Second Order Partial Differential Equation Solver Calculator

Calculator Inputs

This solver uses the model equation: a·u_xx + 2b·u_xy + c·u_yy + d·u_x + e·u_y + f·u = g on a rectangular domain with constant Dirichlet boundaries.

a coefficient

Coefficient for uxx.

b coefficient

Mixed term uses 2b·uxy.

c coefficient

Coefficient for uyy.

d coefficient

First derivative x term.

e coefficient

First derivative y term.

f coefficient

Reaction term multiplier.

g source term

Constant forcing term.

Domain length Lx

Horizontal domain size.

Domain length Ly

Vertical domain size.

Grid points Nx

Horizontal grid count.

Grid points Ny

Vertical grid count.

Relaxation ω

SOR acceleration factor.

Left boundary

x = 0 edge value.

Right boundary

x = Lx edge value.

Top boundary

y = Ly edge value.

Bottom boundary

y = 0 edge value.

Maximum iterations

Upper iteration limit.

Tolerance

Convergence threshold.

Reset

Example Data Table

Parameter	Sample Value	Purpose
a	1	Controls x-direction curvature.
b	0	Removes mixed derivative influence.
c	1	Controls y-direction curvature.
d, e, f	0, 0, 0	No drift or reaction term.
g	-6	Creates interior forcing.
Lx, Ly	1, 1	Uses a unit square domain.
Nx, Ny	17, 17	Builds a moderate grid.
Boundaries	0, 0, 0, 0	Sets all edges to zero.
ω, tolerance, iterations	1.10, 1e-6, 2500	Improves convergence control.

Formula Used

General model:
a·u_xx + 2b·u_xy + c·u_yy + d·u_x + e·u_y + f·u = g

Classification rule:
Discriminant = b² - ac.
If b² - ac < 0, the PDE is elliptic.
If b² - ac = 0, the PDE is parabolic.
If b² - ac > 0, the PDE is hyperbolic.

Central differences:
u_xx ≈ (u_i+1,j - 2u_i,j + u_i-1,j) / Δx²
u_yy ≈ (u_i,j+1 - 2u_i,j + u_i,j-1) / Δy²
u_xy ≈ (u_i+1,j+1 - u_i+1,j-1 - u_i-1,j+1 + u_i-1,j-1) / (4ΔxΔy)

First derivatives:
u_x ≈ (u_i+1,j - u_i-1,j) / (2Δx)
u_y ≈ (u_i,j+1 - u_i,j-1) / (2Δy)

The page solves the discretized linear system iteratively with Gauss-Seidel style updates and SOR relaxation. This makes the calculator useful for constant-coefficient, rectangular-domain, boundary-value experiments.

How to Use This Calculator

Enter coefficients a, b, c, d, e, f, and g.
Set domain lengths Lx and Ly.
Choose grid points Nx and Ny.
Provide left, right, top, and bottom boundary values.
Set relaxation, tolerance, and iteration limit.
Click Solve PDE to generate the numerical field.
Review classification, residual, center value, and graph outputs.
Download the grid as CSV or save the summary as PDF.

Frequently Asked Questions

1. What does this calculator solve?

It solves a constant-coefficient second order PDE on a rectangular region. The page classifies the equation and estimates a grid-based numerical solution using finite differences and iterative relaxation.

2. How is the PDE type identified?

The calculator uses the discriminant b² - ac for the equation a·uxx + 2b·uxy + c·uyy + ... . Negative means elliptic, zero means parabolic, and positive means hyperbolic.

3. Why do I need boundary values?

Boundary values anchor the solution on the domain edges. The interior grid values are then computed from the PDE coefficients, source term, and those fixed edge conditions.

4. What does the relaxation factor do?

The relaxation factor ω speeds or damps iterative updates. Values slightly above 1 often improve convergence, but values set too high can destabilize the iteration.

5. Why might the solver stop without full convergence?

Some coefficient combinations, coarse grids, or strongly hyperbolic cases can resist this steady iterative approach. In such cases, increase iterations, refine the grid, or change the coefficients.

6. What does the center value mean?

The center value is the computed solution near the middle of the domain. It is useful for comparing runs when you change coefficients, source strength, or boundary data.

7. Can I use this for learning canonical forms?

Yes. The rotation angle and PDE class help you inspect canonical behavior. The calculator is especially helpful for visualizing how mixed derivatives and anisotropy affect the solution field.

8. What do the CSV and PDF downloads include?

The CSV file exports the computed grid values with x and y locations. The PDF report saves the main summary metrics and the generated surface graph.