Graduate School of Fundamental Science and Engineering Waseda University
༤
༤㻌 ኈ㻌 ㄽ㻌 ᩥ㻌 ᴫ㻌 せ
Doctoral Thesis Synopsis
ㄽ ᩥ 㢟 ┠ T h e s i s T h e m e
Verified numerical computation for elliptic partial differential equations and related
problems
ᴃᆺ೫ᚤศ᪉⛬ᘧᑐࡍࡿ⢭ᗘಖドࡁ
ᩘ್ィ⟬㛵㐃ࡍࡿၥ㢟
⏦ ㄳ ⪅ (Applicant name)
Kazuaki TANAKA ⏣୰ ୍ᡂ
Department of Pure and Applied Mathematics Research on Numerical Analysis
December, 2016
ຊจͰҎԼͷପԁܕσΟϦΫϨڥք(1)ʹର͢Δਫ਼อূ͖ܭ
ࢉ ๏ Λ ߟ ͑ Δ ɽ
−Δu(x) =f(u(x)), x∈Ω,
u(x) = 0, x∈∂Ω. (1)
͜͜ͰɼΩR2্ͷ༗քͳଟ֯ܗྖҬɼଈͪ༗քͳଟ֯ܗঢ়ͷ։ू߹ͱ͢Δɽ∂Ω
ͦͷڥքΛද͢ɽ·ͨɼΔϥϓϥε࡞༻ૉɼf :R→R༩͑ΒΕͨඇઢܗࣸ૾
ͱ͢Δɽຊจʹ͓͍ͯ(1)ʹର͢Δ“ਫ਼อূ”ͱɼ(1)ͷۙࣅղuˆΛత ʹ ٻ Ί ɼͦ Ε Λ த ৺ ͱ ͢ Δ ໌ ࣔ ͞ Ε ͨ ܘrͷ ด ٿB(ˆu, r) ʹ ਅ ͷ ղ ͕ ଘ ࡏ ͢ Δ
͜ͱΛূ໌͢Δͱ͍͏͜ͱΛࢦ͢ɽͨͩ͠ɼຊจͰH01ϊϧϜ(ଈͪ∇ · L2(Ω)) ͱL∞ϊ ϧ Ϝ ͷ ҙ ຯ Ͱ ͷ ਫ਼ อ ূ Λ ߟ ͑ Δ ɽຊ จ ͷ ର Ҏ Լ ͷ (2)ͷ ղ ʹ ର ͢ Δ ਫ਼ อ ূ ͖ ܭ ࢉ Λ ؚ Ή ɽ
⎧⎪
⎨
⎪⎩
−Δu(x) =f(u(x)), x∈Ω, u(x)>0, x∈Ω, u(x) = 0, x∈∂Ω.
(2)
ଈͪɼඞཁͰ͋Δ߹(1)ͷਫ਼อূ͞Εͨղʹରͯ͠ɼͦͷਖ਼ੑಉ࣌ʹอ
ূ ͢ Δ ͜ ͱ Ͱ(2)ͷ ղ Λ ਫ਼ อ ূ ͢ Δ ɽ (1)ͷ ղu͕ (2)ͷ ղ Ͱ ͋ Δ ͱ
͖ ɼuΛ(1)ͷ ਖ਼ ղ ͱ ͍ ͏ɽ ͷ Մ ղ ੑ ਫ਼ อ ূ ͖ ܭ ࢉ Λ ߦ ͏ ͠ ͞
ඇઢܗࣸ૾fʹେ͖͘ґଘ͢Δ͕ɼຊจͰදతͳྫͱͯ͠ɼf(t) =|t|p−1t (p >1)͓ Α ͼf(t) =ε−2(t−t3)Λ औ Γ ѻ ͏ɽ͜ ͜ Ͱ ɼε ಛ ҟ ઁ ಈ ͱ ݺ Ε Δ ݱ ʹ ؔ ͢ Δ ਖ਼ ͷ খ ͞ ͍ ύ ϥ ϝ ʔ λ ʔ Ͱ ͋ Δ ɽຊ จ ͷ ͍ ͘ ͭ ͔ ͷ Օ ॴ ʹ ͯΩͷ ತ
ੑ ͕ Ճ ͑ ͯ ཁ ٻ ͞ Ε Δ ɽ͜ Ε ਫ਼ อ ূ ͞ Ε ͨ(1)ͷ ऑ ղu ∈H01(Ω)ͷH2ਖ਼ ଇ ੑ
͕L∞ϊ ϧ Ϝ ͷ ҙ ຯ Ͱ ͷ ਫ਼ อ ূ Λ ߦ ͏ ࡍ ʹ ඞ ཁ ͱ ͳ Δ ͨ Ί Ͱ ͋ Δ ɽ
ࠓ ɼ ܭ ࢉ ੜ ֶ ཧ ֶ ʹ ༝ དྷ ͢ Δ ༷ʑͳ ݱ Λ ཧ ղ ͢ Δ ͨ Ί ʹ Պ
ֶ ͓ Α ͼ ֶ ͷ ͍ Ͱ ॏ ཁ ͳ ׂ Λ ୲ͬͯ ͍ Δ ɽ͠ ͔ ͠ ै དྷ ͷ ܭ ࢉ Ұ ൠʹؙΊޡࠩଧΓޡࠩɼࢄԽޡࠩΛ͡Ίͱ͢Δ༷ʑͳޡࠩΛ͍ɼͦΕΒ
͕ ࠷ ऴ త ͳ ݁ Ռ ʹ க ໋ త ͳ Ө ڹ Λ ٴ ΅ ͢ ͜ ͱ ͕ ͋ Δ ͨ Ί ɼ͜ ͷ ͜ ͱ ༧ ͯ Α Γ ࢹ ͞ Ε ͯ ͖ ͨ ɽҰ ํ Ͱ ܭ ࢉ ͷ ա ఔ Ͱ ൃ ੜ ͢ Δ શ ͯ ͷ ޡ ࠩ Λ ߟ ྀ ͠ ɼఆ
ྔ త ʹ ޡ ࠩ ্ ݶ Λ อ ূ ͢ Δ ܭ ࢉ ͷ ͜ ͱ Λ“ਫ਼ อ ূ ͖ ܭ ࢉ”ͱ ݺ Ϳ ɽ͜
͜ Ͱ“ਫ਼ อ ূ”ͱ ͍ ͏ ݴ ༿ ʹ ޡ ࠩ ͷ Ѳ Ҏ ֎ ʹ ɼର ͱ ͢ Δ ͷ ղ ͷ ଘ ࡏ ͷ อ ূ Λ ಉ ࣌ ʹ ߦ ͏ ͱ ͍ ͏ ҙ ຯ Λ ؚ Ή ɽͦ ͷ ͨ Ί ɼ ֶ త ཱ ͔ Β త ݕ ূ ๏ ܭࢉػԉ༻(ଘࡏ)ূ໌ͱݺΕΔ͜ͱ͋Δɽઌߦݚڀͱͯ͠ʰେੴਐҰ,ਫ਼
อ ূ ͖ ܭ ࢉ ,ί ϩ φ ࣾ , 2000ʱʰ த ඌ ॆ ɼ ෦ ળ ོ ,࣮ ྫ Ͱ ֶ Ϳ ਫ਼ อ
ূ ͖ ܭ ࢉ ཧ ͱ ࣮ , α Π Τ ϯ ε ࣾ , 2011ʱ ʹ · ͱ Ί ͯ ͋ Δ ख ๏ ͕ ༗ ໊ Ͱ ͋ Γɼਫ਼ อ ূ ͖ ܭ ࢉ ै དྷ ͷ ܭ ࢉ ʹ ͳ ͍ ಛ Λ ࣋ ͭ ͜ ͱ ͔ Β ۙ
ੈ ք త ʹ Λ ू Ί ͯ ͍ Δ ٕ ज़ Ͱ ͋ Δ ɽ
ಛ ʹ զʑͷ ڵ ຯ ପ ԁ ܕ ڥ ք (1)͓ Α ͼ(2)ʹ ର ͢ Δ ਫ਼ อ ূ ͖ ܭ
ࢉͰ͋Δɽ(1)͓Αͼ(2)ੜֶཧֶʹ༝དྷ͢Δ༷ʑͳϞσϧʹىҼ͢Δॏ
1
ཁ ͳ ภ ඍ ํ ఔ ࣜ Ͱ ͋ Δ ͨ Ί ɼ͜ Ε · Ͱ ղ ੳ త ɼ త ྆ ํ ͷ ଆ ໘ ͔ Β ͘ ݚ ڀ
͞Ε͖ͯͨɽಛʹຊจͰରͱ͢Δf(t) =|t|p−1t(p >1)͓Αͼf(t) =ε−2(t−t3)
(ε >0)ͷͲͪΒͷ߹(2)ͷղͷଘࡏੑɾ།Ұੑʹؔ͢Δ݁Ռ͕ภඍํఔࣜ
ʹΑΓΒΕ͍ͯΔɽf(t) =|t|p−1t(p >1)ͷͱ͖ɼΩ͕Β͔Ͱ͔֤ͭ࣠ʹର͠
ͯ ର শ Ͱ ͋ Ε (2) ། Ұ ղ Λ ࣋ ͭ ɽ· ͨ ɼྖ Ҭ ͕ Β ͔ Ͱ ͳ ͍ ߹ Ω = (0,1)2 ͷ ͷ ղ ͷ े ͳ Β ͔ ͞ ͕ อ ূ ͞ Ε Δ ର শ ྖ Ҭ Ͱ ͋ Ε ಉ ༷ ʹ ਖ਼ ղ ͷ ། Ұ ੑ
͕อূ͞ΕΔɽf(t) =ε−2(t−t3) (ε >0)ͷͱ͖ɼλ1Λ−ΔͷσΟϦΫϨ݅Λ՝
ͨ͠Ω্Ͱͷऑ͍ҙຯͰͷ࠷খݻ༗ͱ͢Δͱ͖ɼε−2> λ1Ͱ͋Ε(2)།Ұղ Λ ࣋ ͭ ͱ ͍ ͏ ͜ ͱ ͕ ҙ ͷ ༗ ք ྖ Ҭ ʹ ର ͠ ͯ ಘ Β Ε Δ ɽ͜ ͷ Α ͏ ͳ ݁ Ռ ͕ Β Ε
ͯ ͍ Δ ʹ ؔ Θ Β ͣɼղ ͷ ఆ ྔ త ͳ ใ(֤ Ͱ ͷ ࠷ େ ɼ࠷ খ ) ͦ ͜
͔ Β ؍ ଌ ͞ Ε Δ ղ ͷ ܗ ঢ় ɼݱ ঢ় ͷ ղ ੳ త ख ๏ ͷ Έ ͔ Β Ͱ ໌ Β ͔ ʹ ͞ Ε ͯ ͍ ͳ ͍ ɽਫ਼ อ ূ ͖ ܭ ࢉ ɼਅ ͷ ղ ͷ ఆ ྔ త ͳ ใ Λ ͦ ͷ త ʹ ಘ ͨ ۙ ࣅ ղ ͱ ͷ ࠩ Λ ද ͢ ෆ ࣜ ͷ ܗ Ͱ ಘ Δ ͜ ͱ ͕ Ͱ ͖ Δ ͱ ͍ ͏ Ͱ ༗ ༻ Ͱ ͋ Δ ͱ ݴ ͑ Δ ɽ
Ҏ Լ ɼຊ จ ͷ ֤ ষ ͷ ֓ ཁ Λ ड़ Δ ɽ
1ষͰຊจͷରͱ͢ΔϞνϕʔγϣϯɼഎܠɼઌߦݚڀʹ͍ͭͯ
ه ड़ ͢ Δ ɽ
2ষ Ͱ · ͣ ຊ จ Ͱ ༻ ͢ Δ ه ߸ ͷ ४ උ Λ ߦ ͏ɽ࣍ ʹ (1)ʹ ର ͢ ΔH01ϊ ϧϜͷҙຯͰͷਫ਼อূ͖ܭࢉͷͨΊʹ༻͍ΔχϡʔτϯɾΧϯτϩϏον ͷఆཧ͓Αͼͦͷվྑ͞ΕͨఆཧΛհ͢Δɽ͜ΕΒͷఆཧ(1)ͷۙࣅղuˆ͕͋
Δछͷ“ྑ͍” ݅Λ ຬ ͨͤ ͦ ͷ ۙ͘ ʹਅ ͷղ ͕ଘ ࡏ͢ Δͱ ͍͏ ͜ͱ Λओ ு͢ Δ
ͷͰ͋Δɽ·ͨɼͦ͜ʹߋʹ݅ΛՃ͑Δ͜ͱʹΑΓ(2)ʹର͢Δ(ଈͪ(1)ͷਖ਼
ղʹର͢Δ)ਫ਼อূՄೳʹͳΔͱ͍͏͜ͱड़ΔɽྖҬ͕ತͰ͋Δ߹
(1)ͷ ऑ ղ ͷH2ਖ਼ ଇ ੑ ͕ อ ূ ͞ Ε Δ ͨ Ί ɼH2(Ω)͔ ΒL∞(Ω) ͷ ຒ Ί ࠐ Έ ఆ
Λ۩ମతʹධՁ͢Δ͜ͱʹΑΓɼ্هख๏ʹΑΓಘΒΕͨH01ϊϧϜͷҙຯͰͷ ղ ͷ แ ؚ ͔ Β ɼL∞(Ω)ͷ ҙ ຯ Ͱ ͷ ղ ͷ แ ؚ ͕ ಘ Β Ε Δ ͜ ͱ ड़ Δ ɽຊ ষ Ͱ հ
͢ Δ ఆ ཧ ͷ ཁ ٻ ͢ Δ ۙ ࣅ ղuˆͷ ຬ ͨ ͢ ͖ ݅ Λ νΣοΫ ͢ Δ ͨ Ί ʹ ɼ (ྖ ҬΩ ඇ ઢ ܗ ࣸ ૾f)uˆʹ ґ ଘ ͢ Δ ͍ ͘ ͭ ͔ ͷ ఆ Λ ۩ ମ త ʹ ධ Ձ ͢ Δ ඞ ཁ ͕ ͋ Δ ɽྫ ͑ ɼ Λ ه ड़ ͢ Δ ࡞ ༻ ૉ ͷuˆʹ ͓ ͚ Δ ઢ ܗ Խ ࡞ ༻ ૉ ͷ ٯ ࡞ ༻ ૉ ϊ ϧ Ϝ
ɼຒΊࠐΈH01(Ω)→Lp(Ω)ͷෆࣜʹݱΕΔఆCp(Ω)Ͱ͋Δɽ2ষͷ࠷ޙͰ
Cp(Ω)ͷ ্ ͔ Β ͷ ૈ ͍ ධ Ձ Λ ಘ Δ ͨ Ί ͷ ݹ య త ͳ ख ๏ Ͱ ͋ Δ λ ϥ ϯ ςΟͷ ࠷ ྑ ఆ ͱ ϔ ϧ μ ʔ ͷ ෆ ࣜ Λ ༻ ͍ ͨ ධ Ձ ํ ๏ ʹ ͭ ͍ ͯ ݴ ٴ ͢ Δ ɽ
3ষ Ͱ ɼର ͱ ͢ Δ (1)Λ ऑ ܗ ࣜ Խ ͠ ࡞ ༻ ૉF : H01(Ω) → H−1(Ω) (u →
−Δu−f(u))Λ༻͍ͯF(u) = 0ͱॻ͍ͨͱ͖ʹɼͦͷۙࣅղuˆʹ͓͚ΔFͷϑϨο γΣඍFuˆͷٯ࡞༻ૉϊϧϜFuˆ−1
B(H−1(Ω),H01(Ω))ͷධՁ๏ʹ͍ͭͯड़Δɽ͜͜
ͰH−1(Ω)H01(Ω)ͷ ڞ ۭ ؒ Λ ද ͠ ɼ௨ ৗ ͷsupϊ ϧ Ϝ Λ ಋ ೖ ͠ ͨ ͷ ͱ ͢ Δ ɽ
͜ ͷ ٯ ࡞ ༻ ૉ ϊ ϧ Ϝ uˆʹ Α Γ ઁ ಈ ͞ Ε ͨ ͋ Δ ପ ԁ ܕ ࣗ ݾ ڞ ࡞ ༻ ૉ ͷ ࠷ খ ݻ ༗
͔ Β ܭ ࢉ ͢ Δ ͜ ͱ ͕ Ͱ ͖ ɼͦ ͷ ࠷ খ ݻ ༗ Λ ཱུɾେ ੴ ͷ ख ๏ Ͱ ্ Լ ͔ Β ۩ ମ త
ʹ ධ Ձ ͢ Δ ͜ ͱ Ͱ ॴ ͷ ϊ ϧ Ϝ ධ Ձ Λ ࣮ ݱ ͢ Δ ɽཱུɾେ ੴ ͷ ख ๏ H01(Ω)ͷ ͋ Δ
༗ݶ࣍ݩ෦ۭؒVN্Ͱۙࣅ͞Εͨݻ༗ͱɼݩͷແݶ࣍ݩۭؒH01(Ω)Ͱܭࢉ͞
Ε ͨ ݻ ༗ ͷ ࠩ Λ ධ Ձ ͢ Δ ɽ͜ ͜ ͰN VN ͷH01(Ω)ʹ ର ͢ Δ ۙ ࣅ Λ ද ͢ ਖ਼ ͷ ύ ϥ ϝ ʔ λ Ͱ ͋ Δ ɽྫ ͑ ຊ จ Ͱ {φi}∞i=1 ΛH01(Ω)Λ ு Δ ج ఈ ͱ ͠ ͨ ͱ ͖ ʹ ɼ VN =span{φi :i= 1,2,· · ·, N}ͷΑ͏ʹVNΛબͼɼ͜ͷͱ͖NVNΛுΔجఈؔ
ͷ Λ ද ͢ ͜ ͱ ʹ ͳ Δ ɽཱུɾେ ੴ ͷ ख ๏ Λ ༻ ͍ ͯ ର ͱ ͢ Δ ݻ ༗ Λ ධ Ձ ͢ Δ ʹVN ͷH01(Ω)ʹର͢ΔۙࣅΛө͢ΔิޡࠩఆCN Λ۩ମతʹධՁ͢Δ ඞཁ͕͋Δɽଈͪɼh∈L2(Ω)ʹରͯ͠ɼͦΕʹରԠ͢ΔϙΞιϯํఔࣜ−Δu=h ͷ σΟϦ Ϋ Ϩ ڥ ք ͷ ऑ ղ Λuhͱ ॻ ͍ ͨ ͱ ͖ ʹ
uh−PNuhV ≤CNhL2(Ω) (3) Λ ҙ ͷh ∈ L2(Ω)ʹ ର ͠ ͯ ຬ ͨ ͢ ਖ਼ ఆ CN Ͱ ͋ Δ ɽ͜ ͜ Ͱ ɼPN H01(Ω)͔ Β VN ͷ ަ ࣹ Ө Ͱ ͋ Γɼ(PNu−u, vN)H01(Ω) = 0 for all u ∈ H01(Ω) andvN ∈ VN Ͱ ఆ ٛ ͞ Ε Δ ɽ3ষ ͷ ࠷ ޙ Ͱ VN Λ ϧ δϟϯ υ ϧ ଟ ߲ ࣜ ͔ Β ߏ ͞ Ε Δ ج ఈ Ͱ ு Β Ε ͨ ۭ ؒ ͱ ͠ ͨ ͱ ͖ ͷCNͷ ධ Ձ ๏ Λ ड़ Δ ɽ
4ষ Ͱ ؔ ۭ ؒ ؒ ͷ ຒ Ί ࠐ ΈH01(Ω)→ Lp(Ω)ͷ ࠷ ྑ ఆ Cp(Ω)ͷ ධ Ձ ๏ ʹ ͭ
͍ ͯ ड़ Δ ɽΑ Γ ਖ਼ ֬ ʹ Cp(Ω)
uLp(Ω)≤Cp(Ω)uH01(Ω) for allu∈H01(Ω). (4) Λ ຬ ͨ ͢ ਖ਼ ఆ ͱ ͠ ͯ ఆ ٛ ͞ Ε Δ ɽ͜ ͷ ࠷ ྑ (ଈ ͪ ۃ )Λ ୡ ͞ ͤ Δ ؔ u ପ ԁ ܕ ڥ ք
⎧⎪
⎨
⎪⎩
−Δu=up−1 in Ω, u >0 in Ω,
u= 0 on ∂Ω
(5)
Λ ຬ ͨ ͢ ͜ ͱ Λ ࣔ ͢ɽߋ ʹ (5)ͷ ղ ͷ ། Ұ ੑ ʹ ͭ ͍ ͯΩ = (0,1)2ͷ ߹ ʹ ٞ
͠ ɼ2ষ Ͱ ड़ ͨ ํ ๏ Ͱ ͦ ͷ ղ ͷ ਫ਼ อ ূ Λ ߦ ͏ ͜ ͱ ʹ Α ΓɼCp(Ω)ͷ ࠷ ྑ Λ λ Π τ ʹ ධ Ձ ͢ Δ ɽಛ ʹ ຊ ষ Ͱ p = 3,4,5,6,7ͷ ߹ ͷ ݁ Ռ Λ ༩ ͑ ɼCp(Ω)ͷ ࠷ ྑ
Λ12ʙ13ܻ ͷ ਫ਼ Ͱ ධ Ձ ͠ ͨ ݁ Ռ Λ · ͱ Ί Δ ɽ
5ষ Ͱ f(t) =ε−2(t−t3)ͷ ߹ ɼଈ ͪ Ξ Ϩ ϯɾΧ ʔ ϯ ํ ఔ ࣜ ͷ ఆ ৗ −Δu=ε−2(u−u3) in Ω,
u= 0 on ∂Ω (6)
ʹର͢Δਫ਼อূ͖ܭࢉΛߟ͑Δɽ͜ͷͷղε >0͕খ͘͞ͳΔʹͭ
Ε ͯ ಛ ҟ ઁ ಈ ͱ ͍ ͏ ݱ Λ ى ͜ ͢ɽ· ͨ ɼෳ ࡶ ͳ ղ ͷ ذ Λ Ҿ ͖ ى ͜ ͢ ͜ ͱ Ͱ
ΒΕΔɽ͜ͷ͜ͱ͔ΒҰൠʹε͕খ͍͞΄Ͳղͷਫ਼อূ͘͠ͳΔɽ5ষͰ
εΛ ม Խ ͞ ͤ ͨ ߹ ʹ ਫ਼ อ ূ ʹ ඞ ཁ ͳ ֤ छ ఆ ղ ͷ ܗ ͕ Ͳ ͷ Α ͏ ʹ ม Խ ͢ Δ͔Λ۩ମతͳྫͱڞʹհ͢Δɽ·ͨɼ2ষͰड़ͨํ๏ΛԠ༻͠ɼͦͷਖ਼ղ ͷ ਫ਼ อ ূ Λ ߦͬͨ ྫ ซ ͤ ͯ հ ͢ Δ ɽ
6ষ Ͱ ຊ จ Ͱ ड़ ͨ ख ๏ɾ݁ Ռ Λ ૯ ׅ ͠ ɼࠓ ޙ ͷ ల Λ ड़ Δ ɽ
3
㹌㹭
᪩
᪩✄⏣Ꮫ ༤ኈ㸦ᕤᏛ㸧 Ꮫ⏦ㄳ ◊✲ᴗ⦼᭩
Ặྡ ⏣୰ ୍ᡂ ༳
㸦 ᖺ ᭶ ⌧ᅾ㸧
✀㢮ู 㢟ྡࠊ Ⓨ⾲࣭Ⓨ⾜ᥖ㍕ㄅྡࠊ Ⓨ⾲࣭Ⓨ⾜ᖺ᭶ࠊ 㐃ྡ⪅㸦⏦ㄳ⪅ྵࡴ㸧
ㄽᩥ
ㅮ₇
[1] Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, Shin'ichi Oishi: Numerical validation of blow-up solutions for ODEs, to appear in Journal of Computational and Applied Mathematics.
[2] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, and Shin'ichi Oishi: Sharp numerical inclusion of the best constant for embedding on bounded convex domain, Journal of Computational and Applied Mathematics, 311, 306–313 (2017), to appear.
Electronically published in doi.org/10.1016/j.cam.2016.07.021.
[3] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, and Shin'ichi Oishi: Estimation of the Sobolev embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and Applications, 1, 1-23 (2015).
[4] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, and Shin'ichi Oishi: Numerical verification of positiveness for solutions to semilinear elliptic problems, JSIAM Letters 7, 73-76 (2015).
[5] Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu, and Shin'ichi Oishi: Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation, Japan Journal of Industrial and Applied Mathematics, 31, 665-679 (2014).
[6] ⏣୰୍ᡂ, 㛵᰿ኴ, ▼㐍୍㸸ᴃᆺᚤศ᪉⛬ᘧࡢṇ್ゎᑐࡍࡿ⢭ᗘಖドࡁᩘ
್ ィ ⟬ ἲ 㸦Verified numerical computation method for positive solutions to elliptic differential equations㸧, RIMS◊✲㞟ࠕ⌧㇟ゎ᫂ྥࡅࡓᩘ್ゎᯒᏛࡢ᪂ᒎ㛤IIࠖ, 2016ᖺ10᭶19᪥㹼10᭶21᪥㸬
[7] 㛵᰿ኴ, ⏣୰୍ᡂ, ▼㐍୍㸸࠶ࡿ↓㝈ḟඖᅛ᭷್ࢆ⏝࠸ࡓᴃᆺ೫ᚤศ᪉⛬ᘧࡢ ゎࡢᏑᅾᛶᑐࡍࡿィ⟬ᶵ⏝ド᫂ἲ 㸦Computer-assisted proof for existence of solutions to PDEs using an infinite eigenvalue㸧, RIMS◊✲㞟ࠕ⌧㇟ゎ᫂ྥࡅࡓᩘ
್ゎᯒᏛࡢ᪂ᒎ㛤IIࠖ, 2016ᖺ10᭶19᪥㹼10᭶21᪥㸬
[8] 㛵᰿ኴ, ⏣୰୍ᡂ, ▼㐍୍㸸᭷⏺࡞ฝ㡿ᇦ࠾ࡅࡿ㐃❧ᴃᆺ೫ᚤศ᪉⛬ᘧࡢゎ ࡢィ⟬ᶵ⏝Ꮡᅾド᫂ἲ㸦Computer assisted existence proof of solutions to system of partial differential equations with bounded convex polygonal domains㸧, The Twenty-Eighth RAMP Symposium, 2016ᖺ10᭶13᪥㹼10᭶14᪥㸬
[9] Kazuaki Tanaka, Kouta Sekine, Shin'ichi Oishi: On verified numerical computation for positive solutions to elliptic boundary value problems, Computer Arithmetic and Validated Numerics, SCAN2016, Sep. 26-29, 2016.
[10] Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi: A norm estimation for an inverse of linear operator using a minimal eigenvalue, Computer Arithmetic and Validated Numerics, SCAN2016, Sep. 26-29, 2016.
[11] Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, and Shin’ichi Oishi: Verified numerical computations for blow-up solutions of ODEs, Computer Arithmetic and Validated Numerics, SCAN2016, Sep. 26-29, 2016.
㹌㹭
᪩
᪩✄⏣Ꮫ ༤ኈ㸦ᕤᏛ㸧 Ꮫ⏦ㄳ ◊✲ᴗ⦼᭩
✀㢮ู 㢟ྡࠊ Ⓨ⾲࣭Ⓨ⾜ᥖ㍕ㄅྡࠊ Ⓨ⾲࣭Ⓨ⾜ᖺ᭶ࠊ 㐃ྡ⪅㸦⏦ㄳ⪅ྵࡴ㸧
ㅮ₇
[12] Kazuaki Tanaka, Shin'ichi Oishi: On verified numerical computation for elliptic Dirichlet boundary value problems using sub- and super-solution method, The fifth Asian conference on Nonlinear Analysis and Optimization, Toki Messe, Niigata, Japan, August 1-6, 2016.
[13] Kouta Sekine, Kazuaki Tanaka, Shin'ichi Oishi: Estimation for optimal constant satisfying an inequality for linear operator using minimal eigenvalue, The fifth Asian conference on Nonlinear Analysis and Optimization, Toki Messe, Niigata, Japan, August 1-6, 2016.
[14] Kaname Matsue, Akitoshi Takayasu, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, and Shin’ichi Oishi: Rigorous numerics of blowup solutions for ODEs, The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, July 1-5, 2016.
[15] Kazuaki Tanaka, Kouta Sekine, Shin'ichi Oishi: Numerically verifiable condition for positivity of solution to elliptic equation, The 11th East Asia SIAM. June, 20-22, 2016.
[16] 㧗Ᏻு⣖, ᯇỤせ, బࠎᮌከᕼᏊ, ⏣୰୍ᡂ, Ỉཱྀಙ, ▼㐍୍㸸ᨺ≀㠃ࢥࣥࣃࢡࢺ
ࢆ⏝࠸ࡿᖖᚤศ᪉⛬ᘧࡢ⇿Ⓨゎࡢᩘ್ⓗ᳨ドἲ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ2015ᖺᗘ㐃ྜ
Ⓨ⾲㸪⚄ᡞᏛ㝔Ꮫ࣏࣮ࢺࣛࣥࢻ࢟ࣕࣥࣃࢫ㸪2016ᖺ3᭶4᪥㹼3᭶5᪥㸬
[17] Kazuaki Tanaka, Kouta Sekine, Makoto Mizuguchi, Shin'ichi Oishi: Numerical verification for positiveness of solutions to self-adjoint elliptic problems, JSST 2015 International Conference on Simulation Technology, Oct. 12-14, 2015.
[18] Kazuaki Tanaka and Shin'ichi Oishi: Computer-assisted analysis for solutions to nonlinear elliptic Neumann problems, JSST 2014 International Conference on Simulation Technology, Oct. 29-31, 2014.
[19] Kazuaki Tanaka and Shin'ichi Oishi: Numerical verification for periodic stationary solutions to the Allen-Cahn equation, The16th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, SCAN2014, Sep. 21-26, 2014.
[20] 㧗Ᏻு⣖, ᯇỤせ㸪బࠎᮌከᕼᏊ㸪⏣୰୍ᡂ㸪Ỉཱྀಙ㸪▼㐍୍㸸Verified numerical
enclosure of blow-up time for ODEs㸪᪥ᮏᩘᏛ2015ᖺᗘᖺ㸪ி㒔⏘ᴗᏛ㸪2015 ᖺ9᭶13᪥㹼9᭶16᪥㸬
[21] ⏣୰୍ᡂ, 㛵᰿ኴ, Ỉཱྀಙ, ▼㐍୍㸸ᴃᆺ೫ᚤศ᪉⛬ᘧࡢゎࡢṇ್ᛶᑐࡍࡿ
ᩘ್ⓗ᳨ドἲ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ2015ᖺᗘᖺ㸪㔠ἑᏛ㸪2015ᖺ9᭶9᪥㹼9᭶ 11᪥㸬
[22] ⱝᒣ㤾ኴ, ⏣୰୍ᡂ, 㛵᰿ኴ, ᑿᓮඞஂ, ▼㐍୍㸸㏲ḟῧຍἲࡼࡿ୕ゅᙧศ
ࡢDelaunay ᛶᑐࡍࡿᩘ್ⓗ᳨ドἲ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ2015ᖺᗘᖺ㸪㔠ἑᏛ㸪
2015ᖺ9᭶9᪥㹼9᭶11᪥㸬
[23] 㧗Ᏻு⣖, ᯇỤせ㸪బࠎᮌከᕼᏊ㸪⏣୰୍ᡂ㸪Ỉཱྀಙ㸪▼㐍୍㸸ᖖᚤศ᪉⛬ᘧࡢ
⇿Ⓨゎᑐࡍࡿ⢭ᗘಖドࡁᩘ್ィ⟬㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ2015ᖺᗘᖺ㸪㔠ἑᏛ㸪 2015ᖺ9᭶9᪥㹼9᭶11᪥㸬
㹌㹭
᪩
᪩✄⏣Ꮫ ༤ኈ㸦ᕤᏛ㸧 Ꮫ⏦ㄳ ◊✲ᴗ⦼᭩
✀㢮ู 㢟ྡࠊ Ⓨ⾲࣭Ⓨ⾜ᥖ㍕ㄅྡࠊ Ⓨ⾲࣭Ⓨ⾜ᖺ᭶ࠊ 㐃ྡ⪅㸦⏦ㄳ⪅ྵࡴ㸧
ㅮ₇
ࡑࡢ
㸦 ࣏ ࢫ ࢱ
࣮Ⓨ⾲㸧
ࡑࡢ
㸦ཷ㈹㸧
[24] 㛵᰿ኴ㸪⏣୰୍ᡂ㸪㧗Ᏻு⣖㸪ᒣᓮ᠇㸸ࢩࢢ࣐ࣀ࣒ࣝࢆ⏝ࡋࡓ⢭ᗘಖドࡁᩘ
್ィ⟬ἲࡢ㐃❧ᴃᆺ೫ᚤศ᪉⛬ᘧࡢᛂ⏝㸪➨㸲㸵ᅇ᪥ᮏᏛ⏕⏘ᕤᏛ㒊Ꮫ⾡ㅮ
₇㸪᪥ᮏᏛ㸪2014ᖺ12᭶6᪥㸬
[25] ⏣୰୍ᡂ㸪Ỉཱྀಙ㸪㛵᰿ኴ㸪▼㐍୍㸸An a priori estimation of the Sobolev embedding constant and its application to numerical verification for solutions to PDEs, ➨10ᅇ᪥ᮏᛂ
⏝⌮Ꮫ◊✲㒊㐃ྜⓎ⾲㸪ி㒔Ꮫྜྷ⏣࢟ࣕࣥࣃࢫ⥲ྜ◊✲㸶ྕ㤋㸪 2014ᖺ3
᭶19᪥㹼3᭶20᪥㸬
[26] Kazuaki Tanaka and Shin'ichi Oishi: Numerical verification for stationary solutions to the Allen-Cahn equation, The International Workshop on Numerical Verification and its Applications, Waseda Univ. Nishiwaseda campus, Japan, March, 2014.
[27] Kazuaki Tanaka, Makoto Mizuguchi, Kouta Sekine, Akitoshi Takayasu, Shin'ichi Oishi:
Estimation of an embedding constant on Lipschitz domains using extension operators, JSST 2013 International Conference on Simulation Technology. Sep. 11-13, 2013.
[28] ⏣୰୍ᡂ, 㧗Ᏻு⣖, 㞷ᓠ, ▼㐍୍㸸⥺ᙧᴃᆺస⏝⣲ࡢNeumann᮲௳ୗ࠾ࡅ
ࡿ⢭ᗘಖドࡁ㏫స⏝⣲ࣀ࣒ࣝホ౯, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ 2013ᖺᗘᖺ, ࢡࣟࢫ⚟
ᒸ, ⚟ᒸ┴⚟ᒸᕷ, 2013ᖺ9᭶9᪥㹼9᭶11᪥.
[29] Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu, Shin'ichi Oishi: Verified norm estimation for the inverse of linear elliptic operators and its application, The 9th East Asia SIAM. June 18-20, 2013.
[30] ⏣୰୍ᡂ, 㧗Ᏻு⣖, 㞷ᓠ, ▼㐍୍㸸㏫స⏝⣲ࣀ࣒ࣝホ౯ࢆ⏝࠸ࡓᴃᆺ
Neumannቃ⏺್ၥ㢟ࡢゎᑐࡍࡿ⢭ᗘಖドࡁᩘ್ィ⟬, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ2013ᖺ
ᗘ㐃ྜⓎ⾲, ᮾὒᏛⓑᒣ࢟ࣕࣥࣃࢫ, 2013ᖺ3᭶14᪥㹼3᭶15᪥.
[31] ⏣୰୍ᡂ, 㧗Ᏻு⣖, 㞷ᓠ, ▼㐍୍㸸⥺ᙧᴃᆺస⏝⣲ࡢNeumann᮲௳ୗ࠾ࡅ
ࡿ⢭ᗘಖドࡁ㏫స⏝⣲ࣀ࣒ࣝホ౯, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ 2012ᖺᗘᖺ, ⛶ෆ᪥✵
࣍ࢸࣝ, ᾏ㐨⛶ෆᕷ, 2012ᖺ8᭶28᪥㹼9᭶2᪥. 㸦ࡑࡢㅮ₇㸳௳㸧
[32] ⱝᒣ㤾ኴ, ⏣୰୍ᡂ, 㛵᰿ኴ, ᑿᓮඞஂ, ▼㐍୍㸸Delaunay ୕ゅᙧศࡢ⢭ᗘಖ
ドࡁᩘ್ィ⟬ᡭἲᑐࡍࡿ⪃ᐹ㸪᪥ᮏᛂ⏝ᩘ⌮Ꮫ2016ᖺᗘᖺ㸪ᕞᅜ㝿
㆟ሙ㸪2016ᖺ9᭶12᪥㹼9᭶14᪥㸬 㸦ࡑࡢ࣏ࢫࢱ࣮Ⓨ⾲㸰௳㸧
[33] ⱝᒣ㤾ኴ, ⏣୰୍ᡂ, 㛵᰿ኴ, ᑿᓮඞஂ, ▼㐍୍, ᪥ᮏᛂ⏝ᩘ⌮Ꮫ2016ᖺᗘᖺ
㸪ඃ⚽࣏ࢫࢱ࣮㈹ ཷ㈹㸬
[34] ⏣୰୍ᡂ㸪2016ᖺᗘᕝຌグᛕ≉ูඃ⚽㈹ ཷ㈹㸬
[35] Kazuaki Tanaka, JSST 2014 International Conference, Student Presentation Award ཷ㈹㸬
[36] Kazuaki Tanaka, JSST 2013 International Conference, Student Presentation Award ཷ㈹㸬