cas picard method for fractional nonlinear differential equation in this paper a computational method for solving the fractional nonlinear differential equation is introduced we proposed a method by utilizing the cas wavelets in conjunction with picard technique we call the method as cas picard method the fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by picard technique and then transformed into a system of algebraic equations by using the operational matrices of cas wavelets the error and supporting analysis of the method are also investigated the comparison analysis of method with other existing numerical methods is also performed

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

the classification of the single traveling wave solutions to the generalized equal width equation abstract the complete discrimination system for polynomial is applied to the generalized equal width ew equation to obtain the classifications of single traveling wave solutions in particular we construct the corresponding solutions for the concrete parameters to show that each solution in the classification can be realized

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

solitary waves of the equal width wave equation a numerical solution of the equal width wave equation based on galerkin s method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves the interaction of two solitary waves is seen to cause the creation of a source for solitary waves usually these are of small magnitude but when the amplitudes of the two interacting waves are equal and opposite the source produces trains of solitary waves whose amplitudes are of the same order as those of the initiating waves the three invariants of the motion are evaluated to determine the conservation properties of the system finally the temporal evolution of a maxwellian initial pulse is studied for small ut uux uxxt 0 only positive waves are formed and the behaviour mimics that of the kdv and filw equations for larger values of d both positive and negative solitary waves are generated

nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces abstract time series averaging in dynamic time warping dtw spaces has been successfully applied to improve pattern recognition systems this article proposes and analyzes subgradient methods for the problem of finding a sample mean in dtw spaces the class of subgradient methods generalizes existing sample mean algorithms such as dtw barycenter averaging dba we show that dba is a majorize minimize algorithm that converges to necessary conditions of optimality after finitely many iterations empirical results show that for increasing sample sizes the proposed stochastic subgradient ssg algorithm is more stable and finds better solutions in shorter time than the dba algorithm on average therefore ssg is useful in online settings and for non small sample sizes the theoretical and empirical results open new paths for devising sample mean algorithms nonsmooth optimization methods and modified variants of pairwise averaging methods

the matrix laguerre transform the laguerre transform introduced by keilson and nunn 1979 and keilson nunn sumita 1981 provides an algorithmic basis for the computation of multiple convolutions in conjunction with other algebraic and summation operations the methods enable one to evaluate numerically a variety of results in applied probability and statistics that have been available only formally for certain more complicated models the formulation must be extended in this paper we establish the matrix laguerre transform appropriate for the study of semi markov processes and markov renewal processes as an extension of the scalar laguerre transform the new formalism enables one to calculate matrix convolutions and other algebraic operations in matrix form as an application a matrix renewal function is evaluated and its limit theorem is numerically exhibited

legendre wavelets method for solving fractional partial differential equations with dirichlet boundary conditions in this paper a new method based on the legendre wavelets expansion together with operational matrices of fractional integration and derivative of these basis functions is proposed to solve fractional partial differential equations with dirichlet boundary conditions the proposed method is very convenient for solving such boundary value problems since the boundary conditions are taken into account automatically convergence of the two dimensional legendre wavelets expansion is investigated illustrative examples are included to demonstrate the validity and applicability of the presented wavelets method

a petrov galerkin method for equal width equation abstract the equal width equation is solved numerically by petrov galerkin method using linear hat function and quadratic b spline function as trial and test functions respectively product approximation has been used in this method a linear stability analysis of the scheme is shown to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient the maxwellian initial condition pulse and the development of an undular bore are also studied

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system in this paper an efficient numerical method for the solution of nonlinear damped van der pol equation based on the haar wavelets approach is proposed the proposed scheme can be used for solving one of the widely studied and challenging equation in nonlinear dynamics like the damped van der pol equation we have applied a numerical procedure involving haar series for the highest order of derivative moreover the numerical result shows that the proposed mechanism is quite reasonable when compare to exact solution

a least squares finite element scheme for the ew equation abstract the equal width ew equation is solved by a least squares technique using linear space time finite elements in simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy and conservation the development of solitary waves from an arbitrary initial condition within an ew environment is examined

a space time least square finite element scheme for advection diffusion equations abstract a space time least square finite element scheme is presented for the advection diffusion problems at moderate to high peclet numbers this scheme is designed to eliminate spurious oscillations and can be used to define the steady state solution as the asymptotic transient solution for large time numerical results using linear elements in a 1d space and bilinear elements in a 2d space demonstrate the accuracy and the stability of the new scheme

scw method for solving the fractional integro differential equations with a weakly singular kernel in this paper based on the second chebyshev wavelets scw operational matrix of fractional order integration a numerical method for solving a class of fractional integro differential equations with a weakly singular kernel is proposed by using the operational matrix the fractional integro differential equations with weakly singular kernel are transformed into a system of algebraic equations the upper bound of the error of the second chebyshev wavelets expansion is investigated finally some numerical examples are shown to illustrate the efficiency and accuracy of the approach

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

kronecker operational matrices for fractional calculus and some applications the problems of systems identification analysis and optimal control have been recently studied using orthogonal functions the specific orthogonal functions used up to now are the walsh the block pulse the laguerre the legendre haar and many other functions in the present paper several operational matrices for integration and differentiation are studied we introduce the kronecker convolution product and expanded to the riemann liouville fractional integral of matrices for some applications it is often not necessary to compute exact solutions approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed finally several systems are solved by the new and other approaches and illustrative examples are also considered

numerical solution of integral equations system of the second kind by block pulse functions abstract this paper endeavors to formulate block pulse functions to propose solutions for the fredholm integral equations system to begin with we describe the characteristic of block pulse functions and will go on to indicate that through this method a system of fredholm integral equations can be reduced to an algebraic equation numerical examples presented to illustrate the accuracy of the method

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

numerical solutions of volterra integral equation with weakly singular kernel using scw method numerical methods for weakly singular volterra integral equations are rarely considered in the literature the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration and bring some difficulties in numerical computation in this paper we present a numerical solution of weakly singular volterra integral equations including the abels equations by the second chebyshev wavelet scw we give the scw operational matrix of fractional integration and combine with the block pulse functions bpfs to derive the procedure of solving this kind integral equations the proposed method is illustrated with numerical examples the results reveal that the method is accurate and easy to implement

improved algorithm about nsfot for the purpose of further saving computing time an improved algorithm about nsfot is provided in this paper that is by introducing the simple operations such as preprocessing or after processing haar and walsh transforms are performed conveniently on the multiprocessor as a result one large size problem is divided into several small size sub problems load on every processor not only decreases greatly but also gets so uniform that much time is saved both the theoretical analysis and experimental results demonstrate the effectiveness of the proposed approach

some new connections between matrix products for partitioned and non partitioned matrices this paper is concerned with a long list of matrix products for partitioned and non partitioned matrices including kronecker khatri rao of first kind hadamard tracy singh khatri rao block kronecker block hadamard and box products sums our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place several new attractive connections are also added to this collection we give as an example two applications in order to show that these connections play a central role in many applications these applications are a block diagonal least squares problem and a generalization of matrix inequalities involving khatri rao products of positive definite matrices

the third kind chebyshev wavelets collocation method for solving the time fractional convection diffusion equations with variable coefficients in this paper a numerical method based on the third kind chebyshev wavelets is proposed for solving a class of time fractional convection diffusion equations with variable coefficients the third kind chebyshev wavelets operational matrices of the integer order integration and the fractional order integration are derived respectively they are utilized to reduce the problem to a system of linear algebraic equations by combining the collocation method the uniform convergence analysis and error estimation for the third kind chebyshev wavelets expansion are investigated illustrative examples are given and the numerical results are presented to demonstrate the efficiency and accuracy of the proposed method

the second kind chebyshev wavelet method for solving fractional differential equations abstract in this paper the second kind chebyshev wavelet method is presented for solving linear and nonlinear fractional differential equations we first construct the second kind chebyshev wavelet and then derive the operational matrix of fractional order integration the operational matrix of fractional order integration is utilized to reduce the fractional differential equations to system of algebraic equations in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method