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By Napalkov V. V.

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Partial differential equations with Fourier series and BVP

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0, 1, 0, . . , 0)t ∈ Rn ✱ i = 1, . . n✳ ❉✐❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ❡rst❡r ❖r❞♥✉♥❣ δf [x]i,− ; x0 − si ei ∈ IRn×n ✈♦♥ f ❛✉❢ [x]i,− ❙❛t③ ✶✳✸✳✸✹ Rn ✱ ❜❡③ü❣❧✐❝❤ x0 − si ei ✉♥❞ δf [x]i,+ ; x0 + ti ei ∈ IRn×n ✈♦♥ f ❛✉❢ [x]i,+ ❜❡③ü❣❧✐❝❤ x0 + ti ei s❡✐❡♥ ❣❡❣❡❜❡♥✱ ✉♥❞ A ∈ Rn×n s❡✐ ❡✐♥❡ ♥✐❝❤ts✐♥❣✉❧är❡ ▼❛tr✐①✳ ❲✐r s❡t③❡♥ [l]i,+ := A f x0 + ti ei i A δf [x]i,+ ; x0 + ti ei + j=i ij · [−sj , tj ] ✉♥❞ [l]i,− := A f x0 − si ei i A δf [x]i,− ; x0 − si ei + j=i ij · [−sj , tj ] . ●✐❧t ❢ür ❥❡❞❡s i ∈ {1, . .

Xj−1 , xj , 0, . . , 0) − fi (x1 , . . , xj−1 , 0, . . , 0) . xj − 0 ❲✐r ❡r❤❛❧t❡♥ f1 (x1 , 0) − f1 (0, 0) x1 − 0 f1 (x1 , x2 ) − f1 (x1 , 0) x2 − 0 = 0, x21 + 1 x2 , = f2 (x1 , 0) − f2 (0, 0) x1 − 0 = 1 s♦✇✐❡ f2 (x1 , x2 ) − f2 (x1 , 0) x2 − 0 = 0. ❉❛♠✐t ❡r❣✐❜t s✐❝❤ ❛❧s ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ♥❛❝❤ ❍❛♥s❡♥s ▼❡t❤♦❞❡ δf (x; x0 ) = x21 + 1 x2 0 0 1 ✉♥❞ ❞❛♠✐t ❛❧s ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] = ([−1, 1] , [−1, 1])T ❜❡③ü❣❧✐❝❤ x0 δf ([x] ; x0 ) = 0 1 [−2, 2] 0 . ❆✉s ❇❡✐s♣✐❡❧ ✶✳✸✳✷✼ ❡r❦❡♥♥❡♥ ✇✐r✿ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥ ✈♦♥ ❋✉♥❦t✐♦♥❡♥ ♠❡❤r❡r❡r ❱❡rä♥❞❡r❧✐✲ ❝❤❡r s✐♥❞ ♥✐❝❤t ❡✐♥❞❡✉t✐❣✳ ❏❡ ♥❛❝❤ ▼❡t❤♦❞❡ ❞❡r ❇❡r❡❝❤♥✉♥❣ ❡r❤ä❧t ♠❛♥ ✐♠ ❆❧❧❣❡♠❡✐✲ ♥❡♥ ✉♥t❡rs❝❤✐❡❞❧✐❝❤❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥❡♥ ✉♥❞ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣❡♥✳ ❆✉ÿ❡r❞❡♠ ❣✐❧t ❡✐♥❡ ❙②♠♠❡tr✐❡❡✐❣❡♥s❝❤❛❢t ✇✐❡ ✐♥ ❙❛t③ ✶✳✸✳✺ ♥✐❝❤t ❢ür ❡✐♥❡ ❙t❡✐❣✉♥❣s✲ ❢✉♥❦t✐♦♥ ❡✐♥❡r ❋✉♥❦t✐♦♥ f : Rn → Rm ♠❡❤r❡r❡r ❱❡rä♥❞❡r❧✐❝❤❡r✳ ■st δf : Rn → Rm×n ✱ x → δf (x; x0 )✱ ❢ür ❥❡❞❡s x0 ∈ [x] ❡✐♥❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ✉♥❞ δf : Rn → Rm×n ♠✐t x → δf (x; x0 ) := δf (x0 ; x) ❞✐❡ ❋✉♥❦t✐♦♥✱ ❞✐❡ ❞✉r❝❤ ❱❡rt❛✉s❝❤❡♥ ✈♦♥ x ✉♥❞ x0 ❡♥tst❡❤t✱ ❞❛♥♥ ❣✐❧t ✐♠ ❆❧❧❣❡♠❡✐♥❡♥ δf (x; x0 ) = δf (x; x0 )✳ ❉✐❡s ③❡✐❣t ❞❛s ❢♦❧❣❡♥❞❡ ❇❡✐s♣✐❡❧✳ ✶✳✸✳ ❙❚❊■●❯◆●❊◆ ❇❡✐s♣✐❡❧ ✶✳✸✳✷✾ ✷✼ ❊s s❡✐ f : [x] ⊆ R2 → R2 , f = (f1 , f2 ) , ✇✐❡ ✐♥ ❇❡✐s♣✐❡❧ ✶✳✸✳✷✼ ✉♥❞ x0 = (x0 )1 , (x0 )2 ✳ ❋ür ❞✐❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 ❡r❤❛❧t❡♥ ✇✐r ♥❛❝❤ ❍❛♥s❡♥s ▼❡t❤♦❞❡ f1 x1 , x2 − f1 (x0 )1 , x2 x1 − (x0 )1 f1 (x0 )1 , x2 − f1 (x0 )1 , (x0 )2 x2 − (x0 )2 x1 + (x0 )1 · x22 , = = (x0 )21 + 1 · x2 + (x0 )2 , f2 x1 , x2 − f2 (x0 )1 , x2 x1 − (x0 )1 = 1, f2 (x0 )1 , x2 − f2 (x0 )1 , (x0 )2 x2 − (x0 )2 = 0.

X0 )n−1 , xn − fi (x0 )1 , . . , (x0 )n fi (x0 )1 , . . , (x0 )j−1 , xj , . . , xn − fi (x0 )1 , . . , (x0 )j , xj+1 , . . , xn       xj − (x0 )j δf (x; x0 )ij := ❢ür xj = (x0 )j        cij ❢ür xj = (x0 )j ❢ür ❜❡❧✐❡❜✐❣❡s cij ∈ R ❡✐♥❡ ❙t❡✐❣✉♥❣s❢✉♥❦t✐♦♥ ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❜❡③ü❣❧✐❝❤ x0 ✳ ❋❛❧❧s f (x0 ) ❡①✐st✐❡rt✱ s♦ s❡t③❡♥ ✇✐r cij := lim xj →(x0 )j δf (x; x0 )ij = ∂fi (x0 )1 , . . , (x0 )j , xj+1 , . . , xn . ∂xj ■♥❞❡♠ ✇✐r ❢ür δf ([x] ; x0 )ij ❥❡✇❡✐❧s ❡✐♥ ■♥t❡r✈❛❧❧ ✇ä❤❧❡♥✱ ❞❛s ❞✐❡ ▼❡♥❣❡ δf (x; x0 )ij | x ∈ [x] ❡✐♥s❝❤❧✐❡ÿt✱ ❡r❤❛❧t❡♥ ✇✐r ❡✐♥❡ ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ δf ([x] ; x0 ) ❡rst❡r ❖r❞♥✉♥❣ ✈♦♥ f ❛✉❢ [x] ❜❡③ü❣❧✐❝❤ x0 ✳ ✷✹ ❑❆P■❚❊▲ ✶✳ ●❘❯◆❉▲❆●❊◆ ❊✐❣❡♥s❝❤❛❢t❡♥ ❋ür ❞✐❡ ❇❡r❡❝❤♥✉♥❣ ❡✐♥❡r ■♥t❡r✈❛❧❧st❡✐❣✉♥❣ δf ([x] ; x0 ) ❡rst❡r ❖r❞♥✉♥❣ ❤❛❜❡♥ ✇✐r ③✉✈♦r ✈❡rs❝❤✐❡❞❡♥❡ ▼ö❣❧✐❝❤❦❡✐t❡♥ ❦❡♥♥❡♥❣❡❧❡r♥t✱ ♥ä♠❧✐❝❤ ❞✐❡ ❆✉s✇❡rt✉♥❣ ❞❡r ❏❛❝♦❜✐✲ ▼❛tr✐①✱ ❞✐❡ ❊✐♥s❝❤❧✐❡ÿ✉♥❣ ✈♦♥ 1 f (x0 + t (x − x0 )) dt 0 ✉♥❞ ❞✐❡ ▼❡t❤♦❞❡ ♥❛❝❤ ❍❛♥s❡♥✳ ❆♥❤❛♥❞ ❞❡s ❢♦❧❣❡♥❞❡♥ ❇❡✐s♣✐❡❧s ✇♦❧❧❡♥ ✇✐r ❞✐❡ ❧❡t③✲ t❡r❡♥ ❜❡✐❞❡♥ ▼❡t❤♦❞❡♥ ✈❡r❣❧❡✐❝❤❡♥✳ ❇❡✐s♣✐❡❧ ✶✳✸✳✷✼ ❲✐r ❜❡tr❛❝❤t❡♥ ❞✐❡ ❋✉♥❦t✐♦♥ f : [x] ⊆ R2 → R2 , f = (f1 , f2 ) , ♠✐t f1 (x1 , x2 ) = x21 + 1 x22 , f2 (x1 , x2 ) = x1 .

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