DZ_2_modul (785502)

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tIPOWOE DOMA[NEE ZADANIE MODULQ 2 “mETOD RAZDELENIQPEREMENNYH. sPECIALXNYE FUNKCII”PO KURSU “urmf I pf” DLQ rl 1, rl 2wARIANT 1rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u(2, ϕ) = 2 cos3 ϕ − sin3 ϕ + sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y − y0 u x=0 =, u0x x=l = 0, 0 ≤ y ≤ y0 ;8y01πx5πx uy y=0 = − sin , uy=y = sin, 0 ≤ x ≤ l.02l2l2lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ < 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ < 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ < 2π. u=vcosϕ,u0z=0z=lRrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u0r (2, ϕ) = 2 cos3 ϕ − 3 sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = 3 cos2 ϑ + sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:72wARIANT 2zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π;u0r (1, ϕ) = 3 cos3 ϕ + sin3 ϕ − sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ; 0 y(2y0 − y) = 0, 0 ≤ y ≤ y0 ;ux x=0 =,u3x=l4y033πx13πx0 u0 =−cos,ucos, 0 ≤ x ≤ l.y y=0y y=y0 =2l2l2l2lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. u=vsinϕ,u0zz=0z=lRzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 4u = 0, 0 ≤ r < 1,u0r (1, ϕ) = cos3 ϕ + sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π,u(2, ϕ, ϑ) = cos2 ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:73wARIANT 3rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 4, 0 < ϕ < 2π;u(4, ϕ) = cos3 ϕ + 4 sin3 ϕ + sin2 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y − y0 u x=0 =, ux=l = 0, 0 ≤ y ≤ y0 ;8y022πxπx u0y =−sin,u=sin, 0 ≤ x ≤ l.y=0y=y0lllzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R , 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r u = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vcosϕ,u0z=0z=lRrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + 9u = 0, 0 ≤ r < 2,u(2, ϕ) = 3 cos3 ϕ − sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = 4 sin ϑ + 3 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:74wARIANT 4rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 3, 0 < ϕ < 2π;u(3, ϕ) = 4 sin3 ϕ − sin2 ϕ + sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ; 0 y − y00ux x=0 =,u0 ≤ y ≤ y0 ;x x=l = 0,22y0πx u, 0 ≤ x ≤ l.=1,u=cosy=0y=y0lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r u=vsinϕ,u0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.0z z=l = 0,z=0RrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 3,u(3, ϕ) = cos2 ϕ − 3 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = 4 cos ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:75wARIANT 5rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 2, 0 < ϕ < 2π;u0r (2, ϕ) = 2 cos3 ϕ + sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y − y0 u x=0 =, u0x x=l = 0, 0 ≤ y ≤ y0 ;2y0πx5πx u=sin=−sin,u, 0 ≤ x ≤ l.y=y0y=02l2lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. uz =vcosϕ,u0 2z=0z=lRrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u0r (2, ϕ) = cos3 ϕ − 2 sin3 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 9u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = sin2 ϑ + 3 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:76wARIANT 6zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 1, 0 < ϕ < 2π;u(1, ϕ) = cos3 ϕ − cos2 ϕ + sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ; 0 y(y − 2y0 )0ux x=0 =,u0 ≤ y ≤ y0 ;x x=l = 0,364y0πx33πx0 u=−cos,ucos, 0 ≤ x ≤ l.y y=y0 =y=0lllzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r uz sinϕ,u0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vz z=l = 0,0 2z=0RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 4u = 0, 0 ≤ r < 2,u0r (2, ϕ) = sin3 ϕ + 5 cos ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 3, 0 ≤ ϕ < 2π,u(3, ϕ, ϑ) = cos ϑ + 2 sin2 ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:77wARIANT 7zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 3, 0 < ϕ < 2π;u0r (3, ϕ) = 3 sin3 ϕ − cos3 ϕ + sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ; y(y − 2y0 ) = 0, 0 ≤ y ≤ y0 ;u x=0 =,u2x=l32y02πx1πx0 u=−sin,usin , 0 ≤ x ≤ l.y y=y0 =y=0lllzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. uz cosϕ,u=v0 2z=lz=0RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 16u = 0, 0 ≤ r < 1,u(1, ϕ) = cos2 ϕ − 3 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π,u0r (2, ϕ, ϑ) = 3 cos ϑ + sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:78wARIANT 8rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 4, 0 < ϕ < 2π;u(4, ϕ) = 2 cos3 ϕ + 4 sin3 ϕ − sin2 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;5πy1πy u x=0 = − cos, u0x x=l =cos, 0 ≤ y ≤ y0 ;2y2y2y000x(x − 2l) u0y =,u= 0, 0 ≤ x ≤ l.y=y0y=064l3zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r uz sinϕ,u0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vz z=l = 0,0 2z=0RrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2,u0r (2, ϕ) = cos3 ϕ + sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 9u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = 2 cos2 ϑ + 7 cos ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:79wARIANT 9zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 1, 0 < ϕ < 2π;u0r (1, ϕ) = −4 cos3 ϕ − sin3 ϕ + 2 sin ϕ + 2 cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX∆u = 0, 0 < x < l, 0 < y < y0 ;1πy 0 ux x=0 = − cos , ux=l = 1, 0 ≤ y ≤ y0 ;y0y0x−l0 u0 , uy y=y0 = 0, 0 ≤ x ≤ l.y y=0 =16l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r u = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.sinϕ,u=v0z=0z=lRzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 3, 0 ≤ ϕ < 2π;u(3, ϕ) = cos3 ϕ + sin2 ϕ − cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = cos ϑ − 5 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:710wARIANT 10rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 2, 0 < ϕ < 2π;u(2, ϕ) = 2 cos3 ϕ − 4 sin3 ϕ + 3 cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;πyπy u x=0 = − cos, ux=l = cos, 0 ≤ y ≤ y0 ;2y02y0x−l u0y =,u= 0, 0 ≤ x ≤ l.y=0y=y02l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r ucosϕ,u0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vz z=l = 0,0z=0RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 9u = 0, 0 ≤ r < 1,u0r (1, ϕ) = sin3 ϕ + 3 cos ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 4, 0 ≤ ϕ < 2π,u0r (4, ϕ, ϑ) = cos ϑ + sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:711wARIANT 11rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 3, 0 < ϕ < 2π;u0r (3, ϕ) = sin3 ϕ − cos3 ϕ + 3 sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;1πy1πy 0 ux x=0 = − sin , u0x x=l =sin , 0 ≤ y ≤ y0 ;y0y03y0y0x(2l − x) u=,u= 0, 0 ≤ x ≤ l.y=0y=04l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. usinϕ,u=v0z=lz=0RrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + 4u = 0, 0 ≤ r < 1,u(1, ϕ) = cos2 ϕ − 2 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 9u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = cos2 ϑ + 3 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:712wARIANT 12rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 6, 0 < ϕ < 2π;u0r (6, ϕ) = 2 cos3 ϕ − 3 cos3 ϕ + cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;πy22πy u x=0 = − sin , u0x x=l = sin, 0 ≤ y ≤ y0 ;yyy000x(x − 2l) u=,u= 0, 0 ≤ x ≤ l.y=0y=y032l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r ucosϕ,u0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vz z=l = 0,0z=0RrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2,u(2, ϕ) = cos2 ϕ + 5 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = cos ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:713wARIANT 13zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 4, 0 < ϕ < 2π;u0r (4, ϕ) = 2 cos3 ϕ + 2 sin3 ϕ + 2 sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;πy33πy1 0 ux x=0 = −sin, u0x x=l =sin,2y02y02y02y0x(2l − x)0 u=,u0 ≤ x ≤ l.y y=y0 = 0,y=04l2zADA^A 3.0 ≤ y ≤ y0 ;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. uz =vsinϕ,u0z=0z=lR2zADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 16u = 0, 0 ≤ r < 2,u0r (2, ϕ) = cos3 ϕ − sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = 4 cos ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:714wARIANT 14rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 1, 0 < ϕ < 2π;u(1, ϕ) = − sin3 ϕ − 2 sin2 ϕ + 3 sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;πy5πy u x=0 = − sin, ux=l = sin, 0 ≤ y ≤ y0 ;2y2y00x−l0 u,u0 ≤ x ≤ l.=y y=y0 = 0,y=02lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r uz =v0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.cosϕ,u0 2z z=l = 0,z=0RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 9u = 0, 0 ≤ r < 1,u(1, ϕ) = cos2 ϕ − 6 sin3 ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = 2 cos ϑ + 3 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:715wARIANT 15zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 2, 0 < ϕ < 2π;u(2, ϕ) = 4 cos3 ϕ − 2 sin3 ϕ + cos2 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;y0 − y u x=0 = 0, u0x x=l =0 ≤ y ≤ y0 ;2 ,16y033πxπx u0y y=0 = − sin, uy=y0 = sin , 0 ≤ x ≤ l.2l2l2lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. uz sinϕ,u=v0 2z=lz=0RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 4u = 0, 0 ≤ r < 2,u(2, ϕ) = cos3 ϕ − sin2 ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = cos ϑ + 4 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:716wARIANT 16rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 3, 0 < ϕ < 2π;u0r (3, ϕ) = 2 cos3 ϕ − 4 sin3 ϕ + cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y(y − 2y0 ) 0 ux x=0 = 0, ux=l =, 0 ≤ y ≤ y0 ;4y021πx33πx0 u0 =−cos,u=cos, 0 ≤ x ≤ l.y y=0y y=y02l2l2l2lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r uz = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vcosϕ,u0zz=0z=lR2rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + 9u = 0, 0 ≤ r < 3,u(3, ϕ) = cos2 ϕ − 3 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = cos ϑ − 2 sin2 ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:717wARIANT 17rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 4, 0 < ϕ < 2π;u(4, ϕ) = 2 cos2 ϕ − sin3 ϕ + cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y0 − y u x=0 = 0, ux=l =, 0 ≤ y ≤ y0 ;8y044πxπx u0 =−sin,u=sin, 0 ≤ x ≤ l.y y=0y=y0lllzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; u r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π. u=υcos2ϕ,u0z=0z=lR2rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u0r (2, ϕ) = 2 cos3 ϕ − sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = 5 cos ϑ − sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:718wARIANT 18rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 5, 0 < ϕ < 2π;u0r (5, ϕ) = 2 sin3 ϕ − sin ϕ + 2 cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y0 − y 0 ux x=0 = 0, u0x x=l =0 ≤ y ≤ y0 ;2 ,2y02πx uy=0 = − cos, uy=y0 = 1, 0 ≤ x ≤ l.lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; u r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 u z=0 = v0 2 sin 2ϕ, uz z=l = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.RrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2,u(2, ϕ) = cos2 ϕ + 5 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 3, 0 ≤ ϕ < 2π,u(3, ϕ, ϑ) = 3 cos ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:719wARIANT 19zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 2, 0 < ϕ < 2π;u(2, ϕ) = 2 cos3 ϕ + 4 sin3 ϕ + 2 sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX∆u = 0, 0 < x < l, 0 < y < y0 ;y0 − y u x=0 = 0, u0x x=l =, 0 ≤ y ≤ y0 ;2y02πx3πx u=sin,u=sin, 0 ≤ x ≤ l.y=0y=y02l2lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 u z=0 = v0 2 cos 2ϕ, uz=l = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 4u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u0r (2, ϕ) = 3 cos3 ϕ + 5 cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π,u0r (2, ϕ, ϑ) = cos ϑ + sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:720wARIANT 20rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 1, 0 < ϕ < 2π;u0r (1, ϕ) = 6 sin3 ϕ − cos3 ϕ + 4 sin ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ; 0 y(2y0 − y)ux x=0 = 0, u0x x=l =, 0 ≤ y ≤ y0 ;64y031πx0 ucos , 0 ≤ x ≤ l.=1,uy y=y0 =y=0llzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 u z=0 = v0 2 sin 2ϕ, uz z=l = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.RrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2,u(2, ϕ) = cos2 ϕ + 4 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = cos ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:721wARIANT 21rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 3, 0 < ϕ < 2π;u(3, ϕ) = cos3 ϕ + sin3 ϕ + sin2 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;y(2y0 − y) u x=0 = 0, ux=l =, 0 ≤ y ≤ y0 ;32y022πx22πx0 u=−sin,u=sin, 0 ≤ x ≤ l.y y=y0y=0lllzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; u r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 uz = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vcos2ϕ,u0z=0z=lR3rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + 9u = 0, 0 ≤ r < 1,u0r (1, ϕ) = cos3 ϕ + sin3 ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 2,u(2, ϕ, ϑ) = cos ϑ − sin2 ϑ.0 ≤ ϕ < 2π,0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:722wARIANT 22zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 4, 0 < ϕ < 2π;u0r (4, ϕ) = 4 sin3 ϕ − sin ϕ − cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;1πyπy u x=0 = − cos, u0x x=l =cos, 0 ≤ y ≤ y0 ;2y02y02y0x(2l − x) u0 ==0,u, 0 ≤ x ≤ l.y y=0y=y032l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; u r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 uz =vsin2ϕ,u0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.0 3z z=l = 0,z=0RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 4u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u0r (2, ϕ) = cos3 ϕ + 12 sin3 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = cos ϑ + 4 sin2 ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:723wARIANT 23rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 1.∆u = 0, 0 ≤ r < 5, 0 < ϕ < 2π;u(5, ϕ) = − cos3 ϕ − sin3 ϕ + cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-zADA^A 2.NIKE (1 BALL):∆u = 0, 0 < x < l, 0 < y < y0 ;22πy 0 ux x=0 = − cos, ux=l = 1, 0 ≤ y ≤ y0 ;y0y0l−x0 u0y , 0 ≤ x ≤ l.=0,u=y y=y0y=016l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; ur r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 uz = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.=vcos2ϕ,u0 3z=0z=lRrE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 4.(2 BALLA):zADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 3, 0 ≤ ϕ < 2π;u0r (3, ϕ) = cos3 ϕ − sin3 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + 4u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = cos ϑ + 2 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:724wARIANT 24zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 2, 0 < ϕ < 2π;u0r (2, ϕ) = 4 cos3 ϕ + 2 sin3 ϕ + sin ϕ − cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;3πyπy u x=0 = − cos, ux=l = cos, 0 ≤ y ≤ y0 ;2y2y00l−x u0 =0,u=, 0 ≤ x ≤ l.y y=0y=y02lzADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; u r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 uz z=0 = v0 3 sin 2ϕ, uz z=l = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + u = 0, 0 ≤ r < 2, 0 ≤ ϕ < 2π;u(2, ϕ) = cos2 ϕ − 3 sin2 ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u0r (1, ϕ, ϑ) = cos ϑ + 12 sin ϑ.0 ≤ ϑ ≤ π;rl1, rl2, urmf I pf, MODULX II, TIPOWOE DOMA[NEE ZADANIEMAX:10MIN:725wARIANT 25zADA^A 1.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGE (1 BALL):zADA^A 2.NIKE (1 BALL):∆u = 0, 0 ≤ r < 1, 0 < ϕ < 2π;u0r (1, ϕ) = 2 cos3 ϕ − sin3 ϕ + 3 sin ϕ − 3 cos ϕ.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W PRQMOUGOLX-∆u = 0, 0 < x < l, 0 < y < y0 ;3πy1πy3 0 ux x=0 = − sin, u0x x=l = sin , 0 ≤ y ≤ y0 ;y0y0y0y0x(x − 2l) u=0,u=, 0 ≤ x ≤ l.y=0y=y04l2zADA^A 3.rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ lAPLASA W KRUGOWOMCILINDRE (3 BALLa):∆u = 0, 0 ≤ r < R, 0 ≤ ϕ ≤ 2π, 0 < z < l; u r=R = 0, 0 ≤ ϕ ≤ 2π, 0 ≤ z ≤ l;r2 u z=0 = υ0 2 sin 2ϕ, uz=l = 0, 0 ≤ r ≤ R, 0 ≤ ϕ ≤ 2π.RzADA^A 4.(2 BALLA):rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W KRUGEzADA^A 5.(3 BALLA):∆u + 16u = 0, 0 ≤ r < 2,u(2, ϕ) = cos2 ϕ + 5 sin ϕ.0 ≤ ϕ < 2π;rE[ITX KRAEWU@ ZADA^U DLQ URAWNENIQ gELXMGOLXCA W [ARE∆u + u = 0, 0 ≤ r < 1, 0 ≤ ϕ < 2π,u(1, ϕ, ϑ) = cos ϑ − 2 sin ϑ.0 ≤ ϑ ≤ π;.

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