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Fig. 1. (a) Concentration values obtained from solution (Eq. (28a)) represented by dotted curves for f(t) = (1 — mt)” and full line curve for f(t) = exp(—mt). (b) Concentration values obtained from solution (Eq. (28b)) represented by dotted curves for f(t) = (1 — mt)~! and full line curve for f(t) = exp(—mt).

Figure 1 (a) Concentration values obtained from solution (Eq. (28a)) represented by dotted curves for f(t) = (1 — mt)” and full line curve for f(t) = exp(—mt). (b) Concentration values obtained from solution (Eq. (28b)) represented by dotted curves for f(t) = (1 — mt)~! and full line curve for f(t) = exp(—mt).

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where T is a new time variable given by the transformation in Eq. (10), Substituting y = 0 in the solutions of Eqs. (28a) and (28b) we get the solutions for a similar dispersion of the uniform input in a medium which has uniformly distributed initial concen- tration. But for a varying input such particular case cannot be obtained from the solution in Eqs. (33a) and (33b). Instead, it can be obtained by putting y = 0 in the solution of Eq. (32) and then proceeding further.
where T is a new time variable given by the transformation in Eq. (10), Substituting y = 0 in the solutions of Eqs. (28a) and (28b) we get the solutions for a similar dispersion of the uniform input in a medium which has uniformly distributed initial concen- tration. But for a varying input such particular case cannot be obtained from the solution in Eqs. (33a) and (33b). Instead, it can be obtained by putting y = 0 in the solution of Eq. (32) and then proceeding further.
Fig. 2. (a) Concentration values obtained from solution (Eq. (33a)) represented by dotted curves for f(t) = (1 — mt)~' and full line curve for f(t) = exp(—mt). (b) Concentration values obtained from solution (Eq. (33b)) represented by dotted curves for f(t) = (1 — mt)~’ and full line curve for f(t) = exp(—mit). As stated in Section 1, the concentration values are evalu- ated from the analytical solutions in the context of pollutants dispersion along a flow in a finite domain 0 < x < 1 (km) for the input data Cp = 1.0, C; = 0.01, Do = 0.14, up = 0.25. It may be noted that the dimensions of Do and up are (km*/year) and (km/year), respectively. In addition to these, m = 0.1(year)' and y = 0.01 (km)~' in Section 2.1 and a = 1.0 (km)! in Section 2.2 are considered. The time of elimination of the source of pollution is considered at f9 = 1.8 years. The solutions in the time domain ¢ < fo are computed at times + (years) = 0.4, 1.0 and 1.6, and those in the time domain ft > fg are computed at ¢ (years) = 1.9, 2.5 and 3.1.
Fig. 2. (a) Concentration values obtained from solution (Eq. (33a)) represented by dotted curves for f(t) = (1 — mt)~' and full line curve for f(t) = exp(—mt). (b) Concentration values obtained from solution (Eq. (33b)) represented by dotted curves for f(t) = (1 — mt)~’ and full line curve for f(t) = exp(—mit). As stated in Section 1, the concentration values are evalu- ated from the analytical solutions in the context of pollutants dispersion along a flow in a finite domain 0 < x < 1 (km) for the input data Cp = 1.0, C; = 0.01, Do = 0.14, up = 0.25. It may be noted that the dimensions of Do and up are (km*/year) and (km/year), respectively. In addition to these, m = 0.1(year)' and y = 0.01 (km)~' in Section 2.1 and a = 1.0 (km)! in Section 2.2 are considered. The time of elimination of the source of pollution is considered at f9 = 1.8 years. The solutions in the time domain ¢ < fo are computed at times + (years) = 0.4, 1.0 and 1.6, and those in the time domain ft > fg are computed at ¢ (years) = 1.9, 2.5 and 3.1.
Fig. 3. (a) Concentration values obtained from solution (Eq. (51a)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1. (b) Concentration values obtained from solution (Eq. (51b)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1.
Fig. 3. (a) Concentration values obtained from solution (Eq. (51a)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1. (b) Concentration values obtained from solution (Eq. (51b)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1.
Fig. 4. (a) Concentration values obtained from solution (Eq. (54a)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1. (b) Concentration values obtained from solution (Eq. (54b)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1.
Fig. 4. (a) Concentration values obtained from solution (Eq. (54a)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1. (b) Concentration values obtained from solution (Eq. (54b)) represented by dotted curves for a = 1.0 and full line curve for a = 0.1.
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EngineeringMathematicsEarth SciencesLaplace TransformSolute transportLaplace transformationSpatial DependenceAdvection diffusion equation
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