The numerical solutions of linear integrodifferential equations of Volterra type have been considered. Power series is used as the basis polynomial to approximate the solution of the problem. Furthermore, standard and Chebyshev-Gauss-Lobatto collocation points were, respectively, chosen to collocate the approximate solution. Numerical experiments are performed on some sample problems already solved by homotopy analysis method and finite difference methods. Comparison of the absolute error is obtained from the present method and those from aforementioned methods. It is also observed that the absolute errors obtained are very low establishing convergence and computational efficiency.
We establish results on multifunction associated with a set of solutions of non-Lipschitz quantum stochastic differential inclusion (QSDI), which still admits a continuous selection from some subsets of complex numbers. The results here generalize existing results.
Screened plant species with potential for green belt development can act as eco-sustainable tools for restoring the polluted ecosystem. Eight plant species from two study locations in Ado-Odo, Ota, Ogun State, Nigeria, were examined to identify their air pollution response and performance by deploying two air pollution indices, namely air pollution tolerance index (APTI) and anticipated performance index (API). APTI results identified all screened plants as sensitive species suitable as bio-indicators of air pollution, with Ficus auriculata (2.42) common to the non-industrial location being the most sensitive. API scores categorized Ficus auriculata (56.25%) as a moderate performer, while Syzygium malaccense (75%) and Mangifera indica (75%) were identified as very good performers, suitable for green belt development. The relationship between each biochemical parameter with APTI was investigated using regression analysis and two-way analysis of variance. The model result showed a significant relationship between each biochemical parameter with APTI, and relative water content had the highest influence on APTI (R2 = 0.99436). Both indices (APTI and API) are suitable for screening and recommending native plant species for cultivation in the polluted environment, thus promoting ecological restoration. Hence, Syzygium malaccense, Mangifera indica and Ficus auriculata, respectively, were recommended for green belts design. Further intensive screening to identify tolerant species and best to excellent performer’s trees suitable for restoring the ecosystem is advised.
Convolution is the sum of independent and identically distributed random variables. Derivatives of the probability density function (PDF) of probability distribution often lead to the construction of ordinary differential equation whose solution is the PDF of the given distribution. Little have been done to extend the construction of the ODE to the PDF, quantile function (QF), survival function (SF), hazard function (HF) and the reversed hazard function (RHF) of convoluted probability distributions. In this paper, three probability distributions were considered namely: Constant parameter convoluted exponential distribution (CPCED), convoluted uniform exponential distribution (CUED) and different parameter convoluted exponential distribution (DPCED). First order ordinary differential equations whose solutions were the PDF, SF, HF and RHF for the probability functions of CPCED by the use of differential calculus. The case of the QF was second order nonlinear differential equations obtained by the use of Quantile Mechanics. Similarly, the same was obtained for CUED for the two cases of the distribution. Some new relationships were obtained for the PDF, SF and HF, and also the RHF, PDF and CDF with their corresponding first derivatives. The difficulty of obtaining the ODE for the probability functions of the DPCED was due to the different parameters that characterize the distribution. The use of partial different equations is not an alternate because the distribution has only one independent variable.
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