316
4 Problems with Variable Boundaries
Obtained a Ph.D. at the University of Göttingen in 1910. Served as professor of
mathematics at the University of Münster in 1919, served as a professor of the
University of Göttingen in 1920. Founded the mathematical institute in Göttingen
and served as the director in 1929. Moved to the United States in 1934, served as a
tenured professor at the New York University in 1936. After the Second World War
to found and lead the Courant Institute of Mathematical Sciences and mechanical
institute of the university. Was an academician of the National Academy of Sciences
USA, an academician of the Soviet Union, Denmark, Italy, Dutch Royal and Berlin
Academy of Sciences. Made the important contribution to mathematical analysis,
function theory, mathematical physics equations, calculus of variations etc. Won the
Distinguished Public Service Award, Navy in 1958, won the outstanding contribution award in mathematics of Mathematical Association of America in 1965. The
works had Methoden der mathematischen Physik (2 volumes, 1924, 1937, 1968),
Differential and Integral Calculus (2 volumes, 1927, 1929), What is Mathematics
(1941, 1978, 1996), Supersonic Flow and Shock Waves (1948, 1976, 1999, 2006),
Dirichlets principle, conformal mappings and minimal surfaces (1950), Introduction
to Calculus and Analysis (1985, 1989, 1999) etc.
Problems 4
4.1 Find the first variation of the functional J [y] =
1
0 y
3 y
2 dx, y(0) = 1.
4.2 Let the functional J [y] =
l
0
1
2
E I y
2
− qy
dx, where, E I = c, q is the
given function of x, y(0) = y(l) = 0. Derive the Euler equation and the
natural boundary condition by the variational equation.
4.3 Let Γ be the fixed boundary for the domain D, find the natural boundary
condition of the functional
J [u] =
¨
D
(u
2
x + u
2
y + u x ϕ + u y ψ)dxdy
where, ϕ, ψ both belong to C
1
(D).
4.4 Find the natural boundary condition of the following the functional obtaining
extremum
(1) J [y] =
x 1
x 0
F(x, y, y
)dx +
1
2
ky
2
(x 1 ), given y(x 0 ) = y 0 ;
(2) J [y] =
x 1
x 0
F(x, y, y
)dx +
1
2
k[y(x 1 ) − y(x 0 )]
2 .
4.5 Find the Euler equation and natural boundary conditions of the functional
J [y] =
1
2
x 1
x 0
[ p(x)y
2
+ q(x)y
2
+ r (x)y
2
− 2s(x)y]dx
4.6 Let the functional J [y] =
1
0 F(x, y, y
)dx, determine the Euler equation and
natural boundary conditions of the following cases.
(1) F = y
2
+ yy
+ y
2 ;
4 Problems with Variable Boundaries
Obtained a Ph.D. at the University of Göttingen in 1910. Served as professor of
mathematics at the University of Münster in 1919, served as a professor of the
University of Göttingen in 1920. Founded the mathematical institute in Göttingen
and served as the director in 1929. Moved to the United States in 1934, served as a
tenured professor at the New York University in 1936. After the Second World War
to found and lead the Courant Institute of Mathematical Sciences and mechanical
institute of the university. Was an academician of the National Academy of Sciences
USA, an academician of the Soviet Union, Denmark, Italy, Dutch Royal and Berlin
Academy of Sciences. Made the important contribution to mathematical analysis,
function theory, mathematical physics equations, calculus of variations etc. Won the
Distinguished Public Service Award, Navy in 1958, won the outstanding contribution award in mathematics of Mathematical Association of America in 1965. The
works had Methoden der mathematischen Physik (2 volumes, 1924, 1937, 1968),
Differential and Integral Calculus (2 volumes, 1927, 1929), What is Mathematics
(1941, 1978, 1996), Supersonic Flow and Shock Waves (1948, 1976, 1999, 2006),
Dirichlets principle, conformal mappings and minimal surfaces (1950), Introduction
to Calculus and Analysis (1985, 1989, 1999) etc.
Problems 4
4.1 Find the first variation of the functional J [y] =
1
0 y
3 y
2 dx, y(0) = 1.
4.2 Let the functional J [y] =
l
0
1
2
E I y
2
− qy
dx, where, E I = c, q is the
given function of x, y(0) = y(l) = 0. Derive the Euler equation and the
natural boundary condition by the variational equation.
4.3 Let Γ be the fixed boundary for the domain D, find the natural boundary
condition of the functional
J [u] =
¨
D
(u
2
x + u
2
y + u x ϕ + u y ψ)dxdy
where, ϕ, ψ both belong to C
1
(D).
4.4 Find the natural boundary condition of the following the functional obtaining
extremum
(1) J [y] =
x 1
x 0
F(x, y, y
)dx +
1
2
ky
2
(x 1 ), given y(x 0 ) = y 0 ;
(2) J [y] =
x 1
x 0
F(x, y, y
)dx +
1
2
k[y(x 1 ) − y(x 0 )]
2 .
4.5 Find the Euler equation and natural boundary conditions of the functional
J [y] =
1
2
x 1
x 0
[ p(x)y
2
+ q(x)y
2
+ r (x)y
2
− 2s(x)y]dx
4.6 Let the functional J [y] =
1
0 F(x, y, y
)dx, determine the Euler equation and
natural boundary conditions of the following cases.
(1) F = y
2
+ yy
+ y
2 ;