Online PDE Calculator

Calculator Inputs

Calculation mode

Boundary type

Grid points

Domain length

dx override

Time steps

A coefficient

B coefficient

C coefficient

D coefficient

E coefficient

F coefficient

G value

Thermal diffusivity alpha

Wave speed c

Velocity v

Diffusion D

Mesh spacing h

Source term S or f

Left value

Center value

Right value

Previous center value

Top value

Bottom value

Example Data Table

Case	Model	Key inputs	Expected use
Heat bar	Heat equation	alpha = 0.25, dx = 0.1, dt = 0.01	Estimate the next temperature value.
String motion	Wave equation	c = 1, dx = 0.1, dt = 0.05	Check a local vibration update.
Plate balance	Laplace or Poisson	left, right, top, bottom, source	Estimate an interior steady value.
Transport spread	Advection diffusion	v, D, dx, dt, source	Combine drift and diffusion effects.

Formula Used

The calculator classifies a second order PDE using the discriminant B² - AC for the form A uxx + 2B uxy + C uyy + D ux + E uy + F u = G.

For heat flow, it uses u_i^{n+1} = u_i^n + r(u_{i-1}^n - 2u_i^n + u_{i+1}^n) + dtS, where r = alpha dt / dx².

For wave motion, it uses u_i^{n+1} = 2u_i^n - u_i^{n-1} + r(u_{i-1}^n - 2u_i^n + u_{i+1}^n) + dt²S.

For Poisson form, it estimates center u = (left + right + top + bottom - f h²) / 4.

How to Use This Calculator

Select the model first. Enter the PDE coefficients when classification matters. Add grid spacing, time step, and local grid values. Use dx override when your mesh spacing is already known. Press Calculate. The result appears below the header and above the form. Use CSV or PDF buttons for reports.

Understanding PDE Calculations

Partial differential equations describe change across more than one variable. They appear in heat flow, waves, fluids, finance surfaces, and image processing. A useful online calculator cannot solve every possible equation. The subject is too wide. It can still give strong help. It can classify a second order equation. It can test a grid step. It can estimate one finite difference update. It can also show the formula behind the step.

Why Classification Matters

The coefficients A, B, and C reveal the main equation type. The calculator uses B² - AC for the discriminant. A negative value usually means elliptic behavior. A zero value means parabolic behavior. A positive value means hyperbolic behavior. This simple check helps choose a numerical method. Elliptic equations often model steady balance. Parabolic equations often model diffusion. Hyperbolic equations often model transport, vibration, or waves.

Numerical Grid Ideas

A grid turns a smooth problem into many small cells. The spacing dx controls detail in space. The step dt controls movement in time. Smaller steps can improve accuracy. They also increase work. Stability matters before accuracy. A heat update needs a safe diffusion ratio. A wave update needs a safe Courant ratio. An advection diffusion update needs both checks. The tool prints these values. It also marks common stability warnings.

Practical Use

Enter local values around one grid point. Use the left, center, right, top, and bottom fields as needed. Choose the model that fits your equation. Then compare the computed value with your class notes or solver output. The result is not a full theorem proof. It is a transparent calculation helper. It supports quick checks during study. It also helps explain each term in reports.

Advanced Options

This page also supports Poisson and local advection diffusion checks. These cases are common in labs. A source term can add heating, forcing, or load. Boundary values guide the interior estimate. The coefficient fields stay visible because classification is useful in many modes. Keep units consistent. Do not mix centimeters with meters. Do not mix seconds with hours. For serious projects, refine the grid and compare several runs. A stable first step is only the beginning. Validation still belongs in your workflow and final review.

FAQs

What does this PDE calculator do?

It classifies second order PDE forms and estimates local finite difference values for heat, wave, Laplace, Poisson, and advection diffusion cases.

Can it solve every partial differential equation?

No. General PDE solving is broad and complex. This tool focuses on classification and common local numerical updates.

What does B² - AC mean?

It is the discriminant for A uxx + 2B uxy + C uyy. It helps identify elliptic, parabolic, or hyperbolic behavior.

When should I use the heat mode?

Use heat mode for diffusion style problems where the next value depends on nearby left and right values and a source term.

When should I use the wave mode?

Use wave mode when a value depends on the current value, the previous time value, nearby points, and wave speed.

What is the Laplace or Poisson estimate?

It estimates an interior center value from four surrounding values. Poisson mode also includes a source term.

Why does stability matter?

An unstable grid can create false growth or oscillation. Stability checks help warn when dx or dt may need adjustment.

Can I export the result?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report of the current calculation.