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S. Abbasb - One of the best experts on this subject based on the ideXlab platform.
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Research Article Modified fractional Euler method for solving Fuzzy sequential Fractional Initial Value Problem under H-differentiability
2016Co-Authors: Roshanfekr H. Varazgahi, S. AbbasbAbstract:Copyright 2015 c ⃝ H. Roshanfekr Varazgahi and S. Abbasbandy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the solution to Fuzzy sequential Fractional Initial Value Problem [FFIVP] under Caputo type fuzzy frac-tional derivatives by a modified fractional Euler method is presented. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylors Formula and a modified trapezoidal rule is used for solving initial value problem under fuzzy sequential fractional differential equation of order 0 < b < 1. Solving two examples of linear and nonlinear FFIVP illustrates the method
Roshanfekr H. Varazgahi - One of the best experts on this subject based on the ideXlab platform.
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Research Article Modified fractional Euler method for solving Fuzzy sequential Fractional Initial Value Problem under H-differentiability
2016Co-Authors: Roshanfekr H. Varazgahi, S. AbbasbAbstract:Copyright 2015 c ⃝ H. Roshanfekr Varazgahi and S. Abbasbandy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the solution to Fuzzy sequential Fractional Initial Value Problem [FFIVP] under Caputo type fuzzy frac-tional derivatives by a modified fractional Euler method is presented. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylors Formula and a modified trapezoidal rule is used for solving initial value problem under fuzzy sequential fractional differential equation of order 0 < b < 1. Solving two examples of linear and nonlinear FFIVP illustrates the method
Carlos Alberto Conceição António - One of the best experts on this subject based on the ideXlab platform.
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Análise Matemática I
2007Co-Authors: Carlos Alberto Conceição AntónioAbstract:A. Cálculo diferencial em R: Revisão de alguns conceitos e resultados. Teorema dos Acréscimos Finitos (Lagrange). Diferenciais de Funções de uma variável - Definição. Regras de cálculo e aplicações. Aproximação Polinominal - Polinómios de Taylor e fórmula de Taylor com resto; aplicações. Série de Taylor como limite dos polinómios de Taylor. Séries numéricas: propriedades das séries, critérios de convergência, séries alternadas. Breve referência às séries de funções. Conceito de intervalo de convergência. B. Integral de Riemann em R: Integração de funções reais de variável real - Integral de Riemann, sua definição e propriedades. Teoremas do valor médio para integrais. Teoremas Fundamentais do Cálculo. O Conceito de Primitiva - Regras de Primitivação por substituição e por partes. Aplicações do integral ao cálculo de áreas em coordenadas cartesianas e polares e ao cálculo de volumes. Primitivação de fracções racionais algébricas. Primitivação de expressões racionais trigonométricas. Primitivação de expressões irracionais por substituição trigonométrica. C. Tópicos adicionais: Funções hiperbólicas. Integrais impróprios. Equações diferenciais de primeira ordem.A. Differential Calculus in R:Review of fundamentals of differentiation.Increments, differentials and linear approximations. The mean-value theorem for derivatives.Polynomial approximations to functions: The Taylor polynomials generated by a function.Taylors Formula with remainder. Estimates for the error in Taylors Formula. The Taylor series as a limit of Taylor polynomials. Numerical series: properties, convergence criteria, alternating series.Reference of functional series. Concept of convergence interval.B. Integral Calculus in R:Riemann sums and the integral. Integrability of bounded monotonic functions. The integrability theorem for continues functions. Properties of the integral. Mean-value theorem for integrals.The derivative of an indefinite integral. The first fundamental theorem of calculus.Primitive functions and the second fundamental theorem of calculus.Integration by substitution. Integration by parts. Areas of plane regions. Polar coordinates. Area calculation in polar coordinates. Volume calculations by the method of cross sections.Integration by rational partial fractions. Rational trigonometric integrals. Integrals containing quadratic polynomials.C. Additional topics:Hyperbolic functions. Improper integral.First order differential equation
Evremar Åsa - One of the best experts on this subject based on the ideXlab platform.
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Avdunstningens höjdberoende i svenska fjällområden bestämd ur vattenbalans och med modellering
SMHI, 1994Co-Authors: Evremar ÅsaAbstract:The evapotranspiration, one of the main parts of the waterbalance, is not well known in theSwedish mountains. The evapotranspiration is difficult to measure reliably and measurementsare only rarely done. To predict the available water quantity for hydro power generation or forthe risks for darnaging high flows, it is important to know the water balance in the mountains.Since many of the driving variables of the evapotranspiration are dependent on the altitude, itwas investigated in this work if the variation of evaporation in the mountains could beapproximated in simple functional form. The aim of this work was to make clear if theevaporation in the Swedish mountains has a dependence on the altitude and to investigate ifsuch a possible dependence could be used to improve the HBV-111odel. This was done bycalculating the evapotranspiration as a residual in the water balance equation for a number ofhigh-altitude drainage basins in the river Ljusnan, the river Indalsälven and the river Luleälven.The calculations were done with longtime mean values from the years 1961-90. The altitudedependence was calculated by linear regression. The result was inserted in the HBV-model as acorrection factor in the calculation of the evapotranspiration. It was also tested if anyimprovements in the model could be achieved by introducing an altitude dependence of thevariables in the Priestley-Taylor Formula. In this case, evapotranspiration was calculated with adaily resolution. Simulations with the HBV-model were performed for the drainage basinsLjusnedal, Torrön and Kultsjön. Evaporation calculated as a water balance residual, decreased24 mm/100 m and year for the river Ljusnan, 54 mm/100 m and year for the river Indalsälvenand 48 mm/ 100 m and year for the river Luleälven. The result of the calculation in the riverLuleälven was condensation in a number of areas. This was probably caused by too fewraingauges in the area. The explained variance <luring simulations with the HBV-modelincreased in the area of Kultsjön if a dependence on the altitude was inserted. The explainedvariance however decreased for the areas of Torrön and Ljusnedal. lf a dependence on thealtitude of the evapotranspiration was inserted in the model the spring flood decreased fasterand the peaks of the water discharge were higher <luring summer. The model produced worseresults if the standard version sirnulated a spring flood that was too short and the dischargepeaks <luring summer that were too high. The Priestley-Taylors Formula led toa calculatedevapotranspiration in the model that decreased with 40 mm/100 m per year in the drainge basinof Torrön and 23 mm/100 m per year of the drainage basin of Kultsjön. The conclusions of thiswork was that the evapotranspiration in the Swedish mountains decreases with about 30 mm/100 m per year and that knowledge may irnprove the HBV-model in some cases.Avdunstningen, en av huvudkomponenterna i vattenbalansen, är dåligt känd i svenska fjälltrakter. Den är svår att mäta tillförlitligt och mätningar görs endast i undantagsfall. För att säkrare kunna förutsäga den tillgängliga tillrinningen för kraftgenerering eller riskerna för skadegörande höga flöden är det viktigt att känna till vattenbalansen i fjällen. Eftersom många av avdunstningens drivvariabler beror av höjden över havet undersöktes i detta arbete om avdunstningens variation i fjällen kunde beskrivas med ett enkelt samband. Målet med arbetet var att klarlägga om avdunstningen i svenska fjälltrakter har ett höjdberoende samt att undersöka om stt eventuellt sådant beroende kunde användas för att förbättra HBV-modellen. Detta gjordes genom att beräkna avdunstningen som en restterm ur vattenbalansekvationen för ett antal högt liggande avrinningsområden i Ljusnan, lndalsälven och Luleälven. Vid beräkningarna användes långtidsårsmedelvärden 1961-90. Genom linjär regression beräknades ett höjdberoende. Resultatet infördes i HBV-modellen som en korrektionsfaktor i avdunstningsberäkningen. I modellen testades även om förbättringar kunde fås genom att införa ett höjdberoende hos ingående variabler i Priestley-Taylors formel och dygnsvis beräkna den potentiella avdunstningen i det simulerade området. Simuleringar med HBV-modellen gjordes för områdena Ljusnedal, Torrön och Kultsjön. Resultatet av beräkningarna ur vattenbalansen blev att avdunstningen i områdena i Ljusnan sjönk med 24 mrn/100 m och år. Avdunstningen i lndalsälven sjönk med54 mm/100 m och år. För områdena i Luleälven ledde beräkningarna till kondensation i flera områden, troligtvis beroende på för få nederbördsmätare i områdena. Trots detta fick avdunstningen även där ett höjdberoende som var 48 mrn/100 m och år. Den förklarade variansen vid simuleringar med HBV- modellen ökade för Kultsjön vid införandet av höjdberoende hos avdunstningen. Den minskade dock för Torrön och Ljusnedal. Skillnaden då man inför ett höjdberoende hos avdunstningen blev att vårfloden avtog snabbare och vattenföringstopparna under sommaren blev större. Modellen försämrades i de fall då modellen i standardutförande simulerade en för kort vårflod och för höga vattenforingstoppar under sommaren. Beräkningarna med Priestley-Taylors formel ledde till att modellens beräknade verkliga avdunstning avtog med 40 mm/100 m och år i Torröns avrinningsområde och 23 mm/100m och år i Kultsjöns avrinningsområde. Slutsatserna blev att avdunstningen avtar med ca 30 mm/100 m och år i svenska fjälltrakter och att HBV-modellen i vissa fall kan förbättras med denna kunskap
