Foundations — Boundary value problems — shooting method, finite difference

4.8.28 · D1 · Maths › Numerical Methods › Boundary value problems — shooting method, finite difference

Isse pehle ki aap parent note ki ek bhi line padh sakein, aapko har woh symbol apna banana hoga jo woh aap par phenkta hai. Hum unhe ek-ek karke build karte hain, har ek us picture se jiska woh naam leta hai.

1. Independent variable $x$ aur interval $[a,b]$

Ek flat rakha hua ruler imagine karo. Bayan (left) tick $x=a$ hai, daayan (right) tick $x=b$ hai. Jo kuch bhi hum karte hain woh is ruler par hota hai.

Figure 1 — ruler par interval. Interval $[a,b]$ ke liye ek seedha horizontal ruler: bayan tick $x=a$ marked hai aur daayan tick $x=b$ plum mein, beech ka shaded band (teal) dikhata hai "har woh position jis par humein dhyan dena hai wahan rahti hai." Abhi koi curve ya heights nahin — woh symbols Sections 2 aur 3 mein aayenge.

Yeh topic ko kyun chahiye: ek BVP is baare mein hai ki do siron ke beech kya hota hai. Woh do sire exactly $x=a$ aur $x=b$ hain.

2. Function $y$ aur uski value $y(x)$

Yeh topic ko kyun chahiye: parent $y(a)$ aur $y(b)$ likhta hai. Woh sirf "rope ki left height" aur "rope ki right height" hain.

3. Greek letters $\alpha$, $\beta$, $\gamma$, $s$

Figure 2 — pinned-end picture. Ab ki $y$, $\alpha$ aur $\beta$ define ho gaye hain, ek teal curve $y(x)$ ruler $[a,b]$ ke upar stretch ki gayi hai, orange dots uske left end ($y(a)=\alpha$) aur right end ($y(b)=\beta$) ko nail kar rahe hain; dono sire plum dashed lines se $x=a$ aur $x=b$ par mark hain.

Yeh topic ko kyun chahiye: IVP aur BVP ke beech poora fark yahi hai ki yeh numbers kahan rehte hain. Left par same $\alpha$, lekin BVP $\beta$ ko door wale end par pin karta hai jabki IVP same end par slope $\gamma$ ko pin karta hai.

4. Derivative $y^{\prime }$ — slope

Prime mark ${}^{\prime }$ ka sirf matlab hai "ki slope." $y^{\prime }$ ko "y-prime" padho.

Figure 3 — derivative as tangent ki steepness. Ek teal curve jis par ek orange tangent arrow ek plum point par touch kar raha hai; ek chhota plum dotted "rise over run" triangle dikhata hai ki $y^{\prime }$ exactly wahi rise divided by run hai.

Yeh topic ko kyun chahiye: shooting $y^{\prime }(a)$ (launch angle) guess karta hai. Agar aapne kabhi slope ko ek tangent arrow ki steepness ki tarah picture nahin ki, toh "missing slope guess karo" phrase meaningless hai.

5. Second derivative $y^{\prime \prime }$ — curvature

Yeh topic ko kyun chahiye: parent jo BVP study karta hai woh har ek second-order ODE hai — governing law $y^{\prime \prime }$ ke terms mein likha jaata hai. Finite-difference method ki core trick $y^{\prime \prime }$ ko teen dots par arithmetic mein badalna hai.

6. Higher derivatives $y^{\prime \prime \prime }$, $y^{\prime \prime \prime \prime }$ aur $y^{(n)}$

Yeh topic ko kyun chahiye: parent ki Taylor expansion mein $y^{\prime \prime \prime }$ aur $y^{\prime \prime \prime \prime }$ remainder terms mein hain. Wahi terms decide karte hain order of accuracy ($O(h^2)$), toh aapko pata hona chahiye ki woh simply "slopes-of-slopes-of-slopes" hain.

7. Differential equation $y^{\prime \prime }=f(x,y,y^{\prime })$

Ise physics ka ek law samjho jo keh raha ho: har point par, aap kitna bend karte ho woh dictated hai ki aap kahan hain aur aap kaise jhuke hain. Rope free nahin hai — use is rule ko apne infinitely many points par ek saath satisfy karna hai.

Yeh topic ko kyun chahiye: yahi problem hai. Dono methods sirf is rule aur dono end conditions maane wali rope dhundhne ke liye hain.

8. Linear vs nonlinear — $y^{\prime \prime }=p(x)y^{\prime }+q(x)y+r(x)$

Yeh topic ko kyun chahiye: parent apna poora treatment is word par split karta hai — "2-shot shortcut" aur tridiagonal setup dono linear form require karte hain.

9. Grid: $h$, $N$, $x_i$, $y_i$, aur subscripts

Figure 4 — rope ek beaded necklace ban jaati hai. Faint teal true rope jis par orange beads $y_0,\dots ,y_N$ ticks $x_0,\dots ,x_N$ par baithti hain; dono plum end-beads fixed boundary values hain, orange interior beads unknowns hain, aur ek labelled bracket step $h$ dikhata hai.

Yeh topic ko kyun chahiye: har finite-difference formula $y_{i-1},y_i,y_{i+1}$ ke saath likha jaata hai — ek bead aur uske do neighbours. Subscripts method ki bhasha hain.

10. Taylor series — woh tool jo stencils banata hai

Yeh topic ko kyun chahiye: central-difference stencils $\frac{y_{i+1}-y_{i-1}}{2h}$ aur $\frac{y_{i+1}-2y_i+y_{i-1}}{h^2}$ Taylor se derive hote hain. Taylor ke bina woh jaadu jaisi lagte hain.

11. Big-O aur order of accuracy: $O(h^2)$

Figure 5 — log–log paper par order of accuracy as a straight line. Error ko step size $h$ ke against plot kiya gaya hai dono axes log scale par: ek first-order $O(h)$ method (teal) slope 1 ki line par girta hai, jabki ek second-order $O(h^2)$ method (orange) do guni tezi se girta hai, slope 2 — $h$ half karna orange dot ko 4 ke factor se neeche shift karta hai.

Yeh topic ko kyun chahiye: "second-order" parent ka headline accuracy claim hai, aur isi wajah se woh saste $O(h)$ forward difference ke against warn karta hai. Dekho Truncation error and order of accuracy.

12. Root finding: $\varphi (s)=0$, notation $y(b;s)$, aur secant idea

Figure 6 — secant method $\varphi$ ka root dhundhta hai. Miss-curve $\varphi (s)$ guessed slope $s$ ke against plotted; do orange tried points ek plum secant line se join hain, aur jahan woh line horizontal axis (teal) ko cross karti hai woh agla, better guess $s_{n+1}$ hai — repeat karo aur aap root $\varphi =0$ par pahunch jaate ho.

Yeh topic ko kyun chahiye: shooting hai root finding stacked on an IVP solver — yeh vocabulary method ka aadha hissa hai.

13. Marching, IVP solvers, aur tridiagonal system $A\mathbf{y}=\mathbf{d}$

Figure 7 — tridiagonal system $A\mathbf{y}=\mathbf{d}$ laid out. Matrix $A$ sirf teen shaded stripes ke saath dikhaya gaya hai (orange diagonal, teal off-diagonals) aur baaki jagah zeros; unknown interior heights $y_1,\dots ,y_{N-1}$ ka vector $\mathbf{y}$; aur right-hand vector $\mathbf{d}$ jiske pehle aur aakhri entries plum boundary values $\alpha ,\beta$ carry karti hain.

Yeh topic ko kyun chahiye: shooting marching (RK4) reuse karta hai; finite difference yeh tridiagonal $A\mathbf{y}=\mathbf{d}$ produce karta hai. $A$, $\mathbf{y}$ aur $\mathbf{d}$ kya hold karte hain yeh jaanna parent ka setup padhne ke liye zaroori hai.

Prerequisite map

Equipment checklist

Daayaan side dhakein aur khud ko test karein. Agar koi bhi jawab fuzzy lage, parent note kholne se pehle uske section par waapis jaayein.