Answer :
The heat equation is a partial differential equation used to study the behavior of heat transfer. Its solution is a sine series with coefficients of 3/2, -1/2, 1/4, and -1/8 for u(x,0). For u(x,t), each coefficient is multiplied by a factor of e^(-2^2t), e^(-4^2t), and e^(-6^2t) corresponding to the respective frequencies of the sine waves, allowing the solution to accurately represent how the temperature distribution changes over time.
u(x,0) = 3/2 - 1/2cos(2x) + 1/4cos(4x) - 1/8cos(6x) and u(x,t) = 3/2 + (1/2)e^(-2^2t)cos(2x) + (1/4)e^(-4^2t)cos(4x) + (1/8)e^(-6^2t)cos(6x)
The heat equation is a partial differential equation that describes how a temperature distribution changes over time in a given space. It is primarily used to study the behavior of heat transfer in a variety of physical systems. The solution to the heat equation is a sine series, which is a series of sine waves with different frequencies and amplitudes. The sine series for u(x,0) includes the first four nonzero terms, each with its own coefficient. The coefficients are all reduced fractions with no trig functions or decimals. These coefficients are 3/2, -1/2, 1/4, and -1/8. The solution to the heat equation, u(x,t), is also a sine series, which includes the same coefficients as the sine series for u(x,0). However, each coefficient is multiplied by a factor of e^(-2^2t) for the 2x frequency sine wave, e^(-4^2t) for the 4x frequency sine wave, and e^(-6^2t) for the 6x frequency sine wave. This factor takes into account the time factor in the heat equation, allowing the solution to accurately represent how the temperature distribution changes over time.
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