Analyze diffusion, electrostatics, and heat equations with confidence. Tune meshes, sources, boundaries, and conductivity quickly. Get clear numerical profiles, gradients, plots, exports, and diagnostics.
This solver models the one-dimensional steady equation d²φ/dx² = -S/k with fixed boundary values using a central-difference stencil.
| Case | L (m) | Nodes | φ(0) | φ(L) | k | S |
|---|---|---|---|---|---|---|
| Heated Rod | 1.00 | 9 | 100 | 300 | 12 | 1500 |
| Electrostatic Slab | 0.50 | 11 | 0 | 50 | 4 | 200 |
| Diffusion Channel | 2.00 | 13 | 1.2 | 0.2 | 0.8 | 0.1 |
The calculator discretizes the domain into equally spaced nodes. Dirichlet boundary values are enforced at both ends. The resulting tridiagonal matrix is solved with the Thomas algorithm, giving the nodal field values. Gradients are then estimated with forward, backward, and central differences.
Finite difference results improve as node count increases because spacing Δx becomes smaller and truncation error declines for the central second derivative. In practical rod or slab problems, moving from 9 nodes to 21 or 41 nodes can smooth curvature and reduce gradient scatter near boundary changes. Users should compare peak value, average value, and flux after each refinement step to confirm convergence.
This calculator applies fixed values at each end of the domain. If φ(0)=100 and φ(L)=300, those limits are enforced exactly, while interior nodes adjust to conductivity and source strength. A larger boundary difference raises slope magnitude. Equal boundary values remove the linear tilt and make the internal source term responsible for most of the curve shape.
The uniform source S controls whether the interior profile departs from a straight line. Positive generation adds curvature and can raise interior temperature or potential above linear interpolation. If S doubles while geometry and conductivity stay fixed, the right side of the algebraic system also doubles, so the discrete solution bends more and end fluxes shift accordingly.
Conductivity k moderates the impact of the source term. Higher k spreads generation more efficiently, flattening the nodal profile. Lower k amplifies curvature and steepens gradients. For example, lowering k from 12 to 6 with identical inputs doubles the forcing term (SΔx²)/k, pushing the solution farther from the no-source linear profile.
The calculator estimates gradients at every node and computes left and right flux-style values from the end slopes. These outputs help evaluate transfer behavior at the boundaries. When the source is nonzero, the two boundary fluxes usually differ because the domain generates or consumes quantity internally. Pairing flux with average field value gives a stronger engineering check than reviewing nodal values alone.
Each interior node creates one equation, forming a tridiagonal matrix with nonzero terms only on three diagonals. The Thomas algorithm solves this structure in linear time, which makes mesh studies practical. Users can change domain length, node count, source level, and conductivity repeatedly while still getting a transparent numerical profile, export-ready table, and plot for review.
It solves a one-dimensional steady boundary value problem using central finite differences, fixed end values, and a tridiagonal linear system for interior nodes.
More nodes reduce grid spacing, usually improving accuracy and revealing whether peak values, gradients, and fluxes are converging toward a stable numerical solution.
Yes. The same mathematical form appears in conduction, diffusion, and electrostatic models when the problem can be written as a one-dimensional steady equation.
A smaller conductivity magnifies the source-term effect, causing stronger curvature and steeper gradients. Extremely small values may also produce very large computed field values.
They summarize transfer behavior at both boundaries using the end gradients. Differences between them often reflect internal generation or absorption across the domain.
CSV exports the nodal results table. PDF captures the calculator page, including inputs, summary metrics, plot, table, example data, formulas, article, and FAQs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.