fuzzy fractional differential equations with the generalized atangana baleanu fractional derivative abstract in this paper we introduce a generalization of atangana baleanu type fractional calculus with respect to the generalized mittag leffler kernel which has been named as the generalized atangana baleanu gab type fractional calculus existence and uniqueness results for the initial value problems of fuzzy differential equations involving a gab fractional derivative in the caputo sense are established by employing the method of successive approximation and by means of fixed point theorems to visualize the theoretical results some examples and numerical simulations are given

a boundary value problem for second order fuzzy differential equations abstract in this paper we interpret a two point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept we present a new concept of solutions and utilizing the generalized differentiability we investigate the problem of finding new solutions some examples are provided for which the new solutions are found

on a class of fuzzy functional differential equations in this paper we establish the local and global existence and uniqueness results for fuzzy functional differential equations for the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle also we apply these results to fuzzy differential equations with distributed delays and fuzzy population models

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

approximations of fractional integrals and caputo fractional derivatives abstract in this paper we propose two algorithms for numerical fractional integration and caputo fractional differentiation we present a modification of trapezoidal rule that is used to approximate finite integrals the new modification extends the application of the rule to approximate integrals of arbitrary order 0 we then using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order 0 where the fractional derivative based on caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points the study is conducted through illustrative examples and error analysis

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

a survey on fuzzy fractional variational problems abstract this paper focuses on introducing and studying generalized fuzzy euler lagrange equation and fuzzy isoperimetric problem according to the concept of caputo type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability we extend and establish some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional euler lagrange equation for both constrained and unconstrained fuzzy fractional variational problems the fuzzy isoperimetric problem is also investigated in relation to the generalized fuzzy euler lagrange finally two examples are given to describe the proposed approach

existence results for extremal solutions of interval fractional functional integro differential equations abstract in this study we introduce interval fractional functional integro differential equations iffides under the caputo generalized hukuhara fractional differentiability we establish some necessary comparison results and use the monotone iterative technique combined with the method of upper and lower solutions to investigate the existence of extremal solutions for iffides an example is presented to illustrate the results

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

generalized differentiability of fuzzy valued functions in the present paper using novel generalizations of the hukuhara difference for fuzzy sets we introduce and study new generalized differentiability concepts for fuzzy valued functions several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them characterization and relatively simple expressions are provided for the new derivatives

fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives in this paper a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated the aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner the optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs the fuzzy fractional dynamical system is described using fuzzy fractional differential equations ffdes in order to achieve the aim an effective approach for solving ffdes should be at disposal due to some restrictions imposed by the previous approaches dealing with ffdes a new approach is proposed the proposed approach is based on the granular derivative and the so called relative distance measure fuzzy interval arithmetic new definitions of fuzzy fractional derivatives and integral called left and right granular riemann liouville fuzzy fractional derivatives left and right granular caputo fuzzy fractional derivatives and the left and right granular fuzzy fractional integral are also presented in addition the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced by the approximations of the granular fuzzy fractional integral and the granular caputo fuzzy fractional derivative the approximation solution to the ffdes is obtained consequently based on the new concepts and theorems the solution to the fuzzy fractional quadratic regulator problem is given by a theorem this paper closes with an example of regulating the motion of boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients

on semi convergence of the parameterized generalized mhss method for singular complex linear systems for solving a class of complex symmetric singular linear systems we propose a parameterized generalized mhss pgmhss iteration method and investigate the semi convergence conditions by analyzing the spectrum of the iteration matrix then we analyze the optimal iteration parameters that minimize the upper bound of the semi convergence factor numerical experiments are used to test the feasibility and the effectiveness of the pgmhss iteration method

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

fuzzy fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

existence theorems for solutions to random fuzzy differential equations abstract we examine a random fuzzy initial value problem with two kinds of fuzzy derivatives for both cases we establish the results of existence and uniqueness of local solutions to random fuzzy differential equations the existence of global solutions is also obtained

second type hukuhara differentiable solutions to the delay set valued differential equations abstract in this paper we consider the delay set valued differential equations with a recently introduced notion of a second type hukuhara derivative under condition that the right hand side of the equation is lipschitzian in the functional variable we obtain the existence and uniqueness of the solution to such the equations some properties of the solutions are established existence of at least one solution is also proved application of the second type hukuhara differentiability concept to the problems of set valued differential equations implies that the diameter of the solution values is a function nonincreasing in time

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

generalized taylor s formula abstract in this paper a new generalized taylor s formula of the kind f x j 0 n a j x a j r n x where a j r x a 0 1 is established such an expression is precisely the classical taylor s formula in case of 1 in addition detailed expressions for r n x and a j that involving caputo fractional derivatives are well constructed some applications including approximation of functions and solutions of fractional differential equations are also given

adaptive t s fuzzy control design for fractional order systems with parametric uncertainty and input constraint abstract thanks to the superiority of fractional differential equations in modeling high technology dynamical real world systems the design and control of fractional order systems have been widely investigated in recent decades on the other hand supposing that dynamic equations of the fractional order systems usually include various parameters and nonlinear terms takagi sugeno t s fuzzy models with if then rules can used to characterize the systems dynamics also the control community knows that there are usually some nonlinearities such as dead zone backlash hysteresis and saturation in control actuators which can decrease the ideal prescribed performance of the fractional systems the saturation nonlinearity is perhaps the most commonly existed one in control applications of fractional nonlinear systems that should be compensated by a proper method so this article proposes a novel adaptive t s fuzzy variable structure control technique for nonlinear fractional order systems in spite of the saturating input and control fluctuations the parameters of the fractional system as well as those of the saturation function are presumed to be unknown accordingly update rules are extracted to estimate such fluctuations then simple adaptive linear like control rules are developed according to the t s fuzzy control theory the stability of the origin equilibrium point of the final closed loop system is assured by lyapunov stability theory at last comparative computer simulations are presented to highlight the accuracy and applicability of the proposed adaptive fuzzy saturated control methodology

a new bernoulli wavelet method for accurate solutions of nonlinear fuzzy hammerstein volterra delay integral equations in this article bernoulli wavelet method has been developed to solve nonlinear fuzzy hammerstein volterra integral equations with constant delay this type of integral equation has a particular case the fuzzy variant of a mathematical model from epidemiology bernoulli wavelets have been generated by dilation and translation of bernoulli polynomials the properties of bernoulli wavelets and bernoulli polynomials are first presented the present wavelet method reduces these integral equations to a system of nonlinear algebraic equations and again these algebraic systems have been solved numerically by newton s method convergence analysis of the present method has been discussed in this article also the results obtained by present wavelet method have been compared with that of by b spline wavelet method some illustrative examples have been demonstrated to show the applicability and accuracy of the present method

neutral fuzzy fractional functional differential equations abstract in this paper a class of fuzzy fractional functional differential equations of neutral type is investigated the existence uniqueness and iterative formula in the case of linear equations are obtained meanwhile the continuous dependence of solutions on initial data parameters and functions involved in right hand side of equation is also discussed several examples are given to illustrate the applications of our results

on a system of fuzzy fractional differential inclusions with projection operators abstract the aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators which captures the desired features of both fuzzy differential inclusions and fractional order projective dynamical systems within the same framework the existence of solutions for the open situation is proved by using continuous selection theorem moreover the existence of solutions for the closed situation is showed by employing lipschitzean selection theorem and fixed point theorem respectively furthermore an application and a numerical example are also given

solving fuzzy differential equations by differential transformation method an extension of the differential transformation method dtm which is an analytical numerical method for solving the fuzzy differential equation fde is given the concept of generalised h differentiability is used this concept is based on an enlargement of the class of differentiable fuzzy mappings to define this the lateral hukuhara derivatives are considered the proposed algorithm is illustrated by numerical examples and some error comparisons are made with other methods for solving a fde

solving complex valued linear systems via equivalent real formulations most preconditioned iterative methods apply to both real and complex valued linear systems at the same time most iterative linear solver packages available today focus exclusively on real valued systems or deal with complex valued systems as an afterthought by recasting the complex problem in a real formulation a real valued solver can be applied to the equivalent real system on one hand real formulations have been dismissed due to their unfavorable spectral properties on the other hand using an equivalent preconditioned real formulation can be very effective we give theoretical and experimental evidence that an equivalent real formulation is useful in a number of practical situations furthermore we show how to use the advanced features of modern solver packages to formulate equivalent real preconditioners that are computationally efficient and mathematically identical to their complex counterparts the effectiveness of equivalent real formulations is demonstrated by solving ill conditioned complex valued linear systems for a variety of large scale applications moreover the circumstances under which certain equivalent real formulations are competitive is more clearly delineated

hyers ulam stability of fuzzy fractional volterra integral equations with the kernel ψ function via successive approximation method abstract in this paper we investigative the hyers ulam stability and the hyers ulam rassias stability of fuzzy fractional volterra integral equations ffvies involving the kernel function we propose conditions for the existence uniqueness and stability of ffvie by employing the method of successive approximation to visualize the theoretical results some examples and numerical simulations are given

on the solutions of first order linear impulsive fuzzy differential equations abstract this paper gives a representation of solutions to first order linear impulsive fuzzy differential equations with constant coefficients using a generalized differentiability concept and an impulsive fundamental matrix we present ideas similar to those in linear fuzzy differential equations for impulsive fuzzy cauchy problems finally numerical examples are given to illustrate the theory

fuzzy delay differential equations under generalized differentiability abstract we interpret a fuzzy delay differential equation using the concept of generalized differentiability in this setting we prove the existence of two fuzzy solutions each one corresponding to a different type of differentiability uniqueness is understood in the sense that the solutions considered do not have switching points the applicability of the theoretical results is illustrated with some real world examples

an mhss like iteration method for two by two linear systems with application to fde optimization problems abstract in this paper we exploit the numerical solvers for a fractional differential equations fde optimization problem with constraints given by fractional elliptic state equations the discretized linear system can be rewritten as a two by two linear system by making use of the strategy of modified hermitian and skew hermitian splitting mhss iteration method proposed by bai we propose an mhss like iteration method comparing with the mhss iteration method the mhss like method only requires one to solve the two by two linear systems by a sparse cholesky factorization and a fast fourier transform hence the mhss like method has less workload than the mhss method the convergence properties and the quasi optimal parameters are analyzed in detail numerical examples are used to testify the efficiency of the new method

analytical and numerical solutions of fuzzy differential equations in this paper we study analytical and numerical solutions of fuzzy differential equations based on the extension principle for linear fuzzy differential equations we state some results on the behaviour of the solutions and study their relationship with the generalised hukuhara derivative in order to approximate the solutions of linear and non linear fuzzy differential equations we propose a new fuzzification of the classical euler method and then incorporate an unconstrained optimisation technique this combination offers a powerful tool to tackle uncertainty in any numerical method an efficient computational algorithm is also provided to guarantee the convexity of fuzzy solutions on the time domain several illustrative examples are given

necessary optimality conditions for fuzzy variational problems the main aims of this study are to derive the fuzzy euler lagrange conditions for both fuzzy unconstrained and constrained variational problems based on the concepts of differentiability and integrability of a fuzzy mapping that may be parameterized by the left and right hand functions of its a level sets

variation of constant formula for first order fuzzy differential equations in this paper we study first order linear fuzzy differential equations by using the generalized differentiability concept and we present the general form of their solutions we also correct and complete some previous results finally some examples are given to illustrate our results

predicting monthly streamflow using data driven models coupled with data preprocessing techniques 1 in this paper the accuracy performance of monthly streamflow forecasts is discussed when using data driven modeling techniques on the streamflow series a crisp distributed support vectors regression cdsvr model was proposed for monthly streamflow prediction in comparison with four other models autoregressive moving average arma k nearest neighbors knn artificial neural networks anns and crisp distributed artificial neural networks cdann with respect to distributed models of cdsvr and cdann the fuzzy c means fcm clustering technique first split the flow data into three subsets low medium and high levels according to the magnitudes of the data and then three single svrs or anns were fitted to three subsets this paper gives a detailed analysis on reconstruction of dynamics that was used to identify the configuration of all models except for arma to improve the model performance the data preprocessing techniques of singular spectrum analysis ssa and or moving average ma were coupled with all five models some discussions were presented 1 on the number of neighbors in knn 2 on the configuration of ann and 3 on the investigation of effects of ma and ssa two streamflow series from different locations in china xiangjiaba and danjiangkou were applied for the analysis of forecasting forecasts were conducted at four different horizons 1 3 6 and 1 2 month ahead forecasts the results showed that models fed by preprocessed data performed better than models fed by original data and cdsvr outperformed other models except for at a 6 month ahead horizon for danjiangkou for the perspective of streamflow series the ssa exhibited better effects on danjingkou data because its raw discharge series was more complex than the discharge of xiangjiaba the ma considerably improved the performance of ann cdann and cdsvr by adjusting the correlation relationship between input components and output of models it was also found that the performance of cdsvr deteriorated with the increase of the forecast horizon

a note on two point boundary value problems associated with non linear fuzzy differential equations in recent papers lakshmikantham et al and o regan et al assert that two point boundary value problem for a fuzzy differential equation is equivalent to an integral equation and the proofs of the main results in the above cited papers use this equivalence we prove by a counterexample that by using h differentiability or hukuhara differentiability this assertion does not hold moreover it is proved that a large class of two point boundary value problems have no solutions at all under h differentiability or hukuhara differentiability

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

a class of efficient parameterized shift splitting preconditioners for block two by two linear systems abstract based on the effective constructing an equivalent block two by two linear systems we establish a class of efficient parameterized shift splitting eps preconditioners for solving block two by two linear systems theoretical analysis shows that the eps iteration method is unconditionally convergent and then we study the spectral properties of the corresponding preconditioned matrix by removing one of the shift term of the eps preconditioners two classes of local eps preconditioners are presented and the eigenvalues distribution of the preconditioned matrices are also studied numerical experiments are provided to demonstrate the feasibility and effectiveness of the proposed preconditioners

a survey on the initial value problems of fuzzy implicit fractional differential equations abstract the purpose of this paper is to present some basic theories of various classes of initial value problems of implicit fractional differential equation involving the caputo fuzzy type riemann liouville fuzzy type and hadamard fuzzy type concepts of fractional derivative in the case of the order 0 1 in particular the existence and uniqueness results of solution for the given initial value problems of implicit fractional differential equations is investigated in addition some results concerning continuous dependence of solutions on perturbations of the given parameters namely of the initial conditions of the function and parameters on the right hand side of the order of fractional derivative are analyzed finally some examples are given to illustrate our main results

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

ulam type stability of first order linear impulsive fuzzy differential equations abstract in this paper we consider ulam stability concepts for first order linear impulsive fuzzy differential equations which generalize standard definitions for impulsive differential equations sufficient conditions are established to guarantee that our equations are ulam stable numerically examples are also provided to illustrate our results

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

solving interval valued fractional initial value problems under caputo gh fractional differentiability abstract in this paper interval valued fractional differential equations ifdes under caputo generalized hukuhara differentiability are introduced we present existence and uniqueness results for ifdes with a krasnoselskii krein type condition the solution to interval valued fractional initial value problems under caputo type interval valued fractional derivatives by a modified fractional euler method mfem and a modified adams bashforth moulton method mabmm is also presented

a note on initial value problems for fractional fuzzy differential equations abstract in this note we present some remarks on solutions of fractional fuzzy differential equation in general a fractional fuzzy differential equation and a fractional fuzzy integral equation are not equivalent in this paper we give an appropriate condition so that this equivalence is valid some examples are provided to illustrate the theory

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

parameterized rotated block preconditioning techniques for block two by two systems with application to complex linear systems in this paper based on the rotated block triangular preconditioning technique we present a class of parameterized rotated block preconditioners to accelerate krylov subspace iterative methods with coefficient matrices of nonsymmetric sub blocks the parameterized rotated block preconditioners can be seen as a generalization of the rotated block triangular preconditioners the eigen properties of the corresponding preconditioned matrices are analyzed numerical experiments are used to demonstrate the feasibility and effectiveness of the parameterized rotated block preconditioners