The temperature in an oven is controlled by a heater which provides heat at rate . The temperature of a pizza in the oven is . Room temperature is the constant value .
and satisfy the coupled differential equations
where and are positive constants. Briefly explain the various terms appearing in the above equations.
Heating may be provided by a short-lived pulse at , with or by constant heating over a finite period , with , where and are respectively the Dirac delta function and the Heaviside step function. Again briefly, explain how the given formulae for and are consistent with their description and why the total heat supplied by the two heating protocols is the same.
For . Find the solutions for and for , for each of and , denoted respectively by and , and and . Explain clearly any assumptions that you make about continuity of the solutions in time.
Show that the solutions and tend respectively to and in the limit as and explain why.