The Inverse Problem of Determination of the Heat Source in the 1D Heat Equation

The Inverse Problem of Determination of the Heat Source in the 1D Heat Equation

Jiichiro U^RABE* Yorimasa O^SHIME** Hideki T^AKUWA*** and Shota S^AKURAI****

(Received January 20, 2011)

First of all, we think the uniform stick that the heat source exists in inside. The heat equation with the heat source is considered in this paper. As assumption, the special conditions are imposed to the heat source. We consider the inverse problem that decides the heat source from the observational data of temperature of the uniform stick at one point. Then, we find the unknown function of the heat source from the observational data of the stick at one point excluded a certain points of limited piece. In the theory of Yamamoto-kim (2008), though the uniqueness of determination of the heat source has been proven, the reconstruction of the heat source is not considered. The reconstruction method of the heat source is considered in this paper. Result, we find the heat source by imposing the special conditions to the heat source.

Key wordsɿ inverse problem, heat source Ωʔϫʔυ ɿٯ໰୊ɼ೤ݯ

Ұ࣍ݩ೤ํఔࣜʹ͓͚Δ೤ݯܾఆٯ໰୊

Ӝ ෦ ࣏ Ұ ࿠ɾ ԡ ໨ པ ণɾ ଟ ٱ ࿨ ӳ थɾ ᓎ Ҫ ᠳ ଠ

1. ॹݴ

͋Δ໰୊Λղ͘ͱ͖ʹ͸ɼॳظ஋ͳͲͷʮݪҼʯɼํఔ

ࣜͳͲͷʮϞσϧʯ͔Βɼʮ݁ՌʯͰ͋ΔղΛٻΊΔॱ໰

୊͕ҰൠతͰ͋ΔɽͦΕʹର͠ɼݱࡏͷ޻ֶɼཧֶɼҩ

ֶͳͲͷ༷ʑͳ෼໺ͰʮԠ౴ʯ΍ʮ݁ՌʯͳͲͷग़ྗ͔

ΒɼʮݪҼʯ΍ʮೖྗʯΛਪఆ͢Δٯ໰୊ͱ͍͏෼໺͕޿

͘औΓѻΘΕΔΑ͏ʹͳ͖͍ͬͯͯΔɽ

ٯ໰୊ͷ۩ମతͳԠ༻ྫͱͯ͠͸ɼͨͱ͑͹ɼҩֶ਍

அͷͨΊͷ̢̧̞ɼൃ਒ػߏɼ͢ͳΘͪ਒ݯʹ͓͚Δμ

* Department of Factory of Culture and Information Science, Doshisha University.

Telephone:+81-774-65-7610, E-mail:[email protected]

** Department of Science and Engineering, Doshisha University.

Telephone:+81-774-65-6494, E-mail:[email protected]

*** Department of Science and Engineering, Doshisha University.

Telephone:+81-774-65-6431, E-mail:[email protected]

ΠφϛΫεͷܾఆɼُ྾ͷ୳஌ɼը૾෮ݩͳͲ༷ʑ΋ͷ

͕͋͛ΒΕΔ¹⁾ɽ͔͠͠ɼٯ໰୊͸Ұൠతʹॱ໰୊ͱ͸

ҧ͍ɼղ͘͜ͱ͸ࠔ೉Ͱ͋ΔͱݴΘΕ͍ͯΔɽ

ͦ͜ͰຊݚڀͰ͸ɼ

• Ұ༷ͳ๮ͷԹ౓Λଌఆ͠ɼݟΔ͜ͱͷͰ͖ͳ͍ঢ়ଶ ͷෆ໌ͳ೤ݯΛਪఆ͢Δ

͜ΕΛѻ͏͜ͱΛ໨తͱ͍ͯ͠Δɽ

ͦ͜Ͱɼ೤ݯͷ৚݅Λ

Q(x, t) =μ(t)f(x), 0< x < π, t >0 ͱ͓͘ɽͨͩ͠ɼ

μ(t) =e^−σt

Ͱ͋Δͱͯ͠ɼ৽ͨʹ೤ݯͷ৚݅Λઃఆͨ͠ɽ๮ͷ۠ؒ

಺ͷҰ఺Ͱͷ؍ଌσʔλ͔Β೤ݯΛܾఆ͢Δٯ໰୊Λߟ

࡯ͨ݁͠Ռɼ೤ݯΛܾఆ͢Δ͜ͱ͕Ͱ͖ͨɽ

2. ݁Ռ

಺෦ʹ೤ݯ͕ଘࡏ͢Δ௕͕͞πͷҰ༷ͳ๮Λߟ͑Δɽ

๮ͷ಺෦ͷ৔ॴΛxɼ࣌ࠁΛtɼ৔ॴx࣌ࠁtʹ͓͚Δ

๮ͷԹ౓Λu(x, t)ͱ͢Δɽ೤ݯͷ͋Δ೤఻ಋํఔࣜΛߟ

͑ɼͦͷͱ͖ͷॳظ৚݅ɼڥք৚݅͸ͱ΋ʹ0ͱ͢Δɽ nx=kπ(n= 1, ..., N, k= 1, ..., N)ͱͳΔ༗ݶݸͷ఺x Λআ͍ͨ๮ͷ۠ؒ಺ͷҰ఺Λx0ͱ͠ɼͦͷ఺Ͱͷ؍ଌ σʔλu(x0, t)͔Βɼະ஌ͷ೤ݯΛܾఆ͢Δٯ໰୊Λߟ

͑Δɽͦͷ݁Ռɼ೤ݯͷܗʹ৚݅Λ༩͑ͯ΍Δ͜ͱͰɼ

೤ݯͷະ஌Ͱ͋ͬͨ෦෼ΛٻΊΔ͜ͱ͕Ͱ͖ͨɽ

3. ೤ݯܾఆٯ໰୊ͷ਺ֶతهड़

ຊདྷॱ໰୊ͱͯ͠ͷܗ͸ɼաڈͷԹ౓ͱɼؔ਺ͷܗ͕

Θ͔͍ͬͯΔ೤ݯ͔ΒɼະདྷͷԹ౓ΛٻΊΔͱ͍͏໰୊

Ͱ͋ΔɽͦΕʹର͠ࠓճѻ͏໰୊͸ɼ๮಺෦ͷҰ఺Ͱͷ Թ౓ͷ؍ଌσʔλu(x0, t)͔Βະ஌ͷ೤ݯQ(x, t)Λਪఆ

͠Α͏ͱ͢Δ໰୊Ͱ͋Δɽ

σΟϦΫϨڥք৚݅ͷԼͰɼ೤ݯͷ͋Δ೤ํఔࣜɼ ut(x, t) =uxx(x, t) +Q(x, t), 0< x < π, t >0 (1) Λߟ͑ɼͦͷͱ͖ॳظ৚݅͸ɼ

u(x,0) = 0, 0< x < π, (2) ڥք৚݅͸ɼ

u(0, t) =u(π, t) = 0, t >0 (3)

ͱ͢Δɽ

͜ͷͱ͖೤ݯͷ৚݅Λ

Q(x, t) =μ(t)f(x), 0< x < π, t >0 (4) ͱ͍͏࣌ؒͷؔ਺ͱۭؒͷؔ਺ͷੵͷܗͰ৽ͨʹߟ͑Δ

͜ͱʹ͢Δɽf(x)͕ܾఆ͍ͨؔ͠਺ͱ͢ΔɽͦͷͨΊ μ(t)͸͋Β͔͡ΊΘ͔͍ͬͯΔͱ͍͏ઃఆͰ͋Δɽ͜ͷ ͱ͖۠ؒ಺ͷҰ఺x0 ∈(0, π) Ͱͷ؍ଌσʔλ͔Βɼ೤

ݯʹදΕΔؔ਺f(x)Λܾఆ͢Δٯ໰୊Λߟ࡯͢Δɽ͜

͜Ͱͷ؍ଌσʔλ͸Ұ఺ʹݶఆ͞Ε͓ͯΓɼ৘ใྔ͸গ ͳ͍͕Թ౓ܭͳͲΛઃஔ͢Δ͜ͱͰ࣮ݱͰ͖Δɽ

ٯ໰୊Ͱ͸ɼॱ໰୊ͷखॱͷٯΛ୧Δ͜ͱͰະ஌ͷؔ

਺f(x)ΛٻΊΔ͜ͱ͕Ͱ͖ͦ͏Ͱ͋ΔɽͦΕΛ͔֬Ί ΔͨΊʹ۩ମతͳܗͱͯ͠ɼ

μ(t) =e^−σt

ͱͯ͠ɼٻΊ͍ͨٯ໰୊ͷղͷܗΛɼ

f(x) = N n=1

bnsinnx (5)

ͱ͢Δɽͨͩ͠bn(n= 1, ..., N)͸ະ஌Ͱ͋Δɽ͜ͷ਺

ྻ{b_n}^N_n=1ΛٻΊ͍ͨɽ

4. ओఆཧ

಺෦ʹ೤ݯͷଘࡏ͢ΔҰ༷ͳ๮ͷ೤఻ಋํఔࣜΛߟ͑

Δɽu(x, t)Λ৔ॴxɼ࣌ࠁtʹ͓͚Δ๮ͷԹ౓ͱ͢Δͱɼ

ut(x, t) =uxx(x, t) +Q(x, t), 0< x < π, t >0 (6) ͱͳΓɼͦͷͱ͖ͷॳظ৚݅͸ɼ

u(x,0) = 0, 0< x < π (7) Ͱ͋Γɼڥք৚݅͸ɼ

u(0, t) =u(π, t) = 0, t >0 (8) ͱ͢Δɽͦͷͱ͖ɼԾఆͱͯ͠೤ݯͷ৚݅Λ

Q(x, t) =μ(t)f(x), 0< x < π, t >0 (9)

ͱ͓͘ɽͨͩ͠μ(t)͸ɼ

μ(t) =e^−σt (10)

ͱΘ͔͍ͬͯΔ΋ͷͱ͢Δɽsinnx = 0(n = 1, ..., N)ɼ

͢ͳΘͪnx=kπ(n= 1, ..., N, k= 1, ..., N)ͱͳΔ༗ݶ ݸͷxΛআ͍ͨ۠ؒ಺ͷҰ఺x0Ͱͷ؍ଌσʔλu(x0, t)

͔Βɼ೤ݯͷٻΊ͍ͨ෦෼Ͱ͋Δf(x)Λܾఆ͢Δٯ໰

୊Λߟ͑Δɽ ٻΊ͍ͨؔ਺Λ

f(x) = N n=1

bnsinnx (11)

ͱઃఆ͢Δɽͨͩ͠bn(n= 1, ..., N)͸ະ஌Ͱ͋Δɽ

͜ͷͱ͖ࠓճͷٯ໰୊͸࣍ͷ༷ͳఆཧͱͳΔɽ

ఆཧ[೤ݯܾఆٯ໰୊]

u(x, t)͕ࣜ(6)ʖ (8)ɼQ(x, t)͕ࣜ(9)Ͱ༩͑ΒΕɼμ(t) Λࣜ(10)ͷ΋ͷͱ͢Δɽ·ͨf(x)Λࣜ(11)ͱ͢Δɽ

͜ ͷ ͱ ͖ ɼ͋ Δ ༗ ݶ ݸ ͷ ఺ Λ আ ͍ ͨ x0 ʹ ର ͠ ͯ u(x0, t)(t > 0)Λ༩͑Ε͹ɼҰҙతʹ਺ྻ{bn}^N_n=1 ͕ ߏ੒Ͱ͖Δɽ

ͭ·Γɼະ஌ͷ೤ݯQ(x, t) =μ(t)f(x)͕ߏ੒Ͱ͖Δɽ

5. ຊݚڀͱઌߦ݁Ռͷൺֱɾٴͼվળ఺

5.1 ઌߦ݁Ռ

ઌߦ݁Ռͱͯ͠ɼࢁຊɾۚ[2]ɼp.60ͷఆཧΛҎԼͰ঺

հ͢Δɽ

೤ݯͷ͋Δ೤ํఔࣜɼ

u_t(x, t) =u_xx(x, t) +Q(x, t), 0< x < π, t >0 (12) Λߟ͑ɼͦͷͱ͖ॳظ৚݅͸ɼ

u(x,0) = 0, 0< x < π (13) Ͱ͋Γɼڥք৚݅͸ɼ

u(0, t) =u(π, t) = 0, t >0 (14)

ͱ͢Δɽ

೤ݯͱͯ͠ɼ

Q(x, t) =μ(t)f(x), 0< x < π, t >0 (15) ͷܗΛߟ͑Δɽ͜ͷͱ͖ɼf(x)Λ۠ؒ಺ͷҰ఺x0∈(0, π) Ͱͷ؍ଌσʔλ͔Βܾఆ͢Δٯ໰୊Λߟ࡯͍ͯ͠Δɽ

͜͜Ͱɼ

μ∈C¹[0,∞), (16)

f ∈C⁵[0, π], (17)

d^jf

dx^j(0) = d^jf

dx^j(π) = 0, j= 0,1, ...,4 (18) ͱ͍ͯ͠Δɽ

μ(t)͸Θ͔͍ͬͯΔͱͯ͠ɼf(x)ΛٻΊΔ໰୊Λߟ

͑Δɽ

͜ͷͱ͖ɼ͜ͷٯ໰୊Ͱ͸࣍ͷ͜ͱ͕Θ͔Δɽ

ఆཧ(ࢁຊɾۚ[2]ɼp.60)

u(x, t)͕ࣜ(12)-(14)ɼQ(x, t)͕ࣜ(15)Ͱ༩͑ΒΕͯ

͍Δɽ·ͨμ(t)͸ࣜ(16)ɼf(x)͸ࣜ(17)(18)ͷ৚͕݅

༩͑ΒΕ͍ͯΔɽͦͯ͠μ(0)= 0ΛԾఆ͢Δɽ

͜ͷͱ͖؍ଌ఺x0͸ɼx0

π ͕ແཧ਺Ͱ͋ΔͱԾఆ͢Δɽ

͜ͷx0ʹରԠ͢Δu(x0, t)Λ༩͑Ε͹ɼf(x)ΛҰҙత ʹٻΊΔ͜ͱ͕Ͱ͖Δɽ

5.2 ઌߦ݁Ռͱຊݚڀͱͷൺֱɾվળ఺

ઌߦ݁Ռͱຊݚڀͷ໨తΛͦΕͧΕ͋͛ͯΈΔɽ

(ઌߦ݁Ռ)

• ೤ݯܾఆٯ໰୊ͷ਺ֶతهड़

• ೤ݯܾఆͷҰҙੑͷূ໌

• f(x)͸ࣜ(17)(18)ͷԾఆͷԼ

(ຊݚڀ)

• ೤ݯܾఆٯ໰୊ͷ਺ֶతهड़

• ೤ݯܾఆͷ۩ମతͳ࠶ߏ੒

• f(x)͸ࣜ(17)(18)ͷԾఆʹ౰ͯ͸·Βͳ͍

ઌߦ݁Ռʹ͓͍ͯ͸ɼ೤ݯܾఆٯ໰୊ͷ਺ֶతهड़͸

ͳ͞Ε͍ͯΔ͕ɼͦ͜Ͱ͸Ұҙੑͷূ໌Λ͢Δ͜ͱ͕໨

తͰ͋Δɽैͬͯ۩ମతͳ೤ݯͷ࠶ߏ੒ͷखॱʹ͍ͭͯ

͸ड़΂ΒΕ͍ͯͳ͍ɽͦΕʹର͠ຊݚڀͰ͸ɼ۩ମతͳ

೤ݯͷ࠶ߏ੒΋ߦͬͨɽ

ͦͯ͠ઌߦ݁ՌͰ͸ɼf(x)͸ࣜ(17)(18)ͷԾఆʹ౰

ͯ͸·Βͳ͚Ε͹ͳΒͳ͔͕ͬͨɼຊݚڀͰ͸f(x)͸

ࣜ(17)(18)ͷԾఆʹ౰ͯ͸·Βͳ͍΋ͷ΋ߟ͑Δ͜ͱ͕

Ͱ͖Δɽ ɹ

6. ೤ݯܾఆٯ໰୊ͷղ๏

6.1 ॱ໰୊Ͱͷղͱͦͷߏ੒खଓ͖

͜ͷ৔߹ͷॱ໰୊͸ɼ೤ݯͷؔ਺ͷܗ͕Θ͔͍ͬͯͯɼ ղͰ͋ΔະདྷͷԹ౓෼෍ΛٻΊΔͱ͍͏໰୊Ͱ͋Δɽ

ॱ໰୊ͷղެࣜ͸ɼ u(x, t) =

∞ n=1

2 π{

_t

0 e^−n2(t−s){ _π

0 μ(s)f(y) sinnydy}ds}sinnx ͱͳΔɽ

an= 2 π

_π

0 f(y) sinnydy (19) ͱ͓͘ͱɼx=x0Λ୅ೖͯ͠u(x0, t)ͱ͢Δͱ

u(x0, t) = _t

0 { ∞ n=1

e^−n2(t−s)ansinnx0}μ(s)ds ͱͳΔɽ͜͜Ͱ

k(t) = ∞ n=1

e^−n2tansinnx0 (20)

ͱ͓͘ͱɼ

u(x0, t) = _t

0 k(t−s)μ(s)ds ͱͳΓɼม਺ม׵ʹΑΓɼ

u(x₀, t) = _t

0 μ(t−s)k(s)ds (21)

ΛಘΔɽ

ݫີʹ͸͜ͷ৔߹ͷॱ໰୊͸u(x, t) ΛٻΊΔ໰୊Ͱ

͋Δɽ͜͜Ͱ͸u(x, t)ʹx=x₀Λ୅ೖͯ͠u(x₀, t)ͱ

͢Δखॱ΋ߟ͍͑ͯΔɽ੔ཧ͢Δͱɼࣜ(19)ʖ (21)ͷख ॱΛॱʹ౿Ή͜ͱͰu(x0, t)ΛٻΊΔ͜ͱ͕Ͱ͖Δɽ

۩ମతʹॱ໰୊Λղ͍͍ͯ͘ɽࣜ(19)ΑΓf →an͸ɼ an=bn (n= 1, ..., N)

ͱͳΔɽ࣍ʹࣜ(20)ΑΓɼan →k(t)͸ɼ

k(t) = N n=1

e^−n2tbnsinnx0

ͱͳΔɽͦͯࣜ͠(21)ΑΓk(t)→u(x0, t)͸ɼ

u(x0, t) = N n=1

bnsinnx0

σ−n² (e^−n2t−e^−σt) ͱͳΓɼ͜ΕͰu(x0, t)͕ٻ·ͬͨɽ

6.2 ٯ໰୊Ͱͷ೤ݯ{b_n}ͷߏ੒खॱ⁶⁾

࣍ʹٯ໰୊Ͱͷ೤ݯͷߏ੒खॱʹ͍ͭͯड़΂Δɽॱ໰

୊ͰٻΊͨu(x0, t)Λ؍ଌσʔλͱߟ͑ͯɼ

f(x) = N n=1

b_nsinnx

ͷະ஌ͷ෦෼Ͱ͋Δb_n(n= 1, ..., N)ΛٻΊ͍ͯ͘ɽ u(x0, t) → k(t)ͱͯࣜ͠(21)ͷٯͷखॱΛߟ͑Δɽ u(x0, t)∈C¹[0,∞)ͳΒ͹ɼ

k(t) =ut(x0, t) +σu(x0, t)

ͱͳΓɼ͜Εʹઌ΄ͲٻΊͨॱ໰୊ͷղΛ୅ೖ͢Δͱɼ

k(t) = N n=1

e^−n2tbnsinnx0 (22)

ͱͳΔɽ

࣍ʹk(t)→anΛߟ͑Δɽdn=bnsinnx0ͱ͓͍ͯd1

͔ΒॱʹٻΊ͍ͯ͘ɽࣜ(22)Λมܗ͢Δͱɼ

d1=e^tk(t)−e^−t N n=2

dne^(−n2+2)t

ͱͳΔɽࣜ(20)ͷk(t)Λ୅ೖͯ͠t→ ∞ͱ͢Δͱୈೋ

߲͸ୈҰ߲ʹൺ΂ແࢹͰ͖ͯɼ d₁= lim

t→∞e^tk(t) =a₁sinx₀

ͱͳͬͯɼa1 =b1ͱͳΔɽ͜͜Ͱsinx0= 0Ͱͳ͚Ε

͹ͳΒͳ͍͜ͱ͕Θ͔Δɽn= 2Ҏ߱΋ಉ༷ͷख๏Ͱղ

͘͜ͱ͕Ͱ͖ɼ

an=bn (n= 1, ..., N)

ͱͳΔɽͦͯ͠ઌ΄Ͳͱಉ༷ʹsinnx0 = 0Ͱͳ͚Ε͹

ͳΒͳ͍ɽΑͬͯ؍ଌ఺Ͱ͋Δx0͸ɼsinnx= 0(n= 1, ..., N)ɼ͢ͳΘͪnx=kπ(n= 1, ..., N, k = 1, ..., N) ͱͳΔxΛআ͍ͨ఺Ͱͳ͚Ε͹ͳΒͳ͍͜ͱ͕Θ͔Δɽ

͜Ε͕؍ଌ఺x0ͷ৚݅Ͱ͋Δɽ

࠷ޙʹan → f ͱͯࣜ͠(19)ͷٯΛߟ͑Δͱɼf(x)

͸ਖ਼ݭڃ਺Ͱ͋ΔͷͰɼ f(x) =

∞ n=1

ansinnx

Ͱ͋Δɽઌ΄Ͳٻ·ͬͨanΛ୅ೖ͢Δͱɼ

f(x) = N n=1

bnsinnx

ͱͳΓٻΊ͍ͨܗͷղΛٻΊΔ͜ͱ͕Ͱ͖ͨɽΑͬͯ

μ(t) =e^−σtͷͱ͖೤ݯΛ࠶ߏ੒͢Δ͜ͱ͕Ͱ͖ͨɽ

6.3 ༗ݶ࣌ؒͷ؍ଌσʔλͷ৔߹

ٯ໰୊ͷk(t)→anͷͱ͖ʹɼ؍ଌσʔλu(x0, t)ͷ t͕༗ݶͷ۠ؒʹؚ·ΕΔ৔߹Λߟ͑Δɽ

T ͸ਖ਼ఆ਺ͱ͢Δͱɼࣜ(22)ΑΓ

k(T) = e^−Td1+e^−4Td2+· · ·+e^−N2TdN

k(2T) = e^−2Td1+e^−8Td2+· · ·+e^−2N2TdN

k(3T) = e^−3Td₁+e^−12Td₂+· · ·+e^−3N2Td_N ...

k(N T) = e^−NTd1+e^−4NTd2+· · ·+e^−N3TdN

Ͱ͋Γɼ͜͜Ͱvn=e^−n2T ͱ͓͘ͱ

⎛

⎜⎜

⎜⎜

⎜⎜

⎝ k(T) k(2T)

... k(N T)

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

=

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

v₁ v₂ . . . v_N v₁² v₂² . . . v²_N ... ... . .. ...

v^N₁ v₂^N . . . v_N^N

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎜⎜

⎜⎜

⎝ d₁ d2

... dN

⎞

⎟⎟

⎟⎟

⎟⎟

⎠ ͱͳΔɽ

V =

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

v1 v2 . . . vN

v²₁ v²₂ . . . v_N² ... ... . .. ...

v^N₁ v^N₂ . . . v_N^N

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

ͱ͓͘ͱɼϰΝϯσϧϞϯυͷߦྻࣜΑΓVͷߦྻࣜ͸ɼ

|V| =

1≤j≤N

vj

1≤j≤i≤N

(vj−vi)

=

1≤j≤N

e^−j2T

1≤j≤i≤N

(e^−j2T −e^−i2T)

ͱͳΓ|V| = 0ΑΓɼV ͸ٯߦྻ͕ଘࡏ͢ΔͷͰɼdnΛ ٻΊΔ͜ͱ͕Ͱ͖Δɽ

͜ͷΑ͏ʹtΛT,2T,3T, . . . , N Tͱ౳ִؒʹऔͬͯ΍

Δ͜ͱͰdnΛٻΊΔ͜ͱ͕Ͱ͖Δɽ

7. ݁ݴ

ຊݚڀͰ͸ɼٯ໰୊Λղ͍ͯɼҰ༷ͳ๮ͷ೤ݯΛܾఆ

͢Δ͜ͱ͕໨తͰ͋ͬͨɽͦͷͨΊʹ೤ݯʹQ(x, t) =

μ(t)f(x)ͱ͍͏৚݅Λ༩͑ͨɽͦͯ͠μ(t)͸Θ͔͍ͬͯ

Δ΋ͷͱͯ͠ɼະ஌ͷؔ਺f(x)ΛٻΊΔ໰୊Λߟ࡯͠

ͨɽ͞Βʹf(x)ʹ f(x) =

N n=1

b_nsinnx

ͱ͍͏ղͷܗΛߟ͑ɼະ஌Ͱ͋Δ{bn}ΛٻΊΔ໰୊ʹ ؼணͨ͠ɽ

ͦͷ݁Ռɼ͋Δ༗ݶݸͷ఺Λআ͍ͨ۠ؒ಺ͷҰ఺Ͱͷ Թ౓ͷ؍ଌσʔλ͔Βɼ೤ݯͷٻΊ͍ͨؔ਺Ͱ͋Δf(x) Λܾఆ͢Δ͜ͱ͕Ͱ͖ͨɽ

ࢁຊɾۚ[2]ɼp.60ͷઌߦ݁Ռʹ͓͍ͯ͸ɼ೤ݯܾఆͷ ٯ໰୊ͷҰҙੑ͸ূ໌͞Ε͍ͯΔ͕ɼ೤ݯܾఆͷ۩ମత ͳղ๏ʹ͍ͭͯ͸৮ΕΒΕ͍ͯͳ͍ɽ

ຊݚڀͰ͸ɼ೤ݯܾఆͷ۩ମతͳղ๏ʹ͍ͭͯߟ࡯͠ɼ ಛघͳ৚݅ͷԼͰ೤ݯΛܾఆ͢Δ͜ͱ͕Ͱ͖ͨɽ

ຊݚڀ͸2009೥౓ಉࢤࣾཧ޻ֶݚڀॴݚڀॿ੒ۚͷ

ิॿΛड͚ͯߦΘΕ·ͨ͜͠ͱΛه͠ɼؔ܎֤Ґʹײँ

ޚྱਃ্͛͠·͢ɽ

ࢀɹߟɹจɹݙ

1) ٱอ࢘࿠,ʠ ٯ໰୊ͷߟ͑ํͱ࿮૊Έ ʡɼ਺ཧՊֶɼ403 רɼ1߸(1998)ɼpp28-33.

2) ࢁຊণ߂ɼۚ੒שɼʮ೤ํఔࣜͰֶͿٯ໰୊ʯ,ʢαΠΤ ϯεࣾɼ౦ژɼ2008ʣ

3) ऱ೭಺໶உ,ʮվగɹؔ਺ղੳೖ໳ʯ,ʢαΠΤϯεࣾɼ౦ ژɼ1975ʣ

4) అਖ਼ٛ,ʮٯ໰୊ͷ਺ֶʯ,ʢڞཱग़൛ɼ౦ژɼ2000ʣ 5) Victor Isakov, Inverse Source Problems, American

Mathematical Society, (1990)

6) Victor Isakov,Inverse Problems for Partial Differential Equations, Springer-Verlag, (2006)

7) ొࡔએ޷,ʮٯ໰୊ͷ਺ཧͱղ๏ʯ,ʢ౦ژେֶग़൛ձɼ౦ ژ1999ʣ

8) ୩ౡݡೋ,ʮϧϕʔάੵ෼ͱؔ਺ղੳʯ,ʢே૔ॻళɼ౦ ژɼ2002ʣ

9) ࢁຊণ޺,ʮٯ໰୊ೖ໳ʯ,ʢؠ೾ॻళɼ౦ژɼ2002ʣ 10) ྛࠀߦ,ʮ਺ֶͷָ͠Έʯ,ʢ೔ຊධ࿦ࣾɼ౦ژɼ2007ʣ