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Increasing demand of vital resources such as water and energy will impose some overwhelming environmental degradation, particularly on wetlands as the most vulnerable bodies of the environment. Consequently, optimization of water and energy portfolios have been widely studied in order to ensure environmental sustainability and to consider the constraints simultaneously. In this study improving the security of portfolios has been performed based on resource security index to find non-dominated portfolios (Pareto Frontier). Non-dominated portfolios as scenarios to remediate water bankruptcy have been developed and probable environmental impacts have been assessed. A new Financial-Environmental Index called Water-Energy-Environment Nexus Security Index (WEENSI) have been utilized through a multi criteria decision making approach to find admiringly compatible non-dominated water and energy portfolios for environmental remediation purposes in the case of the Lake Urmia catchment as an epitome of water bankruptcy situation. Results indicate that portfolio based management approaches could be a desired solution to water bankruptcy and an inspiring option for environmental remediation, particularly in wetland catchments. It has further become known that persisting on the conventional water resources management in this catchment will increase water demand of energy sector up to more than 500 million cubic meters (MCM) in 2060, exacerbating the current critical environmental condition. Also, increasing the share of renewable energies at least up to 40% must be taken into account by managers and policy makers. Broadly speaking, any development of energy sector requires an urgent change in the currently practiced approach and considerable investment in non-conventional energy resources. The increase in primary costs of optimizing and improving the water and energy portfolios may alleviate the anthropogenic impacts with high social costs to the region.

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a b s t r a c tThis paper is concerned with new algorithms which provide the sharp bounds that are guaranteed to contain the exact solutions of nonlinear Volterra integral equations. We develop new enclosure algorithms based on the interval methods which was first introduced by Moore in [24] together with the Taylor polynomials to improve the accuracy of the scheme by reducing the width of interval solutions. The modified methods calculate a priori bound automatically in parallel with the computation of solutions of integral equations. We will show that the accuracy of the proposed algorithms is dependent on the number of interval subdivisions. Some numerical experiments are also included to demonstrate the validity and applicability of the scheme and showing a marked improvement in comparison with the recent existing numerical results.

Abstract.In this paper, we compute piecewise constant bounds on the solution of mixed Mathematical subject classification: 65G20, 45G10, 65G40.