# Applications

Symbolic Math Toolbox provides tools to solve, plot, and manipulate mathematical
expressions, both analytically and numerically with high precision. From the results
of symbolic computations, you can also generate MATLAB^{®} functions, Simulink^{®} Function blocks, and Simscape™ equations to use with other toolboxes. Use these tools to perform
application-specific workflows.

## Teaching and Research

### Mathematics

**Eigenvalues of the Laplace Operator**

This example shows how to solve the eigenvalue problem of the Laplace operator on an L-shaped region.**Solve Partial Differential Equation of Nonlinear Heat Transfer**

This example shows how to solve a partial differential equation (PDE) of nonlinear heat transfer in a thin plate.**Solve Partial Differential Equation of Tsunami Model**

This example simulates a tsunami wave phenomenon by solving the partial differential equations that model the tsunami.**Using Symbolic Mathematics with Optimization Toolbox Solvers**

This example shows how to use the Symbolic Math Toolbox functions`jacobian`

and`matlabFunction`

to provide analytical derivatives to optimization solvers.**Analytical Plotting with Symbolic Math Toolbox**

Symbolic Math Toolbox provides analytical plotting of mathematical expressions without explicitly generating numerical data.

### Physics

**The Physics of the Damped Harmonic Oscillator**

This example explores the physics of the damped harmonic oscillator by solving the equations of motion in the case of no driving forces.**Simulate the Motion of the Periodic Swing of a Pendulum**

Solve the equation of motion of a simple pendulum analytically for small angles and numerically for any angle.**Functional Derivatives Tutorial**

This example shows how to use functional derivatives in Symbolic Math Toolbox using the context of a wave equation.**Solve Differential Equations of RLC Circuit Using Laplace Transform**

Solve differential equations of an RLC circuit by using Laplace and inverse Laplace transforms.**Harmonic Analysis of Transfer Function Output**

This example finds closed-form solutions for the coefficients of frequencies in an output signal.**Units in Physics Calculations**

This example shows how to work with units in physics calculations.

## Mathematical System Modeling

**Animation and Model of Automotive Piston**

This example shows how to model the motion of an automotive piston by using MATLAB and Symbolic Math Toolbox.

**Padé Approximant of Time-Delay Input**

This example shows how to use a Padé approximant in control system theory to model time delays in the response of a first-order system.

**Analytical Model of Cantilever Truss Structure for Simscape**

This example shows how to find parameterized analytical expressions for the displacement of a joint of a cantilever truss structure in both the static and frequency domains.

**Estimate Model Parameters of a Symbolically Derived Plant Model in Simulink**

This example uses Simulink Design Optimization™ to estimate the unknown capacitance and initial voltage of a symbolically derived algebraic model of a simple resistor-capacitor (RC) circuit.

**Customize and Extend Simscape Libraries for a Custom DC Motor**

Create custom-equation-based components for the Simscape library using Symbolic Math Toolbox.

**Finite Element Formulation for Timoshenko Beam Problem**

This example shows how to apply the finite element method (FEM) to solve a Timoshenko beam problem, using both linear and quadratic basis functions for analysis.

## Robotic Control Systems

**Analytical Solutions of the Inverse Kinematics of a Humanoid Robot**

This example shows how to derive analytical solutions for the inverse kinematics of the head chain of a humanoid robot.

**Derive and Apply Inverse Kinematics to Two-Link Robot Arm**

This example shows how to derive and apply inverse kinematics to a two-link robot arm by using MATLAB and Symbolic Math Toolbox.

**Derive Quadrotor Dynamics for Nonlinear Model Predictive Control**

This example shows how to derive a continuous-time nonlinear model of a quadrotor using Symbolic Math Toolbox.

**Derive Equations of Motion and Simulate Cart-Pole System**

This example shows how to derive the equations of motion for the cart-pole system using Symbolic Math Toolbox™ and then simulate the cart-pole system using the `ode45`

solver.

## Quantitative Finance

**The Black–Scholes Formula for Call Option Price**

This example shows how to calculate the call option price using the Black–Scholes formula.

**Explore Single-Period Asset Arbitrage**

Explore basic arbitrage concepts in a single-period, two-state asset portfolio.

**Simulate a Stochastic Process Using the Feynman–Kac Formula**

This example obtains the partial differential equation that describes the expected final price of an asset whose price is a stochastic process given by a stochastic differential equation.

**Markov Chain Analysis and Stationary Distribution**

This example shows how to derive the symbolic stationary distribution of a trivial Markov chain by computing its eigen decomposition.

## Related Information

## Featured Examples

### Solve Partial Differential Equation of Nonlinear Heat Transfer

Solve a partial differential equation (PDE) of nonlinear heat transfer in a thin plate.

### Analytical Plotting with Symbolic Math Toolbox

Symbolic Math Toolbox provides analytical plotting of mathematical expressions without explicitly generating numerical data.

### Simulate the Motion of the Periodic Swing of a Pendulum

Solve the equation of motion of a simple pendulum analytically for small angles and numerically for any angle.

### Analytical Model of Cantilever Truss Structure for Simscape

Find parameterized analytical expressions for the displacement of a joint of a cantilever truss structure in both the static and frequency domains.

### Derive and Apply Inverse Kinematics to Two-Link Robot Arm

Derive and apply inverse kinematics to a two-link robot arm by using MATLAB and Symbolic Math Toolbox.

### Explore Single-Period Asset Arbitrage

Explore basic arbitrage concepts in a single-period, two-state asset portfolio.

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