**Distributed Parameter Systems Modelling:**

We consider several models present in the process and mechanical systems control:

**Transport-reaction systems:**

**1.) Tubular reactor**

with boundary conditions , and initial condition

**2.) Tubular reactor with recycle**

with boundary conditions and initial conditions

**3.) Countercurrent heater**

**4.) Equation of heat conduction **(parabolic PDE)

with , .

**5.) Reaction-diffusion-convection models of dispersion reactor **(parabolic PDE)

with and , .

**Strings & Beams**

**1.) String Equation **(2nd Order Hyperbolic PDE)

Let us consider:

with Derichlet boundary conditions , and and . The change of variable leads to and , which leads to the following and also , from this we obtain and , and therefore:

**2.) Euler-Bernoulli Beam Equation **

Let us consider following version of the beam equation which is a simply supported undamped beam:

where in the mass per unit length, E is the elastic modulus and I is the second moment of area of the beam's cross-section.The x represents deflection of the beam in the direction at some point .

**3.) Rayleigh Beam Equation**

with boundary conditions:

**3.) Shear Beam Equation**

or

**4.) Timoshenko Beam Equation**

where the x is translational displacement of the beam and is angular velocity.

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