1. Applied Partial Differential Equations Haberman Solutions Manual Pdf
2. Applied Partial Differential Equations Haberman
3. Applied Partial Differential Equations Haberman 5th Edition Solutions Manual Pdf

(a) Consider the case where there are no sources of heat. If, on the interval, then heat flux is an increasing function of x. In other words, the heat flux leaving the cross-sectional surface from its right is more than that entering the surface from its left. Therefore, the heat energy decreases on the one-dimensional rod.

Applied Partial Differential Equations Haberman Solutions Manual Pdf

If is negative at x, it implies that the heat flux is a decreasing function of x. In other words, the heat flux leaving the cross-sectional surface from its right is less than that entering the surface from its left.

Thus, the heat energy increases on the one-dimensional rod. Therefore, it is concluded that rate of change in heat flux has a sign opposite to that of the rate of change in heat energy. (b) If is positive, then the temperature is an increasing function of x for fixed t. In other words, temperature on the right of cross-sectional surface at x is more than that on the left side of the surface. Thus, heat flows from right to left across the surface. If is negative, then the temperature is a decreasing function of x for fixed t.

In other words, temperature on the right of cross-sectional surface at x is less than that on the left of the surface. Thus, heat flows from left to right across the surface. Hence, heat flux a sign opposite to the rate of change of temperature. Therefore, it is concluded that the rate of change in temperature has a sign opposite to that of heat flux.

Applied Partial Differential Equations Haberman

(c) If is positive at x, it implies that the heat flux leaving the cross-sectional surface from its right is more than that entering the surface from its left. Therefore, the chemical concentration on the surface decreases with respect to time t. If is negative at x, it implies that the heat flux leaving the cross-sectional surface from its right is less than the surface entering from its left. Thus, the chemical concentration on the surface increases with respect to time t. Therefore, the rate of change in heat flux has a sign opposite to that of the rate of change in chemical concentration.

(d) If is positive, then the chemical concentration to the right of the cross-sectional surface at x is more than that on the left side of the surface. Thus, the particles flow from right to left across the surface.

Applied Partial Differential Equations Haberman 5th Edition Solutions Manual Pdf

If is negative, it implies that the chemical concentration to the right of the cross-sectional surface at x is less than that at the left side of the surface. Thus, the particles flow from left to right across the surface. Hence, the rate of change in chemical concentration has a sign opposite to that of the rate of change of heat flux.

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