Stochastic linear combinations of some random vectors are studied where the dis-tribution of the random vectors and the joint distribution of their coe, cients have Dirichlet distributions. A method is provided for calculating the distribution of these combinations which has been studied before. Our main result is the same as but from a di, erent point of view.

In this paper, we propose a new numerical algorithm for the approximate solution of non-homogeneous fractional di, erential equation. Using this algorithm the fractional di, erential equations are transformed into a system of algebraic linear equations by operational matrices of block-pulse and hybrid functions. Based on our new algorithm, this system of algebraic linear equations can be solved by a proposed (TSI) method. Further, some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm.

In this paper, a new numerical method for solving the fractional Riccati di erential equation is presented. The fractional derivatives are described in the Caputo sense. The method is based upon fractional-order Bernoulli functions approximations. First, the fractional-order Bernoulli functions and their properties are presented. Then, an operational matrix of fractional order integration is derived and is utilized to reduce the under study problem to a system of algebraic equations. Error analysis included the residual error estimation and the upper bound of the absolute errors are introduced for this method. The technique and the error analysis are applied to some problems to demonstrate the validity and applicability of our method.

The purpose of this paper is to present a numerical scheme for solving time-fractional partial di, erential equation based on cubic B-spline quasi-interpolation. For this pur-pose, , rst we will approximate the time-fractional derivative by Laplace transform method and then by using of cubic B-spline quasi-interpolation, the spatial deriva-tives are approximated. Moreover, the stability of this method is studied. Finally, European call and put options are priced and we will show that the results are good agreement with the other methods. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integrodi erential equation (FIDE) with a weakly singular kernel. First, a collocation method based on Haar wavelets (HW), Legendre wavelet (LW), Chebyshev wavelets (CHW), second kind Chebyshev wavelets (SKCHW), Cos and Sin wavelets (CASW) and BPFs are presented for driving approximate solution FIDEs with a weakly singular kernel. Error estimates of all proposed numerical methods are given to test the convergence and accuracy of the method. A comparative study of accuracy and computational time for the presented techniques is given.

In the present work, we focus on solutions of K(p,p) equation which are solitons with compact support called compactons. Such a study of compact solitary waves will help us understanding solitons at a deeper level. One of the interesting feature, they govern is quasi elastic collision and gaining the same coherent shape again after scat-tering. Numerical scheme used to study the compacton solutions of K(p,p) equation is based on reduced di, erential transform method. Both one dimensional di, eren-tial transform method and two dimensional reduced di, erential transform method have been used. Test problems under consideration show the e, cient working of the proposed scheme.

In the present work, we focus on solutions of K(p,p) equation which are solitons with compact support called compactons. Such a study of compact solitary waves will help us understanding solitons at a deeper level. One of the interesting feature, they govern is quasi elastic collision and gaining the same coherent shape again after scat-tering. Numerical scheme used to study the compacton solutions of K(p,p) equation is based on reduced di, erential transform method. Both one dimensional di, eren-tial transform method and two dimensional reduced di, erential transform method have been used. Test problems under consideration show the e, cient working of the proposed scheme.

This paper is aimed to present the Global Thin Plate Spline Differential Quadrature method for the numerical solution of viscous Burgers’ equation. This mesh-less and high-order model is introduced with the motive of diminishing computational effort and dealing with irregular geometries. Thin Plate Spline Radial basis function is used as a test function to determine coefficients of derivatives in differential quadrature. The present algorithm is applied to discretize and solve two-dimensional Burgers’ equation in both rectangular and irregular non-rectangular computational domains with randomly distributed computation nodes. To evaluate the capability of the present model, several problems with different boundary and initial conditions and Reynolds Numbers are solved and the obtained results are compared with the analytical solutions and other previous numerical models. The obtained results show the higher accuracy of the present model for solving Berger's equation with fewer computational nodes compared to the previous models even in irregular domains.

Integro-di erential equations play a fundamental role in various  elds of applied mathematics. The solutions of many engineering problems in general and mechanics and physics in particular, lead to this kind of equations. This paper focuses on the fuzzy Volterra integro-di erential equation of n􀀀 th order of the second-kind with nonlinear fuzzy kernel and initial values. This equation is transformed to a nonlinear fuzzy Volterra integral equation in multi-integrals by application of a certain analytic solution adapted on fuzzy n􀀀 th order derivation under generalized Hakuhara derivative. The derived integral equations are solvable, the solutions of which are unique under certain conditions. The existence and uniqueness of the solutions are investigated in a theorem and an upper boundary is found for solutions. An easily-followed algorithm is provided to illustrate the process. The application of the proposed method helps solving the equation on the basis of the Adomian decomposition method under generalized H-derivation. Comparison of the exact and approximated solutions shows the least error.

One of the considerable strategies for the investigation of integro-di, erential equation is stability. The notion of this strategy shows us that we can rest assured of the numerical results obtained from the computer software. Since there are usually large errors in the numerical results of singular di, erential equations, this strategy will help us to be able to examine singular equations more easily with computer software. In this work, we study the stability of a multi-singular fractional boundary value problem in the sense of Hyers-Ulam stability. We also present three examples and three , gures to illustrate our main result.