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Editorial Fuzzy Linear and Nonlinear Integral Equations...
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Editorial Fuzzy Linear and Nonlinear Integral Equations: Numerical Methods Reza Ezzati, 1 Soheil Salahshour, 2 Ronald R. Yager, 3 and Morteza Khodabin 1 1 Department of Mathematics, College of Basic Sciences, Islamic Azad University, Karaj Branch, Alborz, Iran 2 Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran 3 Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA Correspondence should be addressed to Reza Ezzati; [email protected] Received 17 November 2014; Accepted 17 November 2014; Published 22 December 2014 Copyright © 2014 Reza Ezzati et al. is 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. Integral equations are one of the most useful mathematical tools in both pure and applied mathematics. ey have enormous applications in many real problems. Many initial and boundary value problems associated with ordinary dif- ferential equation (ODE) and partial differential equation (PDE) can be transformed into problems of solving some approximate integral equations. Indeed, modeling such problems using integral equations with the exact parameters is not only easy but also impossible in the real problems. For this purpose, one way is using some uncertainty measures for handling such lack of information. One of the most and recent approaches is using Zadeh’s fuzzy concept. So, instead of using deterministic models, we provide fuzzy integral equations of both linear and nonlinear forms. In fact, obtaining the exact solutions of such fuzzy integral equations is not possible in all cases because of the inherited restrictions form application of fuzzy concepts in these problems. So, in this special issue, we intend to consider the numerical methods to solve fuzzy integral equations and the related topics with real applications. ese topics include fuzzy linear and nonlinear integral equations with numerical methods, investigating the convergence, stability, and con- sistency of numerical approaches, numerically modeling the real problems associated with numerical methods, consider- ing the differences between deterministic and fuzzy numer- ical methods to solve fuzzy integral equations, numerically solving fuzzy differential equations of arbitrary order using the equivalence fuzzy integral equations, obtaining some approximations of the solutions via ranking approaches, and applications in real-world problems with numerical techniques. Our special issue contains few papers in which different numerical techniques are employed. e paper “A simpli- fied Milstein scheme for SPDEs with multiplicative noise” replaces the exponential term with a Pad´ e approximation of order 1 and denotes the resulting scheme by simplified Milstein scheme. e paper “On properties of pseudointegrals based on pseudoaddition decomposable measures” discussed pseudointegrals based on a pseudoaddition decomposable measure. Particularly, the definition of the pseudointegral for a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function was stated. e paper “Quadrature rules and iterative method for numerical solution of two-dimensional fuzzy integral equations” introduced some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy- number-valued functions. Also, it gave error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, it proposed an iterative procedure based on quadrature formula to solve two-dimensional lin- ear fuzzy Fredholm integral equations of the second kind (2DFFLIE2) and presented the error estimation of the pro- posed method. e paper “On solution of integrodifferential equation with delay parameter by Sinc basis functions” is considered. For this purpose, a numerical solution is obtained for an integrodifferential equation with an integral boundary Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 147351, 2 pages http://dx.doi.org/10.1155/2014/147351
Transcript of Editorial Fuzzy Linear and Nonlinear Integral Equations...
EditorialFuzzy Linear and Nonlinear Integral Equations:Numerical Methods
Reza Ezzati,1 Soheil Salahshour,2 Ronald R. Yager,3 and Morteza Khodabin1
1 Department of Mathematics, College of Basic Sciences, Islamic Azad University, Karaj Branch, Alborz, Iran2 Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran3Machine Intelligence Institute, Iona College, New Rochelle, NY 10801, USA
Correspondence should be addressed to Reza Ezzati; [email protected]
Received 17 November 2014; Accepted 17 November 2014; Published 22 December 2014
Copyright © 2014 Reza Ezzati et al. 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.
Integral equations are one of the most useful mathematicaltools in both pure and applied mathematics. They haveenormous applications in many real problems. Many initialand boundary value problems associated with ordinary dif-ferential equation (ODE) and partial differential equation(PDE) can be transformed into problems of solving someapproximate integral equations.
Indeed, modeling such problems using integral equationswith the exact parameters is not only easy but also impossiblein the real problems. For this purpose, one way is using someuncertainty measures for handling such lack of information.One of the most and recent approaches is using Zadeh’sfuzzy concept. So, instead of using deterministic models, weprovide fuzzy integral equations of both linear and nonlinearforms.
In fact, obtaining the exact solutions of such fuzzy integralequations is not possible in all cases because of the inheritedrestrictions form application of fuzzy concepts in theseproblems. So, in this special issue, we intend to considerthe numerical methods to solve fuzzy integral equations andthe related topics with real applications. These topics includefuzzy linear and nonlinear integral equations with numericalmethods, investigating the convergence, stability, and con-sistency of numerical approaches, numerically modeling thereal problems associated with numerical methods, consider-ing the differences between deterministic and fuzzy numer-ical methods to solve fuzzy integral equations, numericallysolving fuzzy differential equations of arbitrary order usingthe equivalence fuzzy integral equations, obtaining some
approximations of the solutions via ranking approaches,and applications in real-world problems with numericaltechniques.
Our special issue contains few papers in which differentnumerical techniques are employed. The paper “A simpli-fied Milstein scheme for SPDEs with multiplicative noise”replaces the exponential term with a Pade approximationof order 1 and denotes the resulting scheme by simplifiedMilstein scheme. The paper “On properties of pseudointegralsbased on pseudoaddition decomposable measures” discussedpseudointegrals based on a pseudoaddition decomposablemeasure. Particularly, the definition of the pseudointegralfor a measurable function based on a strict pseudoadditiondecomposable measure by generalizing the definition ofthe pseudointegral of a bounded measurable function wasstated. The paper “Quadrature rules and iterative methodfor numerical solution of two-dimensional fuzzy integralequations” introduced some generalized quadrature rules toapproximate two-dimensional, Henstock integral of fuzzy-number-valued functions. Also, it gave error bounds formappings of bounded variation in terms of uniformmodulusof continuity. Moreover, it proposed an iterative procedurebased on quadrature formula to solve two-dimensional lin-ear fuzzy Fredholm integral equations of the second kind(2DFFLIE2) and presented the error estimation of the pro-posed method. The paper “On solution of integrodifferentialequation with delay parameter by Sinc basis functions” isconsidered. For this purpose, a numerical solution is obtainedfor an integrodifferential equation with an integral boundary
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 147351, 2 pageshttp://dx.doi.org/10.1155/2014/147351
2 Abstract and Applied Analysis
condition and delay parameter. This type of problems arisesin mathematical physics, mechanics, population growth, andother fields of physics and mathematical chemistry. Then,convergence of this approach is discussed by presenting atheorem which gives exponential type convergence rate andguarantees the accuracy of that. The paper “A new recon-struction of variational iteration method and its application tononlinear Volterra integrodifferential equations” is proposed.Indeed, it reconstructed the variational iteration method,that is, the so-called parametric iteration method (PIM).The proposed method was applied for solving nonlinearVolterra integrodifferential equations (NVIDEs). The paper“Approximating the solution of the linear and nonlinear fuzzyVolterra integrodifferential equations using expansionmethod”is considered. To this end, it introduced an innovativemethodapplying power series to solve numerically the linear andnonlinear fuzzy integrodifferential equation systems.
We hope the papers published in this special issue willprovide a useful guide to a large community of researchersand will give way to development of new innovative theoriesand numerical approaches in the fields of modeling andapproximating fuzzy integral equations and the related topics.
Acknowledgments
We thank all the authors and the honorable reviewers whocontributed to this special issue.
Reza EzzatiSoheil SalahshourRonald R. Yager
Morteza Khodabin
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