Consider the problem of heat conduction in a one dimensional slab (0
where u(x, 1) is the temperature.
Using the separation of variable method seek the solution ) u(x,t) = T t) X (x) and find
i) equation for T (t), X (x) and corresponding conditions on X (x) to solve these equations
ii) the eigenvalues and the corresponding eigenfunctions of the bvp obtained in i) above for ) X (x)
iii) the solution u(x,t) .
We will solve the heat conduction problem using the separation of variables method. --- ### Step 1: Problem Statement We are given the heat equation: \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0 \leq x \leq 1, t > 0 \] with initial and boundary conditions: 1. Initial Condition: \[ u(x,0) = u_0(x) \] 2. Boundary Conditions: \[ \frac{\partial u}{\partial x}(0,t) = 0 \quad \text{(Neumann Condition at \(x=0\))} \] \[ \frac{\partial u}{\partial x}(1,t) + u(1,t) = 0 \quad \text{(Robin Condition at \(x=1\))} \] We will seek a ______ _______ ________ _____ ______ __________ ________ ____ _____ _______ _________.
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Using dimensional analysis, show that the planets obey Kepler's third law.
A raindrop starts falling from the clouds at a considerable height above the surface of the earth. During the fall, the raindrop experiences retardation due to air resistance, which is directly proportional to the instantaneous speed v(t) of the drop.
i) Write the model equations
ii) Is this system static or dynamic? Why?
iii) Obtain an expression for the speed v(t).
iv) Discuss the behaviour of v(t) as the time t changes.
There are N identical firms producing a particular commodity. The cost function for each firm producing q units is q3 + 2q2 + 4q + 6 units of money. Obtain the supply function for each firm. The demand function is . ( Find ) the equilibrium price.
एक टेलीविजन मिस्त्री, जो एक टेलीविजन की मरम्मत में औसतन घंटे लेता है, के पास प्रतिदिन 4 टेलीविज़न की औसत दर से टेलीविजन मरम्मत के लिए यदृच्छया आते हैं। यहां यह मान लिया गया है कि मरम्मत में लगने वाला समय चरघातांकीय बंटित है। वर्कशॉप में टेलीविजन की औसत संख्या क्या है? इस बात की प्रायिकता क्या होगी कि मरम्मत के लिए आने वाले टेलीविजन के पहले कम से कम तीन टेलीविज़न रखे होगें? मिस्त्री प्रति दिन आठ घंटा काम करता है।
A stone is dropped vertically from a tower of height h. At the same time, another stone is thrown vertically upwards from the base of the tower with a velocity u. What is the minimum value of u so that the two stones will meet each other mid- air?
मान लीजिए एक उपभोक्ता का उपयोगिता फलन है। मान लीजिए ,
और इस अवधि में उपभोक्ता की आय रु. 15000/- है। उपभोक्ता के लिए आवश्यक वे मात्रा ज्ञात कीजिए जिससे कि इस संचय हाता उपभोग करने पर उसका उपयोगिता फलन अधिकतम हो जाए।
Let the utility function of a consumer be and that the consumers income for the period is Rs.15000. Obtain the quantities required by the consumer so that his utility function is maximized by consuming this combination.