8.8 Introduction to the Famous Scientists
505
the condition of large deformation, discussed the buckling deformation problem of
plate and shell, published works on the calculus of variations and method of solution
of variational equation.
Kantorovich (Leonid Vitaliyevich, Russian: Leonid Bitalleviq Kantopoviq,
1912.1.19–1986.4.6) Soviet mathematician and economist. Born in Saint Petersburg, died in Moscow. Graduated from the University of Leningrad and took the
teaching post after graduation in 1930. Became a professor in 1934. Received
a doctor’s degree in science of mathematical physics in 1935. Was elected a
communication academician of the Soviet Academy of Sciences in 1958, become
academician in 1964, was also honorary member of the Hungarian Academy
of Sciences and American Academy of Arts and Sciences. Was the winner of
the Stalin prize in 1949, won the Lenin prize in 1959 and the Nobel Prize in
Economics in 1975. One of the founders of the optimization mathematical method
and computational mathematics, the founder of economic mathematics. In functional analysis, program design, function theory, mathematical physics, differential equations, integral equations and calculus of variations etc. many aspects made
some contributions. The works had Matematiqeckie metody opganizacii i
planipovanii ppoizvodctva (Mathematical Methods of Organizing and Planning Production, 1939), Approximate methods of higher analysis (1939), Teopii
vepotnocte (1946), Fynkcionallnyy analiz i ppikladna matematika
(1948), konomiqeckii pacqet nailyqxego icpollzovanii pecypcov (1959)
and Essays in Optimal Planning (1977) etc.
Problems 8
8.1 Using the Euler finite difference method to find the approximate solution of the
minimum problem of the functional J [y] =
1
0 (y
2
+ 2y)dx, the boundary
conditions are y(0) = y(1) = 0.
8.2 Find the approximate solution of the equation u = −1 in the rectangular
domain −a ≤ x ≤ a, −b ≤ y ≤ b, and u on the boundary of the rectangular
domain is equal to zero.
Hint: This problem can be transformed into the discussion of the extremum
for the functional
J [u(x, y)] =
¨
D
(u
2
x + u
2
y − 2u)dxdy
Its approximate solution is searched in the form of u 1 = c 1 (x
2
− a
2
)(y
2
− b
2
).
8.3 Find the approximate solution of extremal problem of the functional J [y] =
1
0 (x
3 y
2
+ 100x y
2
− 20x y)dx, the boundary conditions are y(1) = 0,
y
(1) = 0.
505
the condition of large deformation, discussed the buckling deformation problem of
plate and shell, published works on the calculus of variations and method of solution
of variational equation.
Kantorovich (Leonid Vitaliyevich, Russian: Leonid Bitalleviq Kantopoviq,
1912.1.19–1986.4.6) Soviet mathematician and economist. Born in Saint Petersburg, died in Moscow. Graduated from the University of Leningrad and took the
teaching post after graduation in 1930. Became a professor in 1934. Received
a doctor’s degree in science of mathematical physics in 1935. Was elected a
communication academician of the Soviet Academy of Sciences in 1958, become
academician in 1964, was also honorary member of the Hungarian Academy
of Sciences and American Academy of Arts and Sciences. Was the winner of
the Stalin prize in 1949, won the Lenin prize in 1959 and the Nobel Prize in
Economics in 1975. One of the founders of the optimization mathematical method
and computational mathematics, the founder of economic mathematics. In functional analysis, program design, function theory, mathematical physics, differential equations, integral equations and calculus of variations etc. many aspects made
some contributions. The works had Matematiqeckie metody opganizacii i
planipovanii ppoizvodctva (Mathematical Methods of Organizing and Planning Production, 1939), Approximate methods of higher analysis (1939), Teopii
vepotnocte (1946), Fynkcionallnyy analiz i ppikladna matematika
(1948), konomiqeckii pacqet nailyqxego icpollzovanii pecypcov (1959)
and Essays in Optimal Planning (1977) etc.
Problems 8
8.1 Using the Euler finite difference method to find the approximate solution of the
minimum problem of the functional J [y] =
1
0 (y
2
+ 2y)dx, the boundary
conditions are y(0) = y(1) = 0.
8.2 Find the approximate solution of the equation u = −1 in the rectangular
domain −a ≤ x ≤ a, −b ≤ y ≤ b, and u on the boundary of the rectangular
domain is equal to zero.
Hint: This problem can be transformed into the discussion of the extremum
for the functional
J [u(x, y)] =
¨
D
(u
2
x + u
2
y − 2u)dxdy
Its approximate solution is searched in the form of u 1 = c 1 (x
2
− a
2
)(y
2
− b
2
).
8.3 Find the approximate solution of extremal problem of the functional J [y] =
1
0 (x
3 y
2
+ 100x y
2
− 20x y)dx, the boundary conditions are y(1) = 0,
y
(1) = 0.