the decomposition method applied to systems of fredholm integral equations of the second kind in this paper the adomian decomposition method is applied to solve systems of linear and nonlinear fredholm integral equations of the second kind convergence of the method is proved and some examples are presented to illustrate the method

thermodynamic characterization of networks using graph polynomials in this paper we present a method for characterizing the evolution of time varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial or algebraic characterization of graph structure commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized laplacian matrix we show how the polynomial is linked to the boltzmann partition function of a network this allows us to compute a number of thermodynamic quantities for the network including the average energy and entropy assuming that the system does not change volume we can also compute the temperature defined as the rate of change of entropy with energy all three thermodynamic variables can be approximated using low order taylor series that can be computed using the traces of powers of the laplacian matrix avoiding explicit computation of the normalized laplacian spectrum these polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy energy and temperature we show how these thermodynamic variables can be computed in terms of simple network characteristics e g the total number of nodes and node degree statistics for nodes connected by edges we apply the resulting thermodynamic characterization to real world time varying networks representing complex systems in the financial and biological domains the study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

constructing tables of minimum aberration pn m designs fries and hunter 1980 presented a practical algorithm for selecting standard 2 n m fractional factorial designs based on a criterion they called minimum aberration in this article some simple results are presented that enable the fries hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p 2 is prime examples of minimum aberration 2 n m designs with resolution r 4 are given for m n m 9 a matrix is given for generating 3 n m designs with m n m 6 which have or nearly have minimum aberration

global monotonic solutions of multi term fractional differential equations abstract based on the leray schauder principle a fixed point theorem is established to study the existence of a global monotonic solution for some multi term differential equations of fractional type some existence result for the inclusion problem is proved

minimum aberration designs with two level and four level factors summary for optimal assignment of factors to columns of a 4 2 n design we consider extensions of the minimum aberration criterion for the 2 level designs minimum aberration 412n and 422n designs in 16 and 32 runs are obtained and tabulated some results on the wordlength patterns and design classification are obtained

adomian decomposition a tool for solving a system of fractional differential equations abstract adomian decomposition method has been employed to obtain solutions of a system of fractional differential equations convergence of the method has been discussed with some illustrative examples in particular for the initial value problem d 1 y 1 d n y n t a y 1 y n t y i 0 c i i 1 n where a a i j is a real square matrix the solution turns out to be y x e 1 n 1 x 1 a 1 x n a n y 0 where e 1 n 1 denotes multivariate mittag leffler function defined for matrix arguments and a i is the matrix having ith row as a i 1 a i n and all other entries are zero fractional oscillation and bagley torvik equations are solved as illustrative examples

regular fractional factorial designs with minimum aberration and maximum estimation capacity using the approach of finite projective geometry we make a systematic study of estimation capacity a criterion of model robustness under the absence of interactions involving three or more factors some general results providing designs with maximum estimation capacity are obtained in particular for two level factorials it is seen that constructing a design with maximum estimation capacity calls for choosing points from a finite projective geometry such that the number of lines is maximized and the distribution of these lines among the chosen points is as uniform as possible we also explore the connection with minimum aberration designs under which the sizes of the alias sets of two factor interactions which are not aliased with main effects are the most uniform possible

solution of a system of volterra integral equations of the first kind by adomian method in this article we consider linear and non linear systems of integral equations of the first kind the problem of existence and uniqueness are considered the adomian decomposition method is being used to solve these linear and non linear systems of volterra integral equations of the first kind some examples are prepared to show the efficiency and simplicity of the method

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

solution of the system of ordinary differential equations by adomian decomposition method in this article we use adomian decomposition method which is a well known method for solving functional equations now a days to solve systems of differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one theoretical considerations are being discussed and convergence of the method for theses systems is addressed some examples are presented to show the ability of the method for linear and non linear systems of differential equations

existence of fractional integral equation with hysteresis the existence of solutions to an integral equation of fractional order with hysteresis has been established a fixed point theorem due to schauder is the main tool in carrying out our proof

set valued system of fractional differential equations with hysteresis in this paper we investigate the global solvability in l 1 0 1 of a set valued system of nonlinear fractional differential equations with hysteresis some existence theorems for both single and multivalued systems are proved

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

numerical algorithms for caputo fractional order differential equations abstractthe initial value problems ivps of caputo fractional order differential equations are very important in control systems modelling and simulation a series of numerical algorithms are proposed in the paper in solving systematically various kinds of caputo equations for linear caputo equations the divergent problems of the existing taylor auxiliary function are pointed out and two effective algorithms are presented to transform the nonzero ivps into zero ones where the closed form solutions are available furthermore algorithms for nonlinear caputo equation are also presented aiming at finding numerical solutions to all kinds of caputo equations error analysis for the proposed algorithms is provided and numerical examples are presented to illustrate the accuracy and effectiveness of the algorithms

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

non linear boundary value problems involving caputo derivatives of complex fractional order abstract we study approximate solutions of c d t y t f t y t separately for 0 1 and 1 2 with different boundary data where c d t is the caputo fractional derivative of complex order for this purpose we use the expansion formula for such fractional derivatives and prove the existence and the uniqueness of approximate solutions under certain conditions and their convergence to the original solutions

solving initial and boundary value problems of fractional ordinary differential equations by using collocation and fractional powers abstract this paper presents a novel approach for solving initial and boundary values problems on ordinary fractional differential equations the strategy considered is based on the use of a suitable truncated series expressed in terms of fractional powers of the independent variable and a collocation approach with this strategy accurate numerical approximations can be obtained the error criterion is based on the concepts of residue and relative error two algorithms fitted to the nature linear or nonlinear of the considered fractional differential equation are proposed finally some problems illustrating the efficiency of the proposed method are presented

theory of j characteristics for fractional factorial designs and projection justification of minimum g2 aberration deng tang 1999 introduced the generalised resolution and minimum g aberration criteria for assessing nonregular fractional factorials in tang deng 1999 a relaxed variant of minimum g aberration called minimum g 2 aberration is proposed and studied these criteria are defined using a set of j values called j characteristics in this paper we show that a factorial design is uniquely determined by its j characteristics just as a regular factorial design is uniquely determined by its defining relation the theorem is given through an explicit formula that relates the set of design points to that of j characteristics through this formula projection justification of minimum g 2 aberration is established

some theory for constructing minimum aberration fractional factorial designs summary minimum aberration is the most established criterion for selecting a regular fractional factorial design of maximum resolution minimum aberration designs for n runs and n 2 m n factors have previously been constructed using the novel idea of complementary designs in this paper an alternative method of construction is developed by relating the wordlength pattern of designs to the so called confounding between experimental runs this allows minimum aberration designs to be constructed for n runs and 5n 16 m n 2 factors as well as for n 2 m n

positive solutions of a system of non autonomous fractional differential equations abstract existence of positive solutions for the following system of fractional differential equations d i u i f i t u 1 u 2 u n u i 0 0 0 i 1 1 i n where d i denotes riemann liouville derivative of order i has been studied

homotopy analysis method for solving multi term linear and nonlinear diffusion wave equations of fractional order abstract in this paper we have used the homotopy analysis method ham to obtain solutions of multi term linear and nonlinear diffusion wave equations of fractional order the fractional derivative is described in the caputo sense some illustrative examples have been presented

on fractional differential inclusions with the jumarie derivative in the paper fractional differential inclusions with the jumarie derivative are studied we discuss the existence and uniqueness of a solution to such problems our study relies on standard variational methods

application of the operational matrix of fractional order legendre functions for solving the time fractional convection diffusion equation the operational matrix of fractional order legendre functions method are considered the fractional order convection diffusion problem is solved the problem can be used extensively in science and engineering as in oil reservoir simulations problems which an initially discontinuous profile is propagated by diffusion and convection in this paper the application of the operational matrix of fractional order legendre functions flfs to solve the time fractional convection diffusion equation has been investigated fractional calculus has been applied to model the engineering and physical processes which are best described with other mathematical tools the time variable of the time fractional convection diffusion equation and its space variable have been approximated by flfs and shifted legendre polynomials respectively the fractional derivatives together with product matrices of flfs are employed to convert the solution of this problem to the solution of a system of algebraic equations

new ideas for proving convergence of decomposition methods in this paper we give new formulae which calculate easily the adomian s polynomials used in decomposition methods then the proof of convergence of the adomian s technique becomes almost obvious by using a weak hypothesis on the nonlinear operator of the functional equation

stable evaluations of fractional derivative of the muntz legendre polynomials and application to fractional differential equations abstract the aim of this paper is to present efficient and stable methods to compute caputo fractional derivative cfd of the muntz legendre polynomials based on three term recurrence relations and gauss jacobi quadrature rules this approach with collocation method at chebyshev gauss lobatto points has been applied for solving linear and nonlinear fractional multi order differential equations fdes described in caputo sense the main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations in this work for the first time we present the new rates of convergence for projection error which are more accurate than the rate presented by shen and wang in shen and wang 2016 moreover we present convergence rate for spectral collocation method for linear fdes with initial value on a finite interval and endpoint singularities also we propose an error analysis for jacobi gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper finally the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples

on the fractional differential equations in this paper we are concerned with the semilinear differential equation d ax t dt a f t x t t 0 where a is any positive real number in 3 the author has proved the existence uniqueness and some properties of the solution of this equation when 0 a 1 here we mainly study besides the other properties the continuation of the solution of this equation to the solution of the corresponding initial value problem when a k k 1 2 3 applications of singular integro differential equations are considered

linear differential equations of fractional orders abstract we are concerned here with the linear differential equations of fractional orders we will obtain some results concerning the existence uniqueness and some other properties of the solution analogous and more general to that obtained for the linear ordinary differential equations

on the normalized laplacian energy and general randic index r 1 of graphs in this paper we consider the energy of a simple graph with respect to its normalized laplacian eigenvalues which we call the l energy over graphs of order n that contain no isolated vertices we characterize the graphs with minimal l energy of 2 and maximal l energy of 2 n 2 we provide upper and lower bounds for l energy based on its general randic index r 1 g we highlight known results for r 1 g most of which assume g is a tree we extend an upper bound of r 1 g known for trees to connected graphs we provide bounds on the l energy in terms of other parameters one of which is the energy with respect to the adjacency matrix finally we discuss the maximum change of l energy and r 1 g upon edge deletion

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

numerical solutions for fractional differential equations by tau collocation method the main purpose of this paper is to provide an efficient numerical approach for multi order fractional differential equations based on a tau collocation method to do this multi order fractional differential equations transformed into a system of nonlinear algebraic equations in matrix form thus by solving this system unknown coefficients are obtained the fractional derivatives are described in the caputo sense the rate of convergence for the proposed method is established in the l w p norm some numerical example is also provided to illustrate our results the results reveal that the method is very effective and simple

minimum aberration and model robustness for two level fractional factorial designs summary the performance of minimum aberration two level fractional factorial designs is studied under two criteria of model robustness simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived it is also shown that a minimum aberration design of resolution ilil or higher maximizes the number of two factor interactions which are not aliases of main effects and subject to that condition minimizes the sum of squares of the sizes of alias sets of two factor interactions this roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform it follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here examples are given to show that minimum aberration designs are indeed highly efficient

solving a multi order fractional differential equation using adomian decomposition abstract an algorithm has been developed to convert the multi order fractional differential equation d y t f t y t d 1 y t d n y t y k 0 c k k 0 m where m d denotes caputo fractional derivative of order into a system of fractional differential equations further adomian decomposition method has been employed to solve the system of fractional differential equations some illustrative examples are presented

fractional order legendre laguerre functions and their applications in fractional partial differential equations abstract in this paper we consider a new fractional function based on legendre and laguerre polynomials for solving a class of linear and nonlinear time space fractional partial differential equations with variable coefficients the concept of the fractional derivative is utilized in the caputo sense the idea of solving these problems is based on operational and pseudo operational matrices of integer and fractional order integration with collocation method we convert the problem to a system of algebraic equations by applying the operational matrices pseudo operational matrices and collocation method also we calculate the upper bound for the error of integral operational matrix of the fractional order we illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph we also describe the physical application of some examples

a quadrature method for numerical solutions of fractional differential equations in this article a numerical method is developed to obtain approximate solutions for a certain class of fractional differential equations the method reduces the underlying differential equation to system of algebraic equations an algorithm is presented to compute the coefficient matrix for the resulting algebraic system several examples with numerical simulations are provided to illustrate effectiveness of the method

set valued integral equations of fractional orders the topic of fractional calculus derivative and integral of arbitrary orders is enjoying growing interest not only among mathematicians but also among physicists and engineers see 6 16 18 20 23 25 the set valued integral equations integral inclusions arises in the study of control system see 21 22 26 in this paper we prove the existence of locally bounded variation solution of a volterra type set valued integral equation of arbitrary not necessarily integer order the proof will be based on the measure of weak noncompactness and the existence of caratheodory selectors as a consequence we study the initial value problem for some set valued differential and integro differential equations the corresponding single valued problems will be firstly considered

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

the characteristic polynomial of the laplacian of graphs in a b linear classes in this work we deal with the characteristic polynomial of the laplacian of a graph we present some general results about the coefficients of this polynomial we present families of graphs for which the number of edges m is given by a linear function of the number of vertices n in some of these graphs we can find certain coefficients of the above named polynomial as functions just of n

on the existence of continuous solutions for a singular system of non linear fractional differential equations abstract in the this paper we establish the existence of continuous solutions to some non linear fractional integral and differential equations quadratic system of fractional integral equations cases will be considered our analysis rely on krasnoselskii s fixed point theorem on a cone

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

minimum aberration two level split plot designs the treatment design portion of fractionated two level split plot designs is associated with a subset of the 2 n k fractional factorial designs the concept of aberration is then extended to these splitplot designs to compare designs two methods are presented for constructing two level minimumaberration split plot designs along with examples an extensive catalog of such designs is tabulated extensions to prime level designs and relations to inner outer arrays are also presented

some identities on q sp n m designs with application to minimum aberration designs chen and hedayat and tang and wu studied and characterized minimum aberration 2 n m designs in terms of their complemetary designs based on a new and more powerful approach we extend the study to identify minimum aberration q n m designs through their complementary designs by using macwilliams identities and krawtchouk polynomials in coding theory we obtain some general and explicit relationships between the wordlength pattern of a q n m design and that of its complementary design these identities provide a powerful tool for characterizing minimum aberration q n m designs the case of q 3 is studied in more details

connections between two level designs of resolutions iii and v resolution iii designs are resolution iii designs in which no two factor interactions are confounded with one another they arise naturally in the search for small composite designs and were first suggested by hartley 1959 new results given here link resolution iii designs to resolution v designs enabling passage between the two types of designs to be made this means that previously untabulated resolution iii designs can be derived directly from known resolution v designs for example a hartley type composite design can be based on a 27 1 iii design derived from a 26 1 v design the usual composite design is based on a 27 1 v design double the number of factorial runs the resolution iii designs can also be usefully modified for specific industrial applications box and jones 1989 moreover the maximum number of factors it is possible to accommodate in a resolution iii design of a given size can be obtained directly from the known maximum for resolution v designs of the same size

solving a system of nonlinear fractional differential equations using adomian decomposition adomian decomposition method has been employed to obtain solutions of a system of nonlinear fractional differential equations dxiyi x ni x y1 yn yi k 0 cki 0 k i 1 i n and dxi denotes caputo fractional derivative some examples are solved as illustrations using symbolic computation

multiplicity of positive solutions of a class of nonlinear fractional differential equations this paper is concerned with the nonlinear fractional differential equation l d u f x u u 0 0 0 x 1 where l d dsn an 1dsn 1 a1ds11 0 j 1 2 n 1 some results are obtained for the existence nonexistence and multiplicity of positive solutions of the above equation by using krasnoselskii s fixed point theorem in a cone in particular it is proved that the above equation has n positive solutions under suitable conditions where n is an arbitrary positive integer

fractional resolution and minimum aberration in blocked 2 n k designs systematic sources of variation can be reduced in fractional factorial experiments by grouping the runs into blocks this is accomplished through the use of blocking factors the concepts of resolution and minimum aberration design optimization criteria ordinarily used to rank unblocked fractional factorial designs are extended to such blocked fractional factorial designs by treating the treatment and blocking factors differently in terms of their contribution to word length in the defining contrast subgroup some limited theoretical results are derived and tables of minimumaberration blocked two level fractional factorial designs are presented and considered the relationship between clear effects effects that are estimable when higher order effects are assumed negligible and minimum aberration in the presence of blocking is discussed