PDE Integral Calculator

PDE integral model

Integration method

Initial profile u0

Source term f

Final time T

Position X

Transport velocity a

Heat diffusivity alpha

Initial amplitude

Source scale

Source decay

Wave or shape constant

Heat domain minimum

Heat domain maximum

Time steps

Space steps

Example Data Table

Case	Model	T	X	Profile	Source	Suggested method
Wave transport	Transport	1	2	Sine	Decaying sine	Simpson
Diffusion smoothing	Heat	0.75	1.5	Gaussian	No source	Simpson
Forced diffusion	Heat	1.25	0	Constant	Gaussian pulse	Trapezoid

Formula Used

Transport model: u_t + a u_x = f(t,x). The integral solution is u(T,X) = u0(X - aT) + ∫₀^T f(s, X - a(T-s)) ds.

Heat model: u_t = alpha u_xx + f(t,x). The integral form is u(T,X) = ∫G(T,X-y)u0(y)dy + ∫∫G(T-s,X-y)f(s,y)dyds.

Heat kernel: G(t,z) = 1 / sqrt(4 pi alpha t) multiplied by exp(-z² / (4 alpha t)).

The tool applies Simpson or trapezoid integration. Simpson needs an even step count, so the calculator adjusts odd values automatically.

How to Use This Calculator

Select the PDE integral model.
Choose the initial profile and source term.
Enter final time, position, and constants.
For heat problems, set the spatial integration domain.
Select a numeric integration method and step count.
Press Calculate to show the result above the form.
Use CSV or PDF export for reports.

Understanding PDE Integral Calculations

Why Integral Forms Matter

A PDE integral calculator helps connect differential equations with measurable values. Many practical models are hard to solve by simple algebra. An integral form gives a useful path. It turns change over time and space into summed effects.

Supported Integral Models

This tool focuses on two common integral ideas. The first is the transport characteristic formula. It follows a moving point along a characteristic line. The initial value is read from the foot of that line. Then the source term is added along the path.

The second idea is the heat kernel formula. It spreads the initial profile through a Gaussian kernel. It can also add a time dependent source term. This is useful for diffusion, smoothing, and flow studies.

Accuracy and Exports

Numerical integration is used because real inputs rarely stay simple. Trapezoid rules are stable and easy to understand. Simpson rules are usually more accurate for smooth functions. Higher step counts may improve precision. They also need more processing time.

The calculator offers flexible profiles. You can choose constant, sine, Gaussian, or linear initial data. You can also select no source, constant source, decaying sine source, Gaussian pulse, or polynomial source. These choices make quick testing easier.

Reading the Result

Results include the main solution estimate. They also show key components. For transport, you can see the initial term and path source integral. For heat, you can see the initial convolution and source convolution. This split helps explain the answer.

CSV export is helpful for spreadsheets and records. The PDF button is useful for lessons, reports, and client notes. The example table shows typical values before you start.

Best Practice

Use this calculator for study, planning, and comparison. It is not a replacement for formal proof. Boundary conditions, domains, and exact PDE structure still matter. Always match the selected model to your real problem. Review step counts when values change quickly. A wider heat domain may be needed for wide kernels. Strong sources may also require more time steps.

Good inputs create better estimates. Start with simple profiles first. Then adjust constants one at a time. This makes the result easier to trust. It also makes mistakes easier to find.

FAQs

What does this calculator estimate?

It estimates integral solution forms for selected transport and heat partial differential equation models. It gives numeric values, not symbolic proof.

Which method should I choose?

Use Simpson for smooth functions when you want better accuracy. Use trapezoid for simple checks or rough comparisons.

Why are heat domain limits needed?

The heat formula integrates over space. Domain limits tell the calculator where to approximate that spatial integral.

What happens if I enter an odd Simpson step count?

Simpson integration needs an even number of panels. The calculator automatically increases an odd step count by one.

Can this solve every PDE?

No. It supports selected integral forms. Other equations may need different kernels, boundaries, transforms, or numerical solvers.

Why does step count affect the result?

Step count controls how finely the integral is split. More steps can improve accuracy, but they also increase processing work.

What is the source term?

The source term represents forcing, input, production, or external influence inside the equation. It adds extra contribution over time.

Is the exported result suitable for reports?

Yes. CSV is useful for spreadsheets. PDF is useful for sharing summaries, teaching notes, and quick documentation.