ) ) , p ( x , t ) y x t ( x , t ) + q ( x , t ) y t ( x , t ) + p t ( x , t ) y x ( x , t ) - p x ( x , t ) y t ( x , t ) = f ( x , t , y ( x , t ) ) , p ( x , t ) y x x ( x , t ) + q ( x , t ) y x ( x , t ) = f ( x , t , y ( x , t ) ) . 2000 Mathematics Subject Classification. 26D10; 34K20; 39B52; 39B82; 46B99.

Keywords

• generalized Hyers-Ulam stability
• linear differential equation
• Banach's contraction principle