Table of Contents
- 1 Do solutions to differential equations have to be continuous?
- 2 What does a differential equation represent?
- 3 What is continuous dependence?
- 4 How does continuous simulation modeling differ from process simulation?
- 5 How are differential equations used in real life?
- 6 What is the difference equation in digital control?
- 7 What is the difference between continuous and differentiable functions?
Do solutions to differential equations have to be continuous?
In order to use the theorem both must be continuous on an interval that contains yo=0 y o = 0 and this is problem for us since we do have yo=0 y o = 0 . y(t)=0 y ( t ) = 0 is also a solution to the differential equation and satisfies the initial condition.
What is the significance of differential equation in context of continuous system simulation?
Continuous simulations are based on a set of differential equations. These equations define the peculiarity of the state variables, the environment factors so to speak, of a system. These parameters of a system change in a continuous way and thus change the state of the entire system.
What does a differential equation represent?
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
Can a given differential equation have a unique solution?
Most differential equations that are useful in practice do have unique solutions. If, for example, the equations for the orbits of the planets didn’t have unique solutions, we would not be able to predict their motions.
What is continuous dependence?
(3) Continuous dependence: The solution depends continuously on the data that are present in the problem. Theorem 3.39 Initial value problem for an ODE y = f(x, y), where f is Lipschitz continuuos on a rectangle containining the initial data (x, y0), is well-posed.
Why analog method does not suit for continuous system?
– The analog computer provides limited accuracy because op amps have many assumptions which can never be true in reality. – In reality, the system is of neither a pure continuous nor a pure discrete nature.
How does continuous simulation modeling differ from process simulation?
In continuous simulation all processing is typically performed in time periods that begin at regular intervals of simulated time. As you can see, in the continuous simulation the events occur at regular intervals, while in the discrete-event simulation the events occur at irregular intervals.
What are differential equations used for in engineering?
Differential equations are mathematical tools to model engineering systems such as hydraulic flow, heat transfer, level controller of a tank, vibration isolation, electrical circuits, etc.
How are differential equations used in real life?
Ordinary differential equations applications in real life are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. Also, in medical terms, they are used to check the growth of diseases in graphical representation.
What are differential equations?
An equation relating a function to one or more of its derivatives is called a differential equation. The subject of differential equations is one of the most interesting and useful areas of mathematics. We can describe many interesting natural phenomena that involve change using differential equations.
What is the difference equation in digital control?
From the digital control schematic, we can see that a difference equation shows the relationship between an input signal e ( k) and an output signal u ( k) at discrete intervals of time where k represents the index of the sample. For example, if the sample time is a constant T, then e ( k) represents the value of e at the time kT.
How to find the transfer function of a continuous system?
Recall that a transfer function for a continuous system as we have considered so far is derived by first taking the Laplace transform of a set of differential equations and then rearranging the results into the form Output/Input.
What is the difference between continuous and differentiable functions?
In passing, even if you are not familiar with the logical construct called implication or inference it should be understood that while it is true that if a function is differentiable at then it is continuous at it is not the case that if a function is continuous at then it is differentiable at .