ABSTRACT
A very large portion of mathematical physics is devoted to the study of the class of partial differential equations () a ( x,y,u, ∂ u ∂ x , ∂ u ∂ y ) ∂ 2 u ∂ x 2 + 2 b ( x,y,u, ∂ u ∂ x , ∂ u ∂ y ) ∂ 2 u ∂ x ∂ y + c ( x,y,u, ∂ u ∂ x , ∂ u ∂ y ) ∂ 2 u ∂ y 2 + f ( x,y,u, ∂ u ∂ x , ∂ u ∂ y ) = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1236.tif"/> where a, b, c and f can, as indicated, be functions of x, y, u, ∂ u / ∂ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1237.tif"/> and ∂ u / ∂ y . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1238.tif"/> There is no term ∂ 2 u / ∂ y ∂ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1239.tif"/> in (6.1) because it is assumed, generally, that ∂ 2 u / ∂ x ∂ y = ∂ 2 u / ∂ y ∂ x . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1240.tif"/> In the notation u x = ∂ u / ∂ x, u y = ∂ u / ∂ y, u xx = ∂ 2 u / ∂ x 2 , u yy = ∂ 2 u / ∂ y 2 , u xy = ∂ 2 u / ∂ x ∂ y, … , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1241.tif"/> (6.1) can, and will, be written more simply as () a ( x , y , u , u x , u y ) u xx + 2 b ( x , y , u , u x , u y ) u xy + c ( x , y , u , u x , u y ) u yy + f ( x , y , u , u x , u y ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1242.tif"/> Finally, we assume that at any point of definition () a 2 + b 2 + c 2 ≠ 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493393/adca3d7b-dc1b-488c-907a-7abe95228779/content/eq1243.tif"/> so that at least one of the second-order derivatives in (6.2) is present. Equation (6.2) is called the general, second-order, quasilinear partial differential equation in two independent variables x and y.
