axiomatic-mcp

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AxPDE_derive_source

Derive the source term f = L[u] by applying the PDE operator symbolically to a manufactured solution. Use this forward step of the Method of Manufactured Solutions to obtain an exact forcing term for verification.

Instructions

Derive the source term f = L[u] for a manufactured solution (deterministic, no LLM). Applies the PDE operator symbolically to the supplied solution, producing the forcing term that makes that solution exact. This is the forward step of the Method of Manufactured Solutions; pair it with verify_solution to confirm correctness.

Input Schema

TableJSON Schema

Name	Required	Description
`equations`	Yes	PDE operator(s): [{"name": "pde", "operator_code": "def pde_operator(fields, vars_dict): ..."}]. operator_code defines exactly one function returning a single SymPy expression L[fields], using the L[u] = f convention. Only `sp` (SymPy) is available; no imports, no numpy. For systems give one entry per equation, e.g. "x_momentum", "y_momentum", "continuity".
`variables`	Yes	Coordinate names, e.g. ["x", "t"] or ["x", "y", "t"]. Supported: x, y, t, r, theta, phi.
`solution_exprs`	Yes	Manufactured solution per field as SymPy-parseable strings, e.g. {"u": "sin(pix)exp(-t)"}. Keys must match the field names the operator code reads from `fields`.

Output Schema

TableJSON Schema

Name	Required	Description
`error`	No
`success`	Yes
`source_exprs`	No	Derived source term per equation name, e.g. {"pde": "(-1 + pi*2)exp(-t)sin(pix)"}. Pass this straight to verify_solution as source_exprs.

Schema Changelog

Changes observed during successful MCP inspections.

Addedv0.1.20Sep 15, 2026

TDQS

A4/5.0

Behavior3/5

Does the description disclose side effects, auth requirements, rate limits, or destructive behavior?

No annotations are present, so the description carries the disclosure burden. It adds useful behavioral traits: 'deterministic, no LLM' and 'Applies the PDE operator symbolically,' implying a pure computation with no external calls. It does not explicitly state absence of side effects or error behavior, leaving some transparency gaps.

Agents need to know what a tool does to the world before calling it. Descriptions should go beyond structured annotations to explain consequences.

Conciseness5/5

Is the description appropriately sized, front-loaded, and free of redundancy?

Two tight sentences front-load the core purpose, add workflow context, and avoid repetition of schema details. Every clause earns its place.

Shorter descriptions cost fewer tokens and are easier for agents to parse. Every sentence should earn its place.

Completeness4/5

Given the tool's complexity, does the description cover enough for an agent to succeed on first attempt?

Given the nested schema and existing output schema, the description covers essential orientation: what it computes, its role in MMS, and the companion tool. It does not walk through parameter formats, but the 100%-covered schema already handles those details, so nothing critical is missing.

Complex tools with many parameters or behaviors need more documentation. Simple tools need less. This dimension scales expectations accordingly.

Parameters3/5

Does the description clarify parameter syntax, constraints, interactions, or defaults beyond what the schema provides?

Schema description coverage is 100%, so the schema already documents parameters thoroughly with examples and supported coordinate lists. The description adds little parameter-specific meaning beyond referring to 'the supplied solution' and 'PDE operator,' so a baseline 3 is appropriate.

Input schemas describe structure but not intent. Descriptions should explain non-obvious parameter relationships and valid value ranges.

Purpose5/5

Does the description clearly state what the tool does and how it differs from similar tools?

The description opens with a specific verb and object: 'Derive the source term f = L[u] for a manufactured solution.' It clarifies it is deterministic and 'no LLM,' then explains the symbolic application of the PDE operator. Naming 'verify_solution' and calling this the 'forward step' clearly distinguishes the tool from its siblings.

Agents choose between tools based on descriptions. A clear purpose with a specific verb and resource helps agents select the right tool.

Usage Guidelines4/5

Does the description explain when to use this tool, when not to, or what alternatives exist?

It states the tool is the 'forward step of the Method of Manufactured Solutions' and instructs the agent to 'pair it with verify_solution to confirm correctness,' establishing clear when-to-use context. It does not enumerate exclusions or alternatives for other PDE tasks, so it stops short of a full when-not-to-use guide.

Agents often have multiple tools that could apply. Explicit usage guidance like "use X instead of Y when Z" prevents misuse.

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