Solving Differential-Algebraic Equation Systems: Methods and MATLAB Implementation

Solving differential-algebraic equation systems is highly prevalent in engineering and scientific computations, with MATLAB providing robust tools to address these problems. Differential-Algebraic Equations (DAEs) differ from Ordinary Differential Equations (ODEs) by incorporating both differential equations and algebraic constraints, making the solution process more challenging due to their coupled nature.

While MATLAB's most commonly used solver `ode45` is better suited for ODEs, DAEs require specialized solvers like `ode15s` or `ode23t` designed for stiff problems. These methods employ variable-step numerical integration techniques that efficiently handle the coupling between differential and algebraic components through backward differentiation formulas (BDF) and trapezoidal rule implementations.

During modeling, problems must first be transformed into standard form with clearly separated differential and algebraic components. This involves defining differential variables and algebraic variables while ensuring constraint satisfaction throughout the solution process. MATLAB's `ode` series functions accept mass matrices to handle generalized DAE forms, where the mass matrix defines the relationship between derivatives and state variables using matrix singularity properties.

For high-dimensional or complex DAEs, preliminary simplification using the Symbolic Math Toolbox is recommended to enhance numerical solving efficiency and stability through equation reduction and symbolic differentiation. Finally, solver parameters such as relative error tolerance (`RelTol`) and absolute error tolerance (`AbsTol`) can be adjusted to optimize computational accuracy, with `odeset` function providing comprehensive configuration options for fine-tuning solver behavior.