Diffference Equation vs Differential Equation
Ib yam tshwm sim ntuj yuav raug piav qhia txog lej los ntawm kev ua haujlwm ntawm ntau qhov sib txawv thiab qhov tsis muaj. Tshwj xeeb tshaj yog thaum lawv tau hais los ntawm kev ua haujlwm ntawm spatial txoj haujlwm thiab lub sijhawm nws ua rau muaj kev sib npaug. Qhov kev ua haujlwm yuav hloov nrog kev hloov pauv hauv cov kev hloov pauv ywj pheej lossis cov tsis muaj. Ib qho kev hloov pauv tsis kawg tshwm sim hauv kev ua haujlwm thaum ib qho ntawm nws qhov hloov pauv tau hloov pauv yog hu ua cov txiaj ntsig ntawm qhov haujlwm ntawd.
Ib qho kev sib npaug sib txawv yog ib qho kev sib npaug uas muaj cov txiaj ntsig ntawm kev ua haujlwm nrog rau kev ua haujlwm nws tus kheej. Ib qho kev sib npaug sib txawv yooj yim yog qhov ntawm Newton Txoj Cai Thib Ob ntawm Kev Ua Haujlwm. Yog hais tias ib yam khoom ntawm huab hwm coj m tsiv nrog acceleration 'a' thiab raug ua nrog quab yuam F ces Newton Txoj Cai Thib Ob qhia peb tias F=ma. Ntawm no dua, 'a' sib txawv nrog lub sijhawm, peb tuaj yeem sau 'a' li; ib=dv/dt; v e zog. Kev nrawm yog qhov ua haujlwm ntawm qhov chaw thiab lub sijhawm, uas yog v=ds / dt; yog li ntawd 'a'=d2s/dt2
Ua raws li cov no hauv siab peb tuaj yeem sau Newton txoj cai thib ob raws li kev sib npaug sib txawv;
'F' as a function of v and t – F(v, t)=mdv/dt, or
'F' as a function of s and t – F(s, ds/dt, t)=m d2s/dt2
Muaj ob hom kev sib npaug sib txawv; Cov kab zauv sib txawv zoo tib yam, luv luv los ntawm ODE lossis ib feem ntawm qhov sib txawv, luv luv los ntawm PDE. Ordinary differential equation yuav muaj cov derivatives zoo tib yam (derivatives ntawm ib qho txawv txav) hauv nws. Ib feem sib npaug sib npaug yuav muaj qhov sib txawv ntawm qhov sib txawv (derivatives ntawm ntau tshaj ib qho sib txawv) hauv nws.
eg. F=m d2s/dt2 is ib ODE, whereas α2 d 2u/dx2=du/dt yog PDE, nws muaj derivatives ntawm t thiab x.
Qhov sib txawv sib txawv yog tib yam li qhov sib txawv sib txawv tab sis peb saib nws hauv cov ntsiab lus sib txawv. Hauv kev sib npaug sib txawv, qhov txawv txav txawv xws li lub sijhawm raug suav hais tias nyob rau hauv cov ntsiab lus ntawm lub sijhawm txuas ntxiv. Hauv kev sib cais lub sijhawm, peb hu ua qhov sib txawv raws li qhov sib npaug.
Difference equation is a function of sib txawv. Qhov sib txawv ntawm kev ywj pheej sib txawv yog peb hom; kev ua ntu zus ntawm tus lej, sib cais dynamical system thiab iterated muaj nuj nqi.
Nyob rau hauv cov lej qhov kev hloov pauv yog tsim rov qab siv txoj cai los txheeb xyuas txhua tus lej hauv ib ntu rau cov lej dhau los hauv kab ke.
Qhov sib txawv ntawm qhov sib txawv hauv qhov kev sib cais dynamical siv qee qhov kev tawm tswv yim tawm tswv yim thiab tsim cov teeb liab tawm.
Difference equation yog ib daim ntawv qhia iterated rau iterated muaj nuj nqi. E.g., y0, f(y0), f(f (y0)), f(f(f(y0))),…. Tus f(y0) yog thawj iterate ntawm y0 Tus k-th iterate yuav denoted los ntawm fk (y0).
