A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line

The modified generalized Laguerre-Gauss collocation (MGLC) method is applied to obtain an approximate solution of fractional neutral functional-differential equations with proportional delays on the half-line. The proposed technique is based on modified generalized Laguerre polynomials and Gauss quadrature integration of such polynomials. The main advantage of the present method is to reduce the solution of fractional neutral functional-differential equations into a system of algebraic equations. Reasonable numerical results are achieved by choosing few modified generalized Laguerre-Gauss collocation points. Numerical results demonstrate the accuracy, efficiency, and versatility of the proposed method on the half-line

Integration of Process Modeling, Design, and Optimization with an Experimental Study of a Solar-Driven Humidification and Dehumidification Desalination System

Solar energy is becoming a promising source of heat and power for electrical generation and desalination plants. In this work, an integrated study of modeling, optimization, and experimental work is undertaken for a parabolic trough concentrator combined with a humidification and dehumidification desalination unit. The objective is to study the design performance and economic feasibility of a solar-driven desalination system. The design involves the circulation of a closed loop of synthetic blend motor oil in the concentrators and the desalination unit heat input section. The air circulation in the humidification and dehumidification unit operates in a closed loop, where the circulating water runs during the daytime and requires only makeup feed water to maintain the humidifier water level. Energy losses are reduced by minimizing the waste of treated streams. The process is environmentally friendly, since no significant chemical treatment is required. Design, construction, and operation are performed, and the system is analyzed at different circulating oil and air flow rates to obtain the optimum operating conditions. A case study in Saudi Arabia is carried out. The study reveals unit capability of producing 24.31 kg/day at a circulating air rate of 0.0631 kg/s and oil circulation rate of 0.0983 kg/s. The tradeoff between productivity, gain output ratio, and production cost revealed a unit cost of 12.54 US$/m3. The impact of the circulating water temperature has been tracked and shown to positively influence the process productivity. At a high productivity rate, the humidifier efficiency was found to be 69.1%, and the thermal efficiency was determined to be 82.94%. The efficiency of the parabolic trough collectors improved with the closed loop oil circulation, and the highest performance was achieved from noon until 14:00 p.m

New operational matrices for solving fractional differential equations on the half-line.

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques

Maximum absolute error for <i>γ</i> = 0.01, λ=12 and different values of <i>N</i> and <i>α</i> in <i>x</i> ∈ [0, 100] for Example 3.

<p>Maximum absolute error for <i>γ</i> = 0.01, </p><p></p><p><mi>λ</mi><mo>=</mo></p><p><mn>1</mn><mn>2</mn></p><p></p><p></p> and different values of <i>N</i> and <i>α</i> in <i>x</i> ∈ [0, 100] for Example 3.<p></p

Comparing the exact solution and approximate solutions at <i>N</i> = 4, 6, where <i>α</i> = 0, λ=34 and <i>γ</i> = 0.1, for problem Eq (74).

<p>Comparing the exact solution and approximate solutions at <i>N</i> = 4, 6, where <i>α</i> = 0, </p><p></p><p><mi>λ</mi><mo>=</mo></p><p><mn>3</mn><mn>4</mn></p><p></p><p></p> and <i>γ</i> = 0.1, for problem <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0126620#pone.0126620.e126" target="_blank">Eq (74)</a>.<p></p

The values <i>c</i>₀, <i>c</i>₁, <i>c</i>₂, … and <i>c</i>₆ for different values of <i>α</i> at ν=14 for Example 1.

<p>The values <i>c</i>₀, <i>c</i>₁, <i>c</i>₂, … and <i>c</i>₆ for different values of <i>α</i> at </p><p></p><p><mi>ν</mi><mo>=</mo></p><p><mn>1</mn><mn>4</mn></p><p></p><p></p> for Example 1.<p></p

Graph of the absolute error function for <i>N</i> = 6, <i>α</i> = 0, λ=34 and <i>γ</i> = 0.1, for Example 4.

<p>Graph of the absolute error function for <i>N</i> = 6, <i>α</i> = 0, </p><p></p><p><mi>λ</mi><mo>=</mo></p><p><mn>3</mn><mn>4</mn></p><p></p><p></p> and <i>γ</i> = 0.1, for Example 4.<p></p

Maximum absolute error using FGLC method with various choices of <i>α</i> at <i>N</i> = 4 for Example 5.

<p>Maximum absolute error using FGLC method with various choices of <i>α</i> at <i>N</i> = 4 for Example 5.</p