Example Data Table
| Model |
Main coefficient |
Suggested input |
Typical check |
| Heat |
alpha = 0.5 |
A = 2, L = 10, n = 1, t = 0.4 |
Decay and heat stability ratio |
| Wave |
c = 3 |
A = 1, L = 8, n = 2, t = 0.2 |
Wave value and Courant ratio |
| Transport |
v = 1.2 |
A = 1, L = 12, x = 5, t = 1 |
Shifted profile and residual |
| Laplace |
H = 5 |
A = 4, L = 10, x = 3, y = 2 |
Steady rectangular field mode |
| General |
a, b, c |
u = 1, u_x = 0.2, u_y = 0.1 |
Linear residual check |
Formula Used
Heat equation: u_t = alpha u_xx. This calculator uses u(x,t) = A exp(-alpha k^2 t) sin(kx + phase).
Wave equation: u_tt = c^2 u_xx. This calculator uses u(x,t) = A cos(c k t + phase) sin(kx).
Transport equation: u_t + v u_x = 0. This calculator uses u(x,t) = A sin(k(x - vt) + phase).
Laplace equation: u_xx + u_yy = 0. This calculator uses one separated rectangular mode.
General first order form: a u_x + b u_y + c u = f. The residual is a u_x + b u_y + c u - f.
For sine mode templates, k = n pi / L. The residual shows how closely the equation form is satisfied.
How to Use This Calculator
Select the PDE model first. Enter amplitude, length, position, time, mode number, and the related coefficient. Use alpha for heat, c for wave speed, and v for transport velocity. For the Laplace model, enter the rectangle height and y position. For the general model, enter derivative estimates and coefficients.
Press the solve button. The result appears below the header and above the form. Review the solution value, residual, boundary note, and stability guidance. Use CSV or PDF export when you need to save the calculation.
Understanding PDE Models
A Solve PDE Calculator helps students test common partial differential equation models. It does not replace a symbolic solver. It gives a dependable numeric checkpoint for standard classroom forms.
Partial differential equations describe change across more than one variable. A heat model changes across position and time. A wave model tracks vibration across position and time. A Laplace model studies steady fields. A transport model follows motion through space.
Supported Equation Types
This page uses guided templates. Each template asks for coefficients, location values, and grid data. The calculator then estimates the solution value. It also reports key terms and a residual. A small residual suggests the model and data agree better.
The heat option uses a single sine mode. The solution decays with time. A larger diffusivity makes the mode fade faster. The wave option uses a sine mode with cosine time behavior. A larger speed moves the oscillation faster. The transport option shifts the starting profile by velocity and time. The Laplace option estimates one separated boundary mode in a rectangle.
Numerical Checks
The general first order option checks a linear expression. You enter derivative estimates and coefficients. The tool evaluates a u_x plus b u_y plus c u minus f. This is useful when values come from a table, spreadsheet, or numerical grid.
Stability guidance is included for grid planning. Heat calculations use the ratio alpha times time step divided by space step squared. Wave and transport checks use Courant style ratios. These values help you choose safer step sizes before a long computation.
Good Use Cases
Use this calculator for study, reports, and model review. Enter realistic units. Keep all related values consistent. Review the formula panel before judging the answer. Export the result when you need a record for class notes or technical documentation.
Many partial differential equations need boundary conditions, initial functions, and numerical methods. This tool focuses on approachable templates. It helps learners see how inputs affect terms, residuals, and stability checks.
For best results, compare one case at a time. Change one input, then submit again. This habit makes patterns easier to notice. It also reduces mistakes when checking homework, lab notes, or early design estimates for simple fields.
FAQs
What does this calculator solve?
It estimates selected PDE templates, including heat, wave, transport, Laplace, and a general first order residual check.
Can it solve every partial differential equation?
No. General PDE solving can require symbolic methods, numerical schemes, boundary data, and initial conditions. This tool gives guided template checks.
What does residual mean?
Residual is the difference left after substituting values into the equation. A smaller residual means the selected model agrees better.
What is the mode number?
The mode number controls the sine wave pattern. Higher values create more oscillations across the selected length.
What coefficient should I enter?
Use diffusivity alpha for heat, wave speed c for wave models, and velocity v for transport models.
Why is stability shown?
Stability ratios help when planning numerical grids. They warn when time steps may be too large for simple explicit methods.
Can I export the result?
Yes. After solving, use the CSV or PDF button to save the result table for notes, reports, or review.
Are units required?
The calculator does not force units. Keep all inputs consistent. Mixed units can produce misleading results.