Ngā Whārite Pārōnaki Noa
Pengantar
Ko ngā whārite rerekētanga noa (ODE) he peka o te pāngarau e ako ana i ngā whanaungatanga i waenga i ngā mahi me ō rātou pānga. He mea nui tēnei ariā ki te pūtaiao me te hangarau, nā te mea he maha ngā āhuatanga taiao me ngā āhuatanga i hangaia e te tangata ka taea te whakatauira mā te whakamahi i ngā ODE.
I mua i tā tātou ruku atu, me tīmata tātou me ētahi whakamāramatanga taketake. Ko te GDP he whārite pāngarau e hono ana i tētahi mahi ki ōna pānga. Ko tētahi tauira māmā o te GDP ko:
\[ \frac{dy}{dx} = ky \]
ko te \(y\) he pānga o te taurangi \(x\), ā, ko te \(k\) he pūmau.
Whakarōpūtanga GDP
He maha ngā huarahi e taea ai te whakarōpū i te GDP, i runga i tōna taumata, ahakoa he rārangi, he kore rānei, he ōrite rānei.
Teitei o te GDP
Ka whakatauhia te taumata GDP e te tauwehenga teitei rawa atu e puta ana i roto i te whārite. Hei tauira:
1. GDP Tūnga Tuatahi: \( \frac{dy}{dx} + y = 0 \)
2. GDP Tuarua o te Tūnga: \( \frac{d^2y}{dx^2} – 3\frac{dy}{dx} + 2y = 0 \)
Te rārangitanga
Ka kiia te GDP he rārangi mēnā he rārangi tōna āhua e pā ana ki te mahi me ōna pānga katoa. Hei tauira:
1. GDP Rārangi: \( \frac{dy}{dx} + p(x)y = q(x) \)
2. GDP kore-raina: \( \frac{dy}{dx} + y^2 = x \)
Te ōritetanga
Ko te GDP ōrite he whārite e whakareatia ana ia kupu e uru ana ki tētahi mahi me tōna pānga ki tētahi pūmau. I tētahi atu taha, ki te mea he kupu kāore e rārangi ana ki te mahi, ki tōna pānga rānei, kāore te GDP i te ōrite.
1. GDP ōrite: \( \frac{dy}{dx} + py = 0 \)
2. GDP kore-ōrite: \( \frac{dy}{dx} + py = g(x) \)
Tikanga Otinga GDP
He maha ngā tikanga rerekē hei whakaoti i te GDP, i runga i te momo me ngā āhuatanga o te whārite. Ko ētahi o ngā tikanga noa ko te tikanga wehewehe taurangi, te tikanga tauwehe whakauru, me te panoni Laplace.
Te Wehewehenga o ngā Taurangi
Ka whakamahia tēnei tikanga mō te GDP ina taea te wehewehe i ngā taurangi motuhake me ngā taurangi whakawhirinaki ki ngā taha e rua o te whārite. Hei tauira:
\[ \frac{dy}{dx} = g(x)h(y) \]
Ngā taahiraa hei whakaoti:
1. Wehewehea ngā taurangi: \( \frac{1}{h(y)} dy = g(x) dx \)
2. Whakaurua ngā taha e rua: \( \int \frac{1}{h(y)} dy = \int g(x) dx \)
Tikanga Tauwehe Whakauru
Ka whakamahia tēnei tikanga hei whakaoti i te PDB rārangi tuatahi-raupapa i te āhua paerewa:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
Ngā taahiraa hei whakaoti:
1. Whakatauhia te tauwehenga whakauru \(\mu(x) = e^{\int p(x) dx} \)
2. Whakareatia te whārite taketake ki te \(\mu(x)\)
3. Whakaurua ngā taha e rua kia taea ai te whakaoti i te whārite mō \(y\).
Te panonitanga o Laplace
He taputapu kaha te panonitanga Laplace mō te whakaoti rapanga GDP, inā koa ko ērā e pā ana ki ngā āhuatanga tīmatanga. Ka hurihia e te panonitanga Laplace tētahi whārite rerekētanga i roto i te rohe wā hei whārite taurangi i roto i te rohe auau.
Mō te GDP:
\[ \frac{d^2y}{dt^2} + 5\frac{dy}{dt} + 6y = 0, \quad y(0) = 2, \quad \frac{dy}{dt}(0) = 0 \]
Ka taea e tātou te whakamahi i te panonitanga Laplace:
\[ s^2 Y(s) – sy(0) – y'(0) + 5sY(s) – 5y(0) + 6Y(s) = 0 \]
Kātahi, i muri i te whakamahi i ngā tikanga tīmatanga, ka taea e tātou te whakaoti mō te \(Y(s)\) me te mahi i te panoni Laplace whakamuri hei whiwhi i te \(y(t)\).
Tono GDP
He whānuitia ngā tono a te PDB i roto i ngā momo mara o te pūtaiao taiao me te hangarau.
Ahupūngao
I roto i te ahupūngao, ka whakamahia te GDP hei whakaahua i ngā pūnaha hihiri. Hei tauira, ko te ture tuarua a Newton \( F = ma \), i roto i te āhua GDP ko:
\[ m\frac{d^2x}{dt^2} = F(x,v,t) \]
ko \(x\) te tūnga, ko \(v\) te tere, ko \(m\) te papatipu, ā, ko \(F\) he kaha e whakawhirinaki ana ki te tūnga, te tere, me te wā.
Biology
I roto i te koiora, he maha ngā wā ka whakamahia e ngā tauira tipu taupori te GDP. Ko ngā tauira noa ko te tauira tipu taupū me te tauira tipu logistic:
1. Taupūnga: \( \frac{dP}{dt} = rP \)
2. Te Whakahaere: \( \frac{dP}{dt} = rP\left(1 – \frac{P}{K}\right) \)
ko \(P\) te taupori, ko \(r\) te tere tipu, ā, ko \(K\) te kaha taiao mōrahi.
ōhanga
I roto i te ōhanga, he maha ngā wā ka whakamahia te GDP e ngā tauira tipu ōhanga me ngā tauira whakauru-putanga. Hei tauira, i roto i te tauira Solow:
\[ \frac{dk(t)}{dt} = sf(k) – (n + \delta) k \]
ko \(k(t)\) te whakapaipai mō ia kaimahi, ko \(s\) te reiti penapena, ko \(f(k)\) te mahi whakaputa, ko \(n\) te reiti tipu o te taupori, ā, ko \(\delta\) te reiti whakahekenga uara whakapaipai.
hangarau
I roto i te hangarau hiko, ko te tātari i ngā ara iahiko RC, RL, me te RLC e whakamahi ana i te PDB hei whakatau i te urupare a te ara iahiko ki ngā tāurunga tohu rerekē.
Tauira mō te ara iahiko RC:
\[ V(t) = R \frac{dq}{dt} + \frac{q}{C} \]
ko te ngaohiko te \(V(t)\), ko te ātete te \(R\), ko te utu te \(q\), ā, ko te kaha te \(C\).
Ngā Tikanga Whakatauira me te Tau
Heoi, kāore e taea te whakaoti i ngā GDP katoa mā te tātari. I roto i te nuinga o ngā wā, me whakamahi tātou i ngā tikanga tau hei whiwhi otinga. Ko te tikanga a Euler, ko te tikanga a Runge-Kutta, me te tikanga maha-taahiraa ētahi o ngā tikanga tau rongonui e whakamahia whānuitia ana.
Te tikanga a Euler
Ko te tikanga a Euler te huarahi māmā rawa atu, ā, e whakamahia ana hei whakarato i tētahi whakaaro whānui mō te whanonga o tētahi otinga PDB. Ka whakamahia e tēnei tikanga he whakatata rārangi mō ia taahiraa iti i roto i tētahi wā kua whakaritea.
Tikanga Runge-Kutta
Ko te tikanga Runge-Kutta, inā koa te tikanga tuawhā (RK4), he tika ake, ā, he whānui ake te whakamahinga i roto i ngā mahi whaihua. E whā ngā taahiraa e whakamahia ana e tēnei tikanga mō ia wā hei whakatau tata ake i te otinga.
Te Katinga
He mea nui te mārama ki ngā whārite rerekētanga noa mō te hunga e mahi ana i roto i te pūtaiao, te miihini, te ōhanga, me te maha atu o ngā kaupapa ako. Nā te whānuitanga o ngā tikanga me ngā tono, ka whakaratohia e te PDB he taputapu kaha mō te whakatauira me te mārama ki ngā āhuatanga uaua.
