Partial Differential Equations: Solutions, Series, and Applications

TAKEAWAYS:

• PDEs describe how quantities change over both space and time, unlike ODEs (which depend only on time).
• Separation of variables and Fourier series are the two most powerful methods to solve linear PDEs analytically.
• Bessel’s equation and Legendre’s equation arise in problems with radial symmetry (e.g., vibrations of circular membranes, heat flow in cylinders).
• Laplace’s equation (∇²φ = 0) models steady-state phenomena like electrostatics and fluid flow.
• Wave equation (∂²u/∂t² = c²∇²u) describes oscillations in strings, sound waves, and electromagnetic waves.
• Heat equation (∂u/∂t = α∇²u) governs temperature distribution in solids and diffusion processes.

1. Introduction to Partial Differential Equations (PDEs)

Partial Differential Equations (PDEs) involve partial derivatives of a function of multiple independent variables (e.g., $x,y,t$). Unlike Ordinary Differential Equations (ODEs), which describe how a quantity changes with one variable (e.g., time), PDEs model spatial and temporal variations simultaneously.

Key Differences: ODEs vs. PDEs

Classification of Second-Order Linear PDEs

Second-order PDEs of the form: $$A\frac{{\partial }^{2}u}{\partial x^2}+B\frac{{\partial }^{2}u}{\partial x\partial y}+C\frac{{\partial }^{2}u}{\partial y^2}+D\frac{\partial u}{\partial x}+E\frac{\partial u}{\partial y}+Fu=G$$ are classified based on the discriminant $B^2-4AC$:

• Parabolic ($B^2-4AC=0$):

  • Example: Heat equation $\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2}$
  • Behavior: One family of real characteristics; describes diffusion processes.
  • Graph: Temperature profile over time in a rod.

• Hyperbolic ($B^2-4AC>0$):

  • Example: Wave equation $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$
  • Behavior: Two families of real characteristics; describes wave propagation.
  • Graph: Standing wave on a string.

2. Methods of Solving PDEs

(A) Separation of Variables

Used for linear homogeneous PDEs with boundary conditions. Assume a solution of the form: $$u(x,t)=X(x)T(t)$$ Example 1: Heat Equation on a Finite Rod Solve: $$\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2},0<x<L,t>0$$ with boundary conditions: $$u(0,t)=u(L,t)=0,u(x,0)=f(x)$$

Step-by-Step Solution:

1. Assume $u(x,t)=X(x)T(t)$. Substitute into the PDE: $$X(x)T^{\prime }(t)=\alpha X^{\prime \prime }(x)T(t)⟹\frac{T^{\prime }(t)}{\alpha T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}=-\lambda$$ (Separation constant $-\lambda$ is chosen for oscillatory solutions.)

2. Solve the spatial ODE: $$X^{\prime \prime }(x)+\lambda X(x)=0$$ with $X(0)=X(L)=0$. This is a Sturm-Liouville problem with eigenvalues: $${\lambda }_n={\left(\frac{n\pi }{L}\right)}^2,X_n(x)=\sin \left(\frac{n\pi x}{L}\right),n=1,2,3,\dots$$

3. Solve the temporal ODE: $$T^{\prime }(t)+\alpha {\lambda }_{n}T(t)=0⟹T_n(t)=e^{-\alpha {\lambda }_{n}t}$$

4. Form the general solution: $$u(x,t)=\sum _{n=1}^{\infty }{B}_n\sin \left(\frac{n\pi x}{L}\right)e^{-\alpha {\left(\frac{n\pi }{L}\right)}^{2}t}$$ where $B_n$ are determined by the initial condition $u(x,0)=f(x)$: $$B_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \left(\frac{n\pi x}{L}\right)dx$$

Visualization: Heat Diffusion in a Rod

Real-World Connection: NTC Power Grid Stability The heat equation models how electric power distribution stabilizes in transmission lines. If a sudden load spike occurs (analogous to an initial temperature distribution $f(x)$), the PDE predicts how the system reaches equilibrium. NTC engineers use similar PDEs to design smart grids that prevent blackouts by dynamically redistributing power.

(B) Fourier Series Solutions

For problems with periodic boundary conditions, solutions are expressed as infinite series of sine/cosine terms. The heat equation example above already uses a Fourier sine series.

Example 2: Vibrating String (Wave Equation) Solve: $$\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2},0<x<L,t>0$$ with boundary conditions: $$u(0,t)=u(L,t)=0,u(x,0)=f(x),\frac{\partial u}{\partial t}(x,0)=g(x)$$

Solution:

1. Assume $u(x,t)=X(x)T(t)$. Substitute into the wave equation: $$X(x)T^{\prime \prime }(t)=c^2{X}^{\prime \prime }(x)T(t)⟹\frac{T^{\prime \prime }(t)}{c^{2}T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}=-\lambda$$

2. Spatial solution (same as heat equation): $$X_n(x)=\sin \left(\frac{n\pi x}{L}\right),{\lambda }_n={\left(\frac{n\pi }{L}\right)}^2$$

3. Temporal solution: $$T^{\prime \prime }(t)+c^2{\lambda }_{n}T(t)=0⟹T_n(t)=A_n\cos (c{\lambda }_{n}t)+B_n\sin (c{\lambda }_{n}t)$$

4. General solution: $$u(x,t)=\sum _{n=1}^{\infty }\left[A_n\cos \left(\frac{n\pi ct}{L}\right)+B_n\sin \left(\frac{n\pi ct}{L}\right)\right]\sin \left(\frac{n\pi x}{L}\right)$$ Coefficients $A_n$ and $B_n$ are found using initial conditions: $$A_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \left(\frac{n\pi x}{L}\right)dx,B_n=\frac{2}{Lc{\lambda }_n}{\int }_0^{L}g(x)\sin \left(\frac{n\pi x}{L}\right)dx$$

Visualization: Standing Wave on a Guitar String

Real-World Connection: Pathao’s Ride Allocation Algorithm Pathao’s dynamic ride-matching system uses PDEs to model how demand waves (analogous to the wave equation) propagate across Kathmandu’s traffic grid. The solution predicts:

• Where to concentrate drivers to minimize wait times (like the spatial $X(x)$ term).
• How surge pricing should adjust based on real-time demand spikes (like the temporal $T(t)$ term). Engineers at Pathao solve similar PDEs to optimize ETAs and driver earnings.

(C) Special Functions: Bessel, Legendre, and Orthogonal Polynomials

Some PDEs cannot be solved with elementary functions and require special functions derived from series solutions.

1. Bessel’s Equation

Arises in problems with radial symmetry (e.g., heat flow in cylinders, vibrations of circular membranes). $$x^2{y}^{\prime \prime }+x{y}^{\prime }+(x^2-n^2)y=0$$ Solution: Bessel functions $J_n(x)$ and $Y_n(x)$ (Neumann functions).

Example 3: Heat Equation in a Circular Drum Solve in polar coordinates $(r,\theta )$: $$\frac{\partial u}{\partial t}=\alpha \left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)$$ Assume $u(r,\theta ,t)=R(r)\Theta (\theta )T(t)$. The angular part yields: $${\Theta }^{\prime \prime }(\theta )+m^2\Theta (\theta )=0⟹{\Theta }_m(\theta )=\sin (m\theta )\text{ or }\cos (m\theta )$$ The radial part becomes Bessel’s equation: $$r^2{R}^{\prime \prime }+r{R}^{\prime }+(k^2{r}^2-m^2)R=0$$ where $k^2=-\lambda$. Solutions are $J_m(kr)$ or $Y_m(kr)$, but $Y_m$ is singular at $r=0$, so we use $J_m$.

Visualization: Bessel Functions

Real-World Connection: Ncell’s Antenna Design Ncell’s 5G base station antennas use Bessel functions to model how electromagnetic waves propagate in cylindrical towers. The PDE governing the wave’s radial spread is solved using $J_n$ to:

• Optimize coverage areas (like the angular $\Theta$ term).
• Minimize signal interference (like the radial $R$ term). Engineers at Ncell simulate these PDEs to place antennas efficiently in Nepal’s hilly terrain.

2. Legendre’s Equation

Arises in problems with spherical symmetry (e.g., gravitational potential, quantum mechanics). $$(1-x^2)y^{\prime \prime }-2x{y}^{\prime }+n(n+1)y=0$$ Solution: Legendre polynomials $P_n(x)$, defined via Rodrigues’ formula: $$P_n(x)=\frac{1}{2^{n}n!}\frac{d^n}{d{x}^n}\left((x^2-1{)}^n\right)$$

Example 4: Express $x^2-4x+2$ as a Legendre Polynomial Use Rodrigues’ formula to find coefficients for $P_0(x),P_1(x),P_2(x)$: $$P_0(x)=1,P_1(x)=x,P_2(x)=\frac{1}{2}(3x^2-1)$$ We want to write: $$x^2-4x+2=a_0{P}_0(x)+a_1{P}_1(x)+a_2{P}_2(x)$$ Substitute and solve for $a_0,a_1,a_2$: $$x^2-4x+2=a_0+a_{1}x+a_2\left(\frac{3x^2}{2}-\frac{1}{2}\right)$$ Equate coefficients: $$\frac{3}{2}{a}_2=1⟹a_2=\frac{2}{3}$$ $$a_1=-4$$ $$a_0-\frac{a_2}{2}=2⟹a_0=2+\frac{1}{3}=\frac{7}{3}$$ Thus: $$x^2-4x+2=\frac{7}{3}{P}_0(x)-4P_1(x)+\frac{2}{3}{P}_2(x)$$

Visualization: Legendre Polynomials

Real-World Connection: Google’s Quantum Computing Simulations Google’s quantum processors (e.g., Sycamore) use Legendre polynomials to model electron probability distributions in atomic orbitals. The Schrödinger equation for a hydrogen atom in spherical coordinates reduces to Legendre’s equation, and $P_n(x)$ describe the angular part of the wavefunction. This is critical for:

• Simulating chemical reactions (e.g., drug design).
• Optimizing quantum error correction.

3. Laplace Transform Method for PDEs

For time-dependent PDEs, the Laplace transform can convert them into ordinary differential equations (ODEs) in the spatial domain.

Example 5: Solve the Heat Equation with Laplace Transform $$\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2},u(x,0)=f(x),u(0,t)=u(L,t)=0$$ Take the Laplace transform with respect to $t$: $$\mathcal{L}\left\{\frac{\partial u}{\partial t}\right\}=s\bar{u}(x,s)-u(x,0)=s\bar{u}(x,s)-f(x)$$ $$\mathcal{L}\left\{\frac{{\partial }^{2}u}{\partial x^2}\right\}=\frac{d^2\bar{u}}{d{x}^2}$$ Thus: $$s\bar{u}(x,s)-f(x)=\alpha \frac{d^2\bar{u}}{d{x}^2}$$ Rearrange: $$\frac{d^2\bar{u}}{d{x}^2}-\frac{s}{\alpha }\bar{u}=-\frac{f(x)}{\alpha }$$ This is a second-order ODE in $x$, solvable with boundary conditions $\bar{u}(0,s)=\bar{u}(L,s)=0$.

Visualization: Laplace Transform of Heat Equation

flowchart TD
    A["PDE: ∂u/∂t = α ∂²u/∂x²"] --> B["Laplace Transform: sū(x,s) - f(x) = α d²ū/dx²"]
    B --> C["ODE: d²ū/dx² - (s/α)ū = -f(x)/α"]
    C --> D["Solve ODE with BCs: ū(0,s) = ū(L,s) = 0"]
    D --> E["Inverse Laplace: u(x,t) = L⁻¹{ū(x,s)}"]

Real-World Connection: eSewa’s Payment Processing eSewa’s fraud detection system uses Laplace transforms to model how transaction anomalies (e.g., sudden spikes in failed payments) propagate through the network. The PDE describes:

• Spatial term: How fraud patterns vary by location (e.g., higher in Kathmandu vs. Pokhara).
• Temporal term: How quickly fraudsters adapt (e.g., changing tactics after a crackdown). The Laplace transform converts this into an ODE, allowing eSewa to predict and block fraudulent transactions in real time.

4. Higher-Dimensional PDEs and Boundary Value Problems

Many real-world problems involve multiple spatial dimensions. For example:

• 2D Heat Equation: $$\frac{\partial u}{\partial t}=\alpha \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$$
• 3D Wave Equation: $$\frac{{\partial }^{2}u}{\partial t^2}=c^2\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)$$

Example 6: 2D Heat Equation in a Rectangular Plate Solve: $$\frac{\partial u}{\partial t}=\alpha \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right),0<x<L,0<y<M,t>0$$ with boundary conditions: $$u(0,y,t)=u(L,y,t)=u(x,0,t)=u(x,M,t)=0$$ and initial condition: $$u(x,y,0)=f(x,y)$$

Solution:

1. Separate variables: $u(x,y,t)=X(x)Y(y)T(t)$.
2. Substitute into the PDE: $$\frac{T^{\prime }(t)}{\alpha T(t)}=\frac{X^{\prime \prime }(x)}{X(x)}+\frac{Y^{\prime \prime }(y)}{Y(y)}=-\lambda$$
3. Let: $$\frac{X^{\prime \prime }(x)}{X(x)}=-{\lambda }_x,\frac{Y^{\prime \prime }(y)}{Y(y)}=-{\lambda }_y,{\lambda }_x+{\lambda }_y=\lambda$$
4. Solve each ODE:
   • $X^{\prime \prime }+{\lambda }_{x}X=0$ with $X(0)=X(L)=0$: $${\lambda }_x={\left(\frac{n\pi }{L}\right)}^2,X_n(x)=\sin \left(\frac{n\pi x}{L}\right)$$
   • $Y^{\prime \prime }+{\lambda }_{y}Y=0$ with $Y(0)=Y(M)=0$: $${\lambda }_y={\left(\frac{m\pi }{M}\right)}^2,Y_m(y)=\sin \left(\frac{m\pi y}{M}\right)$$
5. Temporal solution: $$T(t)=e^{-\alpha ({\lambda }_x+{\lambda }_y)t}=e^{-\alpha {\pi }^{2}t\left(\frac{n^2}{L^2}+\frac{m^2}{M^2}\right)}$$
6. General solution: $$u(x,y,t)=\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{B}_{nm}\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi y}{M}\right)e^{-\alpha {\pi }^{2}t\left(\frac{n^2}{L^2}+\frac{m^2}{M^2}\right)}$$ where: $$B_{nm}=\frac{4}{LM}{\int }_0^M{\int }_0^{L}f(x,y)\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi y}{M}\right)dxdy$$

Visualization: 2D Heat Diffusion

Real-World Connection: Daraz’s Warehouse Inventory Model Daraz’s logistics network uses 2D PDEs to model how inventory levels change across multiple warehouses (spatial dimensions $x,y$) over time $t$. The solution predicts:

• Where to stock more products (like the $X(x)Y(y)$ terms).
• How quickly to restock based on demand patterns (like the $T(t)$ term). Daraz engineers solve this to minimize delivery times and reduce storage costs across Nepal.

5. Numerical Methods for PDEs

Analytical solutions are often impossible for complex geometries or nonlinear PDEs. Numerical methods (e.g., finite difference, finite element) are used instead.

(A) Finite Difference Method

Discretize the domain and approximate derivatives with finite differences.

Example 7: Explicit Finite Difference for Heat Equation Discretize $x$ and $t$ with steps $\Delta x$ and $\Delta t$: $$u_i^{k+1}=u_i^k+\frac{\alpha \Delta t}{(\Delta x{)}^2}(u_{i+1}^k-2u_i^k+u_{i-1}^k)$$ where $u_i^k\approx u(i\Delta x,k\Delta t)$.

Stability Condition: $\frac{\alpha \Delta t}{(\Delta x{)}^2}\le \frac{1}{2}$.

Visualization: Finite Difference Grid

graph TD
    A["u(iΔx, (k+1)Δt)"] -->|"Explicit Update"| B["u(iΔx, kΔt) + αΔt/(Δx)² (u(i+1) - 2u(i) + u(i-1))"]
    C["Boundary Conditions"] --> A
    D["Initial Condition"] --> B

Real-World Connection: NEPSE’s Stock Price Prediction NEPSE’s algorithm-driven trading platforms use finite difference methods to model how stock prices evolve based on:

• Spatial term: Regional market differences (e.g., Kathmandu vs. Pokhara).
• Temporal term: Daily/weekly trends. Traders discretize the PDE to predict crashes or rallies and execute trades automatically.

## In the Real World

1. Pathao’s Ride Allocation (Wave Equation) Pathao’s algorithm treats driver availability as a wave propagating through Kathmandu’s grid. The PDE: $$\frac{{\partial }^{2}D}{\partial t^2}=c^2\left(\frac{{\partial }^{2}D}{\partial x^2}+\frac{{\partial }^{2}D}{\partial y^2}\right)$$ models how demand waves (e.g., after a concert) spread. Solutions predict where to send idle drivers to minimize wait times.

2. Ncell’s 5G Signal Optimization (Bessel Functions) Ncell’s antenna arrays solve Bessel’s equation to shape cylindrical coverage patterns. The PDE: $$r^2\frac{{\partial }^{2}E}{\partial r^2}+r\frac{\partial E}{\partial r}+(k^2{r}^2-n^2)E=0$$ ensures signals penetrate hilly terrain efficiently. Engineers use $J_n(kr)$ to design antennas that maximize signal strength in remote areas like Dolpa.

3. eSewa’s Fraud Detection (Laplace Transform) eSewa’s real-time monitoring system converts the PDE governing transaction flows into an ODE via Laplace transform: $$\mathcal{L}\left\{\frac{\partial F}{\partial t}\right\}=s\bar{F}(x,s)-F(x,0)=\alpha \frac{d^2\bar{F}}{d{x}^2}$$ This simplifies fraud pattern detection by filtering noise and isolating anomalies (e.g., sudden spikes in failed transactions).

## Exam Tip

1. Classification is Critical

   • Always classify a PDE as elliptic, parabolic, or hyperbolic before attempting a solution. Examiners often ask for this as a separate part of the question.
   • Example Question: Classify $3u_{xx}+2u_{xy}-u_{yy}+u_x=0$. Solution: $B^2-4AC=(2{)}^2-4(3)(-1)=4+12=16>0$ → Hyperbolic.

2. Separation of Variables is the Workhorse

   • For linear homogeneous PDEs with constant coefficients, separation of variables is the go-to method. Memorize the standard forms:
     • Heat equation: $u_t=\alpha u_{xx}$ → $u(x,t)=X(x)T(t)$.
     • Wave equation: $u_{tt}=c^2{u}_{xx}$ → $u(x,t)=X(x)T(t)$.
     • Laplace’s equation: $u_{xx}+u_{yy}=0$ → $u(x,y)=X(x)Y(y)$.

3. Boundary Conditions Determine Eigenvalues

   • Dirichlet BCs ($u=0$) lead to sine series.
   • Neumann BCs ($u_x=0$) lead to cosine series.
   • Mixed BCs require careful handling (e.g., $u(0,t)=0$, $u_x(L,t)=0$).
   • Example: For $u_{xx}+\lambda u=0$ with $u(0)=0$, $u^{\prime }(L)=0$, the eigenvalues are: $${\lambda }_n={\left(\frac{(2n-1)\pi }{2L}\right)}^2,X_n(x)=\sin \left(\frac{(2n-1)\pi x}{2L}\right)$$

4. Special Functions are Non-Negotiable

   • Bessel’s equation: Arises in cylindrical coordinates. Always check for $x^2{y}^{\prime \prime }+x{y}^{\prime }+\dots$.
   • Legendre’s equation: Arises in spherical coordinates. Recognize $(1-x^2)y^{\prime \prime }-2x{y}^{\prime }+\dots$.
   • Exam Pitfall: Many students forget to use $J_n$ or $Y_n$ and lose marks. Always state the general solution as: $$y(x)=A{J}_n(x)+B{Y}_n(x)$$ unless boundary conditions eliminate $Y_n$.

5. Numerical Methods are Fair Game

   • Questions may ask for finite difference discretizations or stability analysis. For the heat equation:
     • Explicit method: $u_i^{k+1}=u_i^k+r(u_{i+1}^k-2u_i^k+u_{i-1}^k)$, where $r=\alpha \Delta t/(\Delta x{)}^2$.
     • Stability condition: $r\le 1/2$.
   • Example Question: Discretize $u_t=u_{xx}$ with $\Delta x=0.1$, $\Delta t=0.001$, $\alpha =0.5$. Check stability. Solution: $r=0.5\times 0.001/(0.1{)}^2=0.05\le 0.5$ → Stable.

6. Real-World Applications = Extra Marks

   • Examiners love it when you connect PDEs to real systems. For example:
     • Heat equation: "This models how quickly a Daraz package cools down in a truck during winter."
     • Wave equation: "This describes how traffic congestion waves propagate on the Kathmandu Ring Road."
   • Always label axes in your graphs (e.g., "Temperature (°C) vs. Distance (m)") and reference real parameters (e.g., $\alpha =0.1{\text{m}}^2/\text{s}$ for copper).

7. Common Mistakes to Avoid

   • Forgetting initial conditions: The solution is incomplete without $u(x,0)$ or $u_t(x,0)$.
   • Incorrect separation constants: Always double-check signs (e.g., $-\lambda$ vs. $+\lambda$).
   • Ignoring boundary conditions: Eigenvalues and eigenfunctions depend on BCs.
   • Arithmetic errors in Fourier coefficients: Use orthogonality to verify: $${\int }_0^L\sin \left(\frac{n\pi x}{L}\right)\sin \left(\frac{m\pi x}{L}\right)dx=\{\begin{array}{ll}L/2& \text{if }n=m,\\ 0& \text{otherwise.}\end{array}$$

## Practice Problems (Exam-Style)

1. Classify and Solve: Solve $u_t=u_{xx}+u_{yy}$ in $0<x<\pi$, $0<y<\pi$, $t>0$ with: $$u(0,y,t)=u(\pi ,y,t)=u(x,0,t)=u(x,\pi ,t)=0,u(x,y,0)=\sin (x)\sin (y).$$ Hint: Use separation of variables $u(x,y,t)=X(x)Y(y)T(t)$.

2. Bessel’s Equation: Solve $x^2{y}^{\prime \prime }+x{y}^{\prime }+(x^2-4)y=0$. Express the general solution in terms of Bessel functions.

3. Legendre Polynomials: Express $x^3-3x$ as a series of Legendre polynomials $P_0(x),P_1(x),P_2(x),P_3(x)$.

4. Laplace Transform: Solve $u_t=u_{xx}$ for $0<x<1$, $t>0$ with $u(0,t)=u(1,t)=0$, $u(x,0)=\sin (\pi x)$, using Laplace transforms.

5. Numerical Method: Discretize $u_t=u_{xx}$ with $\Delta x=0.2$, $\Delta t=0.01$, $\alpha =0.3$. Write the explicit update formula and check stability.

## Summary Table: Key PDEs and Methods