Calculator inputs
Use a responsive three-column grid on large screens, two on tablets, and one on mobile.
Plotly graph
The surface shows the computed solution field. The line tracks the characteristic through the evaluation point.
Example data table
Use these sample cases to verify your implementation and understand supported boundary configurations.
| Scenario | PDE | Boundary condition | Point | Computed result |
|---|---|---|---|---|
| Horizontal transport case | 2ux + 1uy = 3 | u(x, 0) = sin(x) | (2, 1) | 3 |
| Vertical transport case | 1ux - 2uy = 4 | u(0, y) = y^2 | (3, 1) | 61 |
| Linear ramp input | 3ux + 2uy = 5 | u(x, 1) = x + 4 | (4, 3) | 7 |
Formula used
This solver handles the constant-coefficient first-order linear PDE a·ux + b·uy = c with either a horizontal or vertical boundary line.
The residual shown in the results table is estimated numerically from the computed surface and helps confirm that the solution satisfies the PDE closely.
How to use this calculator
- Enter the coefficients a, b, and c from your PDE model.
- Select whether your boundary data lies on a horizontal line or a vertical line.
- Provide the boundary line value and the boundary expression, such as sin(x), x^2, or y + 3.
- Set the target point (x, y) where you want the solution value.
- Adjust graph ranges and grid density to control the plotted solution field.
- Submit the form to see the closed-form mapping, foot point, residual, and the 3D Plotly surface.
- Download the current result as CSV or PDF for reports and sharing.
FAQs
1) What kind of equations does this solver handle?
It solves first-order linear PDEs with constant coefficients in the form a·ux + b·uy = c. It uses the method of characteristics with boundary data on either y = constant or x = constant.
2) Why does the calculator ask for a boundary type?
The formula depends on where the known data is supplied. Horizontal and vertical boundary lines lead to different foot-point mappings, so the solver needs that choice before building the closed-form expression.
3) When is a horizontal boundary invalid?
A horizontal boundary becomes unsuitable when b = 0. In that case, the characteristics do not cross y = constant properly, so the data does not define a valid transverse starting curve for this solver.
4) When is a vertical boundary invalid?
A vertical boundary becomes unsuitable when a = 0. Then the characteristic flow does not cross x = constant in a usable way, so the solver asks you to switch to a horizontal boundary instead.
5) What does the characteristic foot point mean?
It is the point on the chosen boundary line that connects to your target point through the characteristic curve. The solver first traces back to that foot point, then carries the solution forward along the characteristic.
6) Why is a residual shown in the results?
The residual estimates how closely the computed surface satisfies the PDE after numerical differentiation. Small values near zero indicate strong agreement between the plotted solution and the differential equation.
7) Can I use trigonometric or exponential boundary expressions?
Yes. The expression field accepts common functions such as sin, cos, tan, sqrt, log, exp, abs, and powers with ^. Use x for horizontal data and y for vertical data.
8) Is this a general symbolic PDE engine?
No. It is a focused transport-style solver for constant-coefficient first-order linear PDEs. That makes it fast, clear, and useful for many teaching, checking, and engineering-style advection problems.