Results for “differential equations”

Expert-level differential equations knowledge. Use when working with ordinary differential equations, partial differential equations, systems of ODEs, Laplace transforms, Fourier methods, boundary value problems, stability analysis, or numerical methods for ODEs. Also use when the user mentions 'ODE', 'PDE', 'initial value problem', 'boundary value problem', 'Laplace transform', 'separation of variables', 'eigenvalue problem', 'heat equation', 'wave equation', 'Laplace equation', 'stability', or 'phase plane'.

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Write MATLAB or GNU Octave scripts for matrix calculations, data analysis, signal processing, and scientific plots. The skill also helps explain syntax and functions, convert code between MATLAB and Python, and run scripts when an interpreter is installed.

• Analyzing experimental CSV data and saving plots
• Solving matrix equations and differential equations
• Analyzing signal frequencies and applying filters

Calculus, series, numerical methods, and optimization for computational mathematics. Covers limits (epsilon-delta, L'Hopital, squeeze theorem), derivatives (rules, optimization, related rates), integrals (Riemann sums, Fundamental Theorem of Calculus, integration techniques), series (convergence tests, Taylor and Maclaurin expansions), differential equations (separable, linear, exact), numerical methods (Newton-Raphson, Euler method, Simpson's rule), and computational pitfalls (floating-point arithmetic, catastrophic cancellation, numerical stability). Use when computing derivatives, integrals, limits, series, solving differential equations, or analyzing numerical accuracy.

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Calculus and continuous transformation — derivatives, integrals, ODEs, Taylor series, and optimization. Computes rates of change, accumulates quantities over intervals, solves differential equations, and classifies critical points. Use when computing derivatives or integrals, solving ordinary differential equations, performing Taylor series approximations, finding critical points, or analyzing continuous change and motion.

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Work with the full set of Maxwell's equations in integral and differential form to analyze electromagnetic fields, waves, and energy transport. Use when applying Gauss's law, Faraday's law, or the Ampere-Maxwell law to boundary value problems, deriving the electromagnetic wave equation, computing Poynting vector and radiation pressure, solving for fields at material interfaces, or connecting electrostatics and magnetostatics to the unified electromagnetic framework.

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Solve ordinary differential equations (initial and boundary value problems). Supports stiff/non-stiff systems, event detection, Hamiltonian/symplectic integration, parameter sweeps, and phase space analysis. Use for any ODE system in physics, engineering, or applied math.

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Analyze the dynamics of diffusion processes using stochastic differential equations, Fokker-Planck equations, first-passage time distributions, and parameter sensitivity analysis. Use when deriving probability density evolution for a continuous-time diffusion process, computing mean first-passage times for bounded diffusion, analyzing how drift and diffusion parameters affect process behavior, or validating closed-form solutions against stochastic simulation.

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MATLAB and GNU Octave numerical computing for matrix operations, data analysis, visualization, and scientific computing. Use when writing MATLAB/Octave scripts for linear algebra, signal processing, image processing, differential equations, optimization, statistics, or creating scientific visualizations. Also use when the user needs help with MATLAB syntax, functions, or wants to convert between MATLAB and Python code. Scripts can be executed with MATLAB or the open-source GNU Octave interpreter.

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Generate correct MATLAB code using the Symbolic Math Toolbox. Use when the user asks for symbolic computations, analytical solutions, symbolic differentiation/integration, equation solving, or converting symbolic results to numeric MATLAB functions. Also use when converting differential equations to transfer functions or state-space form, extracting PDE coefficients (pdeCoefficients), symbolic-to-C code generation (matlabFunction + codegen), symbolic matrix variables (symmatrix), physical units and constants (symunit, newUnit), or rewriting/combining algebraic expressions (rewrite, combine).

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Expert in ODEs and PDEs covering solution methods, qualitative analysis, and applications to physics and engineering

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Expert-level calculus knowledge. Use when working with limits, derivatives, integrals, multivariable calculus, vector calculus, differential equations, series, or optimization. Also use when the user mentions 'derivative', 'integral', 'limit', 'chain rule', 'gradient', 'divergence', 'curl', 'Taylor series', 'Fourier series', 'partial derivative', 'double integral', 'line integral', 'Stokes theorem', or 'Lagrange multipliers'.

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Solve partial differential equations — finite differences, spectral methods, and physics-informed neural networks (PINNs via DeepXDE). Supports 1D/2D/3D, steady/transient, linear/nonlinear PDEs with Dirichlet, Neumann, and periodic boundary conditions.

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