derive diffrential equations
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Do you know the law of cooling? Please state it for us.
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Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings). Stated in diifrential equation, dT/dt = k T-Ta, where T is the temperature of the object and Ta is the temperature of the surroundings. Now my question is that what is the procedure to derive the above equation. Why we need differential equation at all. By experimentation we get data and plotting, results in exponential curve and i already have a solution for predicting the temperature of the object at any time.
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I would say the differential equation is the cornerstone of the physical sciences. Experimental data is fine, but it is the mathematical model which allows us to gain further insight into why the experiment behaves the way it does.
The differential equation describing Newton's Law of Cooling comes directly from its statement in words.
The differential equation describing Newton's Law of Cooling comes directly from its statement in words.
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With enough data from many different experiments you might begin to notice that the slope of the temperature curve is always very nearly proportional to the difference between the temperature of the object begin studied and the ambient temperature. You might also notice that for the same object, but different initial and ambient temperatures the constant of proportionality remains the same. You might then notice that objects made of different materials have their own unique constant of proportionality. So you may begin to think of this constant as a heat transfer coefficient.
Hi MarkFL
One of my question is still unanswered. What is the reason behind expressing these physical laws through differential equations. Since we have generalized solution in the first place, what is the need to find out instantaneous rate of change (in this case). At least in speed time graph, I have reason to find the slope to estimate velocity and acceleration. But I do not find the utility of derivative in the expression of law of cooling.
One of my question is still unanswered. What is the reason behind expressing these physical laws through differential equations. Since we have generalized solution in the first place, what is the need to find out instantaneous rate of change (in this case). At least in speed time graph, I have reason to find the slope to estimate velocity and acceleration. But I do not find the utility of derivative in the expression of law of cooling.
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First, we have the observations of some system made by a scientist. Careful measurements are taken and patterns begin to emerge.
The scientist then takes these observations and tries to model the system mathematically, usually in the form of an initial value problem (IVP) containing a differential equation, where all assumptions and boundaries are stated explicitly.
If the scientist is lucky, there is a solution for the differential equation. If not, then numerical techniques are used to approximate the solution, or the equation is linearized.
From the solution, predictions can be made to either verify the model, or discredit it. Once it is verified by experimentation over time, it may eventually become known as a law.
Because differential equations relate rates of change, and things change with time, position, or other variables, the resulting model will frequently be in the form of a differential equation.
garimella,
The differential equation for newton's law of cooling is the simplified form of energy equation with specific boundary conditions. The solution involving the exponential that you have is obtained by integrating the differential equation and it has been verified experimentally. You can use the solution directly, but remember that it has been obtained by integrating the differential equation.
The basics conservation laws are typically coupled partial differential equations. The energy equation is usually uncoupled an ODE. Sometimes, these equations also have an integral form depending on how they are applied.
These equations are typically solved numerically as MarkFL explained.
Cheers,
Sai.
The differential equation for newton's law of cooling is the simplified form of energy equation with specific boundary conditions. The solution involving the exponential that you have is obtained by integrating the differential equation and it has been verified experimentally. You can use the solution directly, but remember that it has been obtained by integrating the differential equation.
The basics conservation laws are typically coupled partial differential equations. The energy equation is usually uncoupled an ODE. Sometimes, these equations also have an integral form depending on how they are applied.
These equations are typically solved numerically as MarkFL explained.
Cheers,
Sai.