1. Gabatarwa

Wannan aikin yana magance matsalar Cauchy don ma'auni na ci gaba na oda na biyu a cikin sararin Hilbert, wanda ke wakiltar babban juzu'in ma'auni na Ball integro-differential. Ma'auni yana fasalta masu aiki masu zaman kansu masu ingantaccen aiki a cikin babban sashi, waɗanda ƙila ba su da iyaka. Babban manufa ita ce haɓaka da bincika tsarin sama-uku mai daidaitaccen rabewa don kimanin mafita ga wannan matsala, tare da kimanin sharuɗɗan rashin layi ta amfani da hanyoyin haɗin kai.

Ma'aunin da ake la'akari da shi ya haɓaka ma'auni na katako na J.M. Ball, wanda kansa ya faɗaɗa nau'in ma'auni na Kirchhoff don katako wanda asalin S. Woinowsky-Krieger ya samo. Gudunmawar Ball ta gabatar da sharuɗɗan damping don yin la'akari da tasirin damping na waje da na ciki. Binciken ma'auni na Kirchhoff ya fara ne da babban aikin Bernstein kuma tun daga lokacin an faɗaɗa shi ta ɗimbin masu bincike ciki har da Arosio, Panizzi, Berselli, Manfrin, D'Ancona, Spagnolo, Medeiros, Matos, Nishihara, da sauransu.

Binciken da ya gabata ya mayar da hankali kan sassa daban-daban ciki har da ingantaccen tsari, iyawar warwarewa, da wanzuwar mafita masu ƙarancin ƙa'ida don nau'in ma'auni na Kirchhoff. Kwatankwacin da ake la'akari da shi a wannan aikin yana amfana da halartar murabba'in babban mai aiki a cikin sashin layi, wanda ke sauƙaƙe samun mahimman kimantawa na farko.

2. Tsarin Lissafi

An tsara matsalar Cauchy a cikin sararin Hilbert H don ma'auni na ci gaba na oda na biyu mai rashin layi:

u''(t) + A u(t) + M(||B u(t)||²) u(t) = f(t, u(t), u'(t)), t ∈ (0,T]

tare da sharuɗɗan farko:

u(0) = u₀, u'(0) = u₁

inda A da B su ne masu aiki masu zaman kansu masu ingantaccen aiki a cikin H, mai yuwuwa marasa iyaka, kuma M aiki ne mai rashin layi wanda ke wakiltar kimanin ma'anar haɗin kai. Kalmar ||B u(t)||² tana nuna murabba'in ma'auni na gradient a cikin saitin da ba a taɓa gani ba.

Masu aikin A da B sun gamsu da wasu sharuɗɗan gani waɗanda ke tabbatar da ingantaccen tsarin matsala. Ana ɗaukar rashin layi na M a matsayin mai ci gaba da Lipschitz a cikin gida kuma ya gamsar da sharuɗɗan girma masu dacewa don tabbatar da wanzuwar da keɓantaccen mafita.

3. Tsarin Sama-Uku Mai Rabewa

An ba da tsarin sama-uku mai rabewa na lokaci-lokaci ta hanyar:

(u^{n+1} - 2u^n + u^{n-1})/τ² + A u^n + M(||B u^n||²) u^n = f(t_n, u^n, (u^{n+1} - u^{n-1})/(2τ))

inda τ ke wakiltar girman matakin lokaci, u^n yana kimanin u(t_n) a lokacin t_n = nτ, kuma an kima sharuɗɗan rashin layi da suka haɗa da gradient ta amfani da hanyoyin haɗin kai.

Tsarin yana da daidaito kuma an tsara shi don kiyaye wasu kaddarorin makamashi na ci gaba da matsala. Kimantawar sharuɗɗan rashin layi ta amfani da hanyoyin haɗin kai yana tabbatar da mafi kyawun kaddarorin kwanciyar hankali idan aka kwatanta da hanyoyin daidaitawa kai tsaye.

Sharuɗɗan farko na rabewa sune:

u⁰ = u₀, u¹ = u₀ + τ u₁ + (τ²/2)(-A u₀ - M(||B u₀||²) u₀ + f(0, u₀, u₁))

4. Binciken Kwanciyar Hankali

Binciken kwanciyar hankali yana ci gaba a matakai da yawa. Na farko, mun kafa iyakar daidaitaccen mafita ga matsala mai rabewa mai rashin layi da kwatankwacinta na bambanci na asalin oda na farko.

Theorem 4.1 (Iyakar Daidaitacce): Ƙarƙashin zato masu dacewa akan masu aiki A, B da rashin layi na M, mafita {u^n} na matsala mai rabewa mai rashin layi da ma'anar bambanci {(u^{n+1} - u^n)/τ} suna da iyaka daidai gwargwado game da ma'auni na rabewa τ.

Don matsala mai rabewa mai layi da ta dace, mun samo kimantawa na farko na babban oda ta amfani da polynomials na Chebyshev masu canji biyu. Waɗannan kimantawa suna da mahimmanci don kafa kwanciyar hankali na matsala mai rabewa mai rashin layi.

Theorem 4.2 (Kwanciyar Hankali): Tsarin sama-uku mai rabewa yana da kwanciyar hankali, ma'ana cewa ƙananan rikice-rikice a cikin bayanan farko da gefen dama suna haifar da ƙananan canje-canje a cikin mafita ta lamba, tare da mai haɓaka factor ɗin da ake sarrafa ta hanyar ma'auni na rabewa.

Hujja ta dogara ne akan kimantawan makamashi da kulawar sharuɗɗan rashin layi ta hanyar kimanin haɗin kai.

5. Sakamakon Haɗuwa

Don mafita masu santsi, muna ba da kimantawan kuskure don mafita mai kima. An taƙaita babban sakamakon haɗuwa a cikin ka'idar mai zuwa:

Theorem 5.1 (Kimantawar Kuskure): A ɗauka cewa ainihin mafita u(t) tana da isasshen santsi. Sa'an nan akwai akai C > 0, mai zaman kanta daga τ, irin wannan kuskuren e^n = u(t_n) - u^n ya gamsar da:

max₀≤n≤N ||e^n|| ≤ C τ²

inda N = T/τ shine adadin matakan lokaci.

Hujja tana amfani da binciken daidaito, sakamakon kwanciyar hankali, da kaddarorin kima na haɗin kai don sharuɗɗan rashin layi. An sami daidaiton oda na biyu saboda daidaiton tsarin sama-uku da kulawar rashin layi da kyau.

6. Hanyar Maimaitawa

An yi amfani da hanyar maimaitawa don nemo mafita mai kima ga kowane mataki na lokaci. Tsarin maimaitawa don warware matsala mai rabewa mai rashin layi a matakin lokaci n+1 an ba da shi ta hanyar:

(u^{n+1,k+1} - 2u^n + u^{n-1})/τ² + A u^n + M(||B u^n||²) u^n = f(t_n, u^n, (u^{n+1,k} - u^{n-1})/(2τ))

inda k ke nuna ma'auni na maimaitawa.

Theorem 6.1 (Haɗuwar Tsarin Maimaitawa): Ƙarƙashin sharuɗɗan da suka dace akan matakin lokaci τ da akai na Lipschitz na f, tsarin maimaitawa yana haɗuwa zuwa keɓantaccen mafita na matsala mai rabewa mai rashin layi a kowane matakin lokaci.

Hujja tana amfani da hujjojin ƙayyadaddun maki kuma tana amfani da kaddarorin kwanciyar hankali na masu aiki masu daidaitawa.

7. Muhimman Fahimta

8. Ƙarshe

Wannan aikin yana gabatar da cikakken bincike na tsarin sama-uku mai rabewa don kwatankwacin ma'auni na Ball integro-differential. Manyan gudunmawar sun haɗa da:

• Haɓaka tsarin sama-uku mai daidaito tare da kimanin haɗin kai don sharuɗɗan rashin layi
• Hujja na iyakar daidaitacce don mafita mai rabewa mai rashin layi da ma'anar bambancinta
• Samun kimantawa na farko na babban oda ta amfani da polynomials na Chebyshev
• Kafa kwanciyar hankali ga matsala mai rabewa mai rashin layi
• Bayar da kimantawan kuskure don mafita masu santsi
• Hujja na haɗuwa don hanyar maimaitawa da aka yi amfani da ita don warware tsarin rashin layi a kowane matakin lokaci

Sakamakon ya nuna cewa tsarin da aka gabatar yana da tasiri don kimanin mafita ga wannan ajin ma'auni na ci gaba mai rashin layi, tare da kiyaye kwanciyar hankali da daidaiton oda na biyu. Tsarin zance yana sa sakamakon ya dace da kewayon matsaloli na kankare a cikin ilimin lissafi na zahiri da aka kwatanta ta irin wannan ma'auni na integro-differential.

Hanyoyin bincike na gaba sun haɗa da faɗaɗawa zuwa tsarukan cikakken rabewa, dabarun matakan lokaci masu daidaitawa, da aikace-aikace ga takamaiman samfurori na zahiri kamar katako na viscoelastic da faranti.