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Legendre wavelets and their exact Riemann-Liouville fractional integrals are used to compute numerical solutions to fractional delay differential equations, by reducing the problem to algebraic equations. An error bound of Legendre wavelet approximation is analytically computed, which implies the approximation converges when the degree of the Legendre polynomials or the number of wavelets approaches infinity. Finally, several numerical examples are considered.

We give an effective method for solving fractional Riccati differential equations. We first define the fractional-order Boubaker wavelets (FOBW). Using the hypergeometric functions, we determine the exact values for the Riemann-Liouville fractional integral operator of the FOBW. The properties of FOBW, the exact formula, and the collocation method are used to transform the problem of solving fractional Riccati differential equations to the solution of a set of algebraic equations. These equations are solved via Newton's iterative method. The error estimation for the present method is also determined. The performance of the developed numerical schemes is assessed through several examples. This method yields very accurate results. The given numerical examples support this claim.

Recently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously.This study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations are defined using this piecewise fractional derivative.A new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations.The accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically.This study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods.

In this article, the non-singular variable-order fractional derivative in the Heydari-Hosseininia concept is used to formulate the variable-order fractional form of the coupled nonlinear Ginzburg-Landau equations. To solve this system, a numerical scheme is constructed based upon the shifted Vieta-Lucas polynomials. In this method, with the help of classical and fractional derivative matrices of the shifted Vieta-Lucas polynomials (which are extracted in this study), solving the studied problem is transformed into solving a system of nonlinear algebraic equations. The convergence analysis and the truncation error of the shifted Vieta-Lucas polynomials in two dimensions are investigated. Numerical problems are demonstrated to confirm the convergence rate of the presented algorithm.

In this paper, we propose a numerical method for solving distributed-order fractional partial differential equations (FPDEs). For this method, we first introduce fractional-order generalized Taylor wavelets (FOGTW). An estimation for the error of the approximation is also studied. In addition, by using the regularized beta function we give a formula for determining the Riemann-Liouville fractional integral operator for the FOGTW. Combining this formula with the Gauss-Legendre quadrature, we obtain a numerical method for solving distributed-order FPDEs. Several illustrative examples are given to show the applicability and the accuracy of the proposed method.

In this study, a new numerical method for the solution of the linear and nonlinear distributed fractional differential equations is introduced. The fractional derivative is described in the Caputo sense. The suggested framework is based upon Legendre wavelets approximations. For the first time, an exact formula for the Riemann–Liouville fractional integral operator for the Legendre wavelets is derived. We then use this formula and the properties of Legendre wavelets to reduce the given problem into a system of algebraic equations. Several illustrative examples are included to observe the validity, effectiveness and accuracy of the present numerical method.

A numerical method for solving polynomial partial differential equations in two independent variables, defined in the paper, is presented. The technique is based on polynomial approximation. Properties and the operational matrices for partial derivatives for a polynomial in two variables are presented first. These properties are then used to reduce the solution of partial differential equations in two independent variables to a system of algebraic equations. Five illustrative examples are presented to prove the effectiveness of the present method. Results show that the numerical scheme is very convenient for solving polynomial partial differential equations.

In this paper, a variable-order fractional version of the Benney–Lin equation is defined using the variable-order fractional derivative in the Caputo type. A collocation method based on the shifted Jacobi polynomials is applied to deal with this problem. Some matrix relationships related to these polynomials are extracted and used in constructing the established method. The obtained relations cause the method calculations to be significantly reduced, which reduce the method execution time. The established technique converts solving the problem under study into solving an algebraic system of equations. The high accuracy and low computations of the presented scheme are investigated by solving some numerical examples.

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.

This paper presents an efficient numerical method for solving the initial and boundary value problems of the Bratu-type. In the proposed method, the Taylor wavelets are introduced, for the first time. An operational matrix of integration is derived and is utilized to reduce the Bratu-type initial and boundary value problems to a system of algebraic equations. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed wavelets method. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique and a comparison is made with the existing results.

In this study, a kind of piecewise fractional derivatives based on the Caputo fractional derivative is used to define a novel category of fractional optimal control problems. The piecewise Chebyshev cardinal functions as an appropriate family of basis functions are considered to construct a numerical method for solving such problems. The classical and piecewise fractional derivative matrices of these basis functions are derived and used in constructing the proposed technique. The established scheme transforms obtaining the solution of such problems into finding the solution of algebraic systems of equations by approximating the state and control variables using the mentioned basis functions. The accuracy of the expressed approach is investigated by solving some examples.

In this paper, a new set of basis functions called the piecewise Chebyshev cardinal functions is generated to investigate a class of constrained fractional optimal control problems. These basis functions possess many useful properties, such as orthogonality, cardinality and spectral accuracy. The fractional integral matrix of these functions is obtained. A direct scheme based on the these basis functions together with their fractional integral matrix is developed for solving the problem under consideration. The established method transforms solving the original problem into solving a constrained minimization problem by approximating the state and control variables in terms of the piecewise Chebyshev cardinal functions. Some numerical examples are given to show the efficiency of the proposed technique.

We provide a new effective method for the two-dimensional distributed-order fractional differential equations (DOFDEs). The technique is based on fractional-order Chebyshev wavelets. An exact formula involving regularized beta functions for determining the Riemann-Liouville fractional integral operator of these wavelets is given. The given wavelets and this formula are utilized to find the solutions of the given two-dimensional DOFDEs. The method gives very accurate results. The given numerical examples support this claim.

In this paper, a novel fractal-fractional derivative operator with Mittag-Leffler function as its kernel is introduced. Using this differentiation, the fractal-fractional model of the coupled nonlinear Schrodinger-Boussinesq system is defined. The orthonormal shifted discrete Chebyshev polynomials are generated and used for constructing a computational matrix method to solve the defined system. In the established method, using the matrices of the ordinary and fractal-fractional differentiations of these polynomials, the fractal-fractional system transformed into a system of algebraic equations, which is solved readily. Practicability and precision of the method are examined by solving two numerical examples.

In this paper, the numerical technique with the help of the Lucas wavelets (LWs) and the Legendre–Gauss quadrature rule is presented to study the solution of fractional Fredholm–Volterra integro-differential equations. The modified operational matrices of integration and pseudo-operational of fractional derivative for the proposed wavelet functions are calculated. These matrices in comparison to operational matrices existing in other methods are more accurate. The Lucas wavelets and their operational matrices provide the precise numerical scheme to get the approximate solution. Also, we exhibit the upper bound of error based on the method. We illustrate the behavior of the new scheme in several numerical examples with the help of tables and figures. The results confirm the accuracy and applicability of the numerical approach.

In this paper, a new numerical method for solving the distributed fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

The main idea of this paper is to establish the novel fractional Gegenbauer functions (FGFs) for solving three kinds of fractional differential equations generated by the variable-order fractional derivatives in the Atangana-Baleanu-Caputo (ABC) sense. The numerical scheme is discussed based on the modified operational matrices (MOMs) of Atangana-Baleanu variable-order (AB-VO) fractional integration and the delay operational matrix. The methodology of obtaining the MOMs of integration is calculated with high accuracy. So that the precision of the computation method is influenced directly by the proposed matrix. In addition, we investigate the error analysis of the proposed approach. At last, several numerical experiments are employed to clarify the performance and efficiency of the method.

In this paper, a new numerical method for solving nonlinear fractional integro-differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann–Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the nonlinear fractional integro-differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this paper, a numerical method for solving the linear initial problems for partial differential equations with constant coefficients and analytic initial conditions in two and three independent variables is presented. The technique is based upon the Taylor series expansion. Properties and the operational matrices for partial derivatives for the Taylor series in two and three variables are first presented. These properties are then used to reduce the solution of partial differential equations in two and three independent variables to a system of algebraic equations. The procedure can be extended to linear partial differential equations with more independent variables. The Taylor series may not converge if the solution is not analytic in the whole domain, however, the present method can be applied to boundary-value problems for linear partial differential equations, when the solution is analytic in the interior of the domain and also on some open subsets for each distinct part of the boundary. The method is computationally very attractive and applications are demonstrated through illustrative examples.

In this paper, a new numerical method for solving the optimal control of a class of systems described by integro-differential equations with quadratic performance index is presented. This optimization problem plays an important role in describing the dynamics of an elastic aircraft with allowance for non-steady flow past its profile. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of optimization problem to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this paper, a new numerical method for solving the nonlinear constrained optimal control with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrix of integration is introduced. This matrix is then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra–Fredholm–Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples.

This paper presents a numerical method for solving nonlinear optimal control problems including state and control inequality constraints. The method is based upon rationalized Haar functions. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A numerical method for solving nonlinear initial-value problems is proposed. The Lane–Emden type equations which have many applications in mathematical physics are then considered. The method is based upon hybrid function approximations. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials are presented and are utilized to reduce the computation of nonlinear initial-value problems to a system of non-algebraic equations. The method is easy to implement and yields very accurate results.

In this paper, we find the asymptotic behavior of solutions of the third order difference equation x n + 1 = x n - 2 1 + px n + qx n - 1 + x n - 2 , n = 0 , 1 , 2 , … for all admissible non-negative values of the parameters p , q where the initial conditions x −2 , x −1 , x 0 are positive. We show that the solutions do not exhibit a periodic attitude for all parameters of the above mentioned difference equation. It is worth to mention that this difference equation was an open problem introduced by Kulenovic and Ladas. Note that we also generalize and extend the above mentioned equation and we investigate the same arguments as the third difference equation for the zero equilibrium point of the higher case.

A numerical method for solving the classical Blasius’ equation is proposed. The Blasius’ equation is a third order nonlinear ordinary differential equation , which arises in the problem of the two-dimensional laminar viscous flow over a semi-infinite flat plane. The approach is based on a modified rational Legendre tau method. The operational matrices for the derivative and product of the modified rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of Blasius’ equation to the solution of a system of algebraic equations. A numerical evaluation is included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

In this paper, nonclassical pseudospectral method is proposed for solving the classic brachistochrone problem. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Properties of nonclassical pseudospectral method are presented, these properties are then utilized to reduce the computation of brachistochrone problem to the solution of algebraic equations. Using this method, the solution to the brachistochrone problem is compared with those in the literature.

In this work we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the Fokker-Planck equation and some similar equations. This method can successfully be applied to a large class of problems. The Adomian decomposition method needs less work in comparison with the traditional methods. This method decreases considerable volume of calculations. The decomposition procedure of Adomian will be obtained easily without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent series with easily computed components. In this work we are concerned with the application of the decomposition method for the linear and nonlinear Fokker-Planck equation. To give overview of methodology, we have presented several examples in one and two dimensional cases.

In this paper, we study the qualitative behavior of solutions of the class of delay difference equation x n + 1 = β x n - k + 1 + γ x n - 2 k + 1 A + Bx n - k + 1 , n = 0 , 1 , 2 , … , where the initial conditions x −2 k +1 , … , x −1 , x 0 are positive, k ∈ {1, 2, …}, and the parameters β , γ , A , B are positive. Our concentration is on invariant intervals and the global stability of the above mentioned equation. We obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients. Furthermore, the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed. It is also illustrated that for some special case of parameters, the solution will be either a period-2 solution, or will converge to the equilibrium point, or will have unbounded solutions.

We investigate the global attractivity, the periodic behavior, and the invariant interval with the semicycles character of solutions of the equation x n + 1 = α + γ x n - 2 k + 1 Bx n - k + 1 + Cx n - 2 k + 1 , n = 0 , 1 , 2 , … for all admissible non-negative values of the parameters α , γ , B , C and the initial conditions x −2 k +1 ,… , x 1 , x 0 . We show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other.

The main purpose of the current paper is to prove that every positive solution of the delay difference equationx"n=Ax"n"-"k+Bx"n"-"3"k,where A,B@?(0,~), the initial conditions x"-"3"k"+"1,x"-"3"k"+"2,...,x"0@?(0,~) and k@?{1,2,3,...}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion.

The Sinc-Galerkin method is presented for solving nonlinear two-point boundary value problems for second order differential equations. A problem arising from chemical reactor theory is then considered. Properties of the Sinc-Galerkin method are utilized to reduce the computation of nonlinear two-point boundary value problems to some algebraic equations. The method is computationally attractive and applications are demonstrated through an illustrative example.

This paper is concerned with the qualitative behavior of solutions to the difference equation x n + 1 = p + qx n - k 1 + x n , n = 0 , 1 , 2 , … , where the initial conditions x−k, …, x−1, x0 are non-negative, k ∈ {1, 2, 3, …}, and the parameters p, q are non-negative. We start by establishing the periodicity, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. We also present solutions that have unbounded behavior. It is worth to mention that this difference equation is a special case of an open problem was introduced by M.R.S. Kulenovic and G. Ladas [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002]. Several computational examples are given to support our theoretical discussions. The presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation.

A numerical method for solving the nonlinear Volterra-Fredholm integral equations is presented. The method is based upon Legendre wavelet approximations. The properties of Legendre wavelet are first presented. These properties together with the Gaussian integration method are then utilized to reduce the Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

An effective method based upon Alpert multiwavelets is proposed for the solution of Hallen's integral equation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of Hallen's integral equation to the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of resulted matrix equation.

A method for the solution of Volterra integro-differential equations by using single-term Walsh series is presented. Properties of single-term Walsh series are utilized to reduce the computation of Volterra integro-differential equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.

A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The operational matrices of integration, delay and product are utilized to reduce the solution of optimal control to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A method for finding the solution of time-delay systems using a hybrid function is proposed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrices of product and delay are introduced. These matrices together with the operational matrix of integration are utilized to reduce the solution of time-delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

An approximate method for solving Volterra's population model for population growth of a species in a closed system is proposed. Volterra's model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. The approach is based on a rational Chebyshev tau method. The Volterra's population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterra's model to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results.

A numerical method for solving the controlled Duffing oscillator is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions which consists of block-pulse functions plus Legendre polynomials are given. The method is based upon expanding various time functions in the system as their truncated hybrid functions. The operational matrix of product is introduced. This matrix together with the operational matrix of integration are utilized to reduce the solution of controlled Duffing oscillator to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A numerical method for solving time varying linear quadratic optimal control problems with inequality constraints is presented. The method is based upon hybrid functions approximations. The properties of hybrid functions consisting of block-pulse functions and Legendre polynomials are presented. The operational matrices of integration and product are then utilized to reduce the optimal control problem to the solution of algebraic equations. The inequality constraints are first converted to a system of algebraic equalities. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A numerical scheme is devised to develop a discrete bidirectional reflectance model. A pseudospectral method is utilized with which the discrete solution of the radiative transfer equation is made part of this development. To produce discrete bidirectional reflectance values, a set of algebraic equations is solved. Illustrative examples are given to demonstrate the performance of the developed model.

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

A numerical technique for solving the nonlinear problems of the calculus of variations is presented. Two nonlinear examples are considered. In the first example, the brachistochrone problem is formulated as a nonlinear optimal control problem, and in the second example, a higher-order nonlinear problem is given. An operational matrix of integration is introduced and is utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement, and yields very accurate results.

Rationalized Haar functions are developed to approximate the solution of the differential equations. Properties of rationalized Haar functions are first presented, the operational matrix of integration together with product operational matrix are utilized to reduce the computation of differential equations into algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.

A direct method for solving variational problems using Legendre wavelets is presented. An operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.