Question

1. Explain the concept of an initial-value problem for a differential equation (You may use examples to support your answer). 2. Find an example of a boundary value problem for a partial differential equation (you can write this in the same formas described in the lecture notes) 3. Consider the following equation: som AS (a) Identify the equation (by name). (b) Identify the variables in the equation: f and t. (C) Explain what the left and right hand sides of the equation represent physically. 4. Recall, a wave is defined as a disturbance originating at a point in a medium and propagating! at a finite speed within the medium. Give three examples of wave phenomena in nature of mathematics. 5. Briefly explain the concept of Huygen's principle. You may use an example to support your explantations, 6. Find an example of a (a) linear partial differential equation. (b) nonlinear partial differential equation. 7. Explain the concept of a reference frame. You may use examples to support your explanta tions 8. What is an inertial reference frame? You may use examples to support your explantations 9. What is a conservation law? Provide an example to support your answer 10. Briefly explain the ideas behind Noethor's Theorem. What mathematical structures are kry to this result?

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Answer 1

1. Tritial value problem to be solveel ander ito derivative Variable, then the When an ordinary differential equation ? Conditiomo povolving dependent variable and at the same value of the independent problem under consideration is said to be an initial value problem . For example: for First order ordinary differential equation : home - - with y13) = 4 is an initial valve problem For second oncler ordinary differential equation: - is an initial value problema for nth order ordinary differential equatim: & yt the one + Oures over het to Dalus dan net een + Conrad y = 4(a) with 9 10) = Qo, y'(a) = 4, , yº(a) = 4, --. , yu(a) = 4me -tarot Onlar dan-a Where ai (x1) and dire) are defined and continuous on azxeb is an initial value problem. go all abuse example y(a) is ebolution of their respective clifferential equation !

climension o clifferential equations : Simibry , we can consider higher ge: 78. -88 +62n = with -lok, tuig 12x2 2,(0) = -4 ya (0) = 4 37(0) = dt -2x + xat H3 = da is an initial value problem.