Transport Equation Solver Calculator

Calculator Inputs

Transport velocity c

Reaction coefficient λ

Constant source S

Evaluation point x

Evaluation time t

Initial profile type

Primary amplitude / constant term

Offset / lower state

Shape parameter 1

Shape parameter 2

Grid spacing Δx

Time step Δt

Parameter guide: sinusoidal uses shape1 = k and shape2 = φ; gaussian uses shape1 = σ and shape2 = μ; step uses shape1 = x0; polynomial uses shape1 = a and shape2 = b; exponential uses shape1 = r.

Example Data Table

Profile	c	λ	S	x	t	Key parameters	Approx. u(x,t)
Sinusoidal	2.0	0.15	1.5	4.0	1.2	A = 5, B = 1, k = 1.1, φ = 0	6.584389
Gaussian	1.8	0.10	0.8	3.0	1.0	A = 6, B = 0.5, σ = 0.9, μ = 1.2	6.642744
Step	1.5	0.05	0.2	2.5	0.8	A = 4, B = 1, x0 = 1.0	4.000000

Formula Used

This calculator solves the constant-coefficient one-dimensional transport equation ∂u/∂t + c∂u/∂x + λu = S using the method of characteristics. The characteristic path is x = x₀ + ct, so the foot of the characteristic is ξ = x - ct.

When λ ≠ 0: u(x,t) = e^-λt f(x - ct) + (S/λ)(1 - e^-λt)

When λ = 0: u(x,t) = f(x - ct) + St

The initial function f(ξ) depends on the selected profile. The calculator supports sinusoidal, gaussian, step, polynomial, and exponential initial conditions. It also computes the Courant number C = cΔt/Δx to give a practical numerical stability hint.

How to Use This Calculator

Enter the transport velocity, reaction coefficient, and constant source term.
Set the position and time where you want the solution evaluated.
Select an initial profile that best matches your transport problem.
Fill in the profile parameters using the guide shown below the form.
Provide Δx and Δt if you want a Courant number check.
Press the solve button to display the exact characteristic solution.
Review the local sample table, then export the result as CSV or PDF.

Frequently Asked Questions

1) What does this transport equation solver calculate?

It evaluates the one-dimensional linear transport equation with constant advection speed, reaction coefficient, and source term. The result is computed at a chosen position and time using an exact characteristic-based formula.

2) Why is the characteristic foot ξ important?

The characteristic foot identifies where the current solution value originated on the initial profile. For constant transport speed, the foot is ξ = x - ct, which shifts the initial condition along the flow direction.

3) What happens when λ equals zero?

When the reaction coefficient is zero, attenuation disappears. The solution becomes the shifted initial value plus linear source accumulation over time, so the formula simplifies to u(x,t) = f(x - ct) + St.

4) How should I choose the initial profile?

Use sinusoidal for waves, gaussian for localized pulses, step for fronts, polynomial for smooth curves, and exponential for growth or decay patterns. Choose the profile that best matches your starting field.

5) What is the Courant number used for?

The Courant number compares physical transport distance during one time step with the grid spacing. It helps judge whether many explicit numerical schemes are likely stable, accurate, or potentially unstable.

6) Is this an exact or approximate solver?

For the supported constant-coefficient equation, the calculator uses an exact closed-form characteristic solution. The sample stability note is advisory only and relates to possible numerical implementations using Δx and Δt.

7) Can I use negative transport velocity values?

Yes. A negative velocity simply shifts the characteristic in the opposite direction. The formula remains valid, and the characteristic foot ξ automatically adjusts to reflect reverse advection.

8) What do the export buttons include?

The CSV button downloads the sample solution table around your query point. The PDF button creates a compact report containing the key result summary and the visible table shown after solving.