4.7.9 · D1Separation of variables se heat equation solve karna
Heat failti hai kyunki har point apne neighbours ke average ki taraf drift karta hai, aur jitna zyada temperature kisi point par bend karti hai, utna hi tezi se wo point change hota hai. Separation of variables ek mushkil equation — jo space-aur-time-ek-saath ke baare mein hai — ko do aasan equations mein badal deta hai — ek purely shape in space ke baare mein, aur ek purely time mein fading ke baare mein.
Parent derivation ki ek bhi line padhne se pehle, tumhe usmein har squiggle ko khud se jaanna hoga. Yeh page unhe ek-ek karke, jis order mein woh ek doosre par build hote hain, walk-through karta hai. Koi bhi cheez use hone se pehle draw ki jaati hai.
u ( x , t ) u temperature hai. Yeh do inputs par depend karta hai:
x position (left end se kitni door), 0 L t time (kitne seconds guzre hain), ek number ≥ 0 u ( x , t ) "Jo point x t
Picture ek curve nahi hai — yeh ek poora landscape hai. Socho rod zameen par rakh di gayi hai (x t
Intuition Do variables kyun hain?Ek akeli hot rod ko already x move karti hai, toh jaise t
Definition Derivative — ek curve ki steepnessEk curve lo. Ek point par uska derivative woh line ka slope hai jo wahan curve ko sirf kiss karti hai (tangent): right ki taraf ek step mein height kitni tezi se badhti hai.
Slope > 0
Slope < 0
Slope = 0
Slope ki zarurat kyun hai? Kyunki heat temperature ke slopes se neeche flow karti hai — hot se cold ki taraf. Slope literally "heat ki hill ki steepness" hai, aur physics kehta hai heat downhill roll karti hai.
Kyunki u do inputs par depend karta hai, "slope" ambiguous hai — kaunsi direction mein slope? Toh hum aaise slopes invent karte hain jo ek input ko frozen rakhte hain.
∂ ∂ partial derivative likhte hain, ordinary straight d d ∂ yeh input change ho raha hai jabki baaki frozen hain."
∂ t ∂ u sirf time ko nudge karke aur x u t ∂ x ∂ u sirf position ko nudge karke, t u x x ∂ x 2 ∂ 2 u u xx u t u x u xx ∂ u / ∂ t bilkul same cheez hai — subscript sirf shorthand hai.
u t u t = ∂ t ∂ u ek position x Yeh us spot par temperature-versus-time graph ka slope hai. Positive u t
u xx bump (concave-down) par woh point apne do neighbours ke average se zyada hot hota hai, toh heat dono taraf leak hoti hai — yeh cool hota hai. Ek dip (concave-up) par yeh apne neighbours se zyada thanda hota hai, toh heat andar flow hoti hai — yeh warm hota hai. Curvature u xx u xx u x
u xx Galat kyun hai: u x kis direction mein flow hoti hai; lekin ek seedhi tilted line (u x = 0 u xx = 0 bending (u xx = 0
α 2 α 2 positive number hai jo batata hai yeh material heat conduct karne mein kitna achha hai . Bada α 2 α 2 length 2 / time α 2
α 2 α square likhne se guarantee hoti hai ki yeh positive hai (ek real number ka square kabhi negative nahi hota). Physically, diffusion always positive hoti hai, toh notation yeh bake in karta hai.
Equation akele infinitely many solutions rakhti hai. Hum ek ko extra facts se pin down karte hain.
L L rod ki length . Position x ∈ [ 0 , L ] 0 L included hain — hume unhe ends par rules batane ke liye chahiye honge). Time t ≥ 0
Definition Boundary conditions (BCs) — ends par kya hota haiRules jo x = 0 x = L har waqt fixed hain. Yahan:
u ( 0 , t ) = 0 , u ( L , t ) = 0. 0 ∘ Dirichlet type hai: value fixed. Contrast: slope u x = 0 Neumann type hai. Dekho Boundary Conditions — Dirichlet vs Neumann .
Definition Initial condition (IC) — starting snapshotEk rule jo t = 0 sabhi positions ke liye fixed hai:
u ( x , 0 ) = f ( x ) . f ( x ) starting temperature profile hai — rod ki heat ki shape clock shuru hone se pehle.
Intuition BC vs IC — figure padhoStrip ki do vertical red edges boundary conditions hain (har waqt 0 Bottom horizontal edge initial condition hai (t = 0
X ( x ) T ( t ) guess hai answer ki form ke baare mein:
u ( x , t ) = X ( x ) T ( t ) .
X ( x ) sirf position ka function hai — rod ki ek fixed shape (jaise ek hump). Use time ke baare mein pata nahi.T ( t ) sirf time ka function hai — ek akela dial jo us shape ko upar ya neeche scale karta hai jaise clock chalta hai. Use position ke baare mein pata nahi.
Toh guess kehta hai: rod same shape rakhti hai aur sirf time ke saath use brighten ya dim karti hai.
Intuition Hum yeh guess karne ki permission kyun rakhte hain?Hum yeh claim nahi kar rahe ki har solution ek shape × ek dial jaisi dikhti hai — zyatir nahi dikhti. Lekin sabse simple possible form try karne mein kuch nahi jaata. Agar kaam kare, toh ek mushkil do-variable equation do aasan ek-variable equations mein split ho jaati hai (ek X T Superposition Principle humein kai aisi simple pieces add karne deta hai kisi bhi starting profile ko build karne ke liye. Picture: shape X ( x ) T ( t )
sin sin θ 0 1 0 − 1 cos θ cos 0 = 1
Yahan sines kyun? Kyunki ends zero hone chahiye. sin x = 0 0 x = L 0
n sin L nπ x X n ( x ) = sin L nπ x , n = 1 , 2 , 3 , …
n = 1 n = 2 n = 3 L nπ sin ( 0 ) = 0 aur sin L nπ L = sin ( nπ ) = 0 π L X ( x )
Intuition Cosines yahan banned kyun haincos 0 = 1 = 0 u ( 0 , t ) = 0 hand-pick karte hain sines. (Insulated ends par flip karo aur cosines wapas aate hain — eigenfunction hamesha boundary rule se match karti hai.)
∑ ∑ n = 1 ∞ n = 1 n = 2 n = 3 X n ( x ) = sin L nπ x f ( x ) Fourier Series hai.
e − λ t T ( t ) e ≈ 2.718 e − λ t 1 t = 0 0 λ hi har shape ke liye time-dial T ( t )
λ n = ( nπ / L ) 2 λ n = ( nπ / L ) 2 n 2 3 1 9 × u xx n = 1
Definition Superposition — solutions ka sum bhi solution hota haiKyunki heat equation linear hai (koi u homogeneous hai ("= 0 Superposition Principle hai: yeh humein ek hump-times-dial ek waqt mein solve karne deta hai, phir add karne deta hai. Zero ends ke saath X Sturm-Liouville Theory ka ek baby case hai.
B n f ( x ) = ∑ n = 1 ∞ B n sin L nπ x , B n number hai jo batata hai hump n — f ( x ) n B n = L 2 ∫ 0 L f ( x ) sin L nπ x d x , f n B n e − λ n α 2 t
Self-test: kya tum parent note padhne se pehle har ek ko ek saansh mein bol sakte ho?
u ( x , t ) Rod ke saath position x t
Curly ∂ ∂ u / ∂ t u t Yeh partial derivative mark karta hai — slope jab sirf ek input change hota hai aur baaki frozen hain; u t ∂ u / ∂ t
u t Woh rate jisme ek fixed point ki temperature time mein change hoti hai (frozen x
u xx Snapshot ki curvature; u xx > 0 u xx < 0
α 2 Use positive force karne ke liye (diffusivity always positive hoti hai) aur units length 2 / time
Boundary condition aur initial condition mein kya difference hai? BC ends ko har waqt fix karta hai (x = 0 , L t = 0
Separation guess kya hai, aur X ( x ) T ( t ) u ( x , t ) = X ( x ) T ( t ) X T
Hum separation guess karne ki permission kyun rakhte hain? Sabse simple form try karne mein kuch nahi jaata; agar kaam kare toh PDE do aasan ODEs mein split ho jaata hai, aur superposition humein kisi bhi profile ke liye kai aisi pieces add karne deta hai.
Zero-value (Dirichlet) ends sine eigenfunctions kyun force karte hain, cosines kyun nahi? sin 0 = 0 cos 0 = 1 = 0
Sine ke andar L nπ Taaki sin L nπ x x = 0 x = L sin nπ = 0
Mode n n 2 Uska decay constant λ n = ( nπ / L ) 2
Coefficient B n f ( x ) n f sin L nπ x
Kaunsi property humein single-hump solutions ko saath add karne deti hai? Heat equation ki linearity + homogeneity → superposition principle.
