A Numerical Scheme for Solving Time-Fractional Bessel Differential Equations

Purpose The purpose of this paper is to develop a new method based on operational matrices of two-dimensional delta functions for solving two-dimensional nonlinear quadratic integral equations (2D-QIEs) of fractional order, numerically. Design/methodology/approach For this aim, two-dimensional delta functions are introduced, and their properties are expressed. Then, the fractional operational matrix of integration based on two-dimensional delta functions is calculated for the first time. Findings By applying the operational matrices, the main problem would be transformed into a nonlinear system of algebraic equations which can be solved by using Newton's iterative method. Also, a few results related to error estimate and convergence analysis of the proposed method are investigated. Originality/value Two numerical examples are presented to show the validity and applicability of the suggested approach. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

An efficient collocation technique based on operational matrix of fractional-order Lagrange polynomials for solving the space-time fractional-order partial differential equations

In this paper, a numerical method has been proposed for solving several two-dimensional porous medium equations (2D PME). The method combines Newton and Explicit Group MSOR (EGMSOR) iterative method namely four-point NEGMSOR. Throughout this paper, an initialboundary value problem of 2D PME is discretized by using the implicit finite difference scheme in order to form a nonlinear approximation equation. By taking a fixed number of grid points in a solution domain, the formulated nonlinear approximation equation produces a large nonlinear system which is solved using the Newton iterative method. The solution vector of the sparse linearized system is then computed iteratively by the application of the four-point EGMSOR method. For the numerical experiments, three examples of 2D PME are used to illustrate the efficiency of the NEGMSOR. The numerical result reveals that the NEGMSOR has a better efficiency in terms of number of iterations, computation time and maximum absolute error compared to the tested NGS, NEG and NEGSOR iterative methods. The stability analysis of the implicit finite difference scheme used on 2D PME is also provided.

ABSTRACTThis study introduces an indirect method for deriving the approximate fractional solutions using fractional Bernoulli functions, within the framework of Liouville‐Caputo's fractional derivative, for the crossover lumpy skin disease model. A formula for fractional derivatives is formulated approximately. We use this formula with the least square approximation technique to convert the model into a set of nonlinear algebraic equations (SNAEs). We resolve this SNAEs by employing Newton's iteration method. The convergence examination and error estimation for the suggested scheme are also included by computing the order of convergence. Numerical examples illustrate the good impact of the strategies and the rightness of the theoretical framework of the suggested method.

In this paper, Newton's iterative method to solve the nonlinear matrix equation is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given.

In this paper, three iterative methods (Stokes, Newton and Oseen iterative methods) based on finite element discretization for the stationary micropolar fluid equations are proposed, analyzed and compared. The stability and error estimation for the Stokes and Newton iterative methods are obtained under the strong uniqueness conditions. In addition, the stability and error estimation for the Oseen iterative method are derived under the uniqueness condition of the weak solution. Finally, numerical examples test the applicability and the effectiveness of the three iterative methods.

During the 1986--1989 project period, two major areas of research developed into which most of the work fell: matrix-free'' methods for solving linear systems, by which we mean iterative methods that require only the action of the coefficient matrix on vectors and not the coefficient matrix itself, and Newton-like methods for underdetermined nonlinear systems. In the 1990 project period of the renewal grant, a third major area of research developed: inexact Newton and Newton iterative methods and their applications to large-scale nonlinear systems, especially those arising in discretized problems. An inexact Newton method is any method in which each step reduces the norm of the local linear model of the function of interest. A Newton iterative method is any implementation of Newton's method in which the linear systems that characterize Newton steps (the Newton equations'') are solved only approximately using an iterative linear solver. Newton iterative methods are properly considered special cases of inexact Newton methods. We describe the work in these areas and in other areas in this paper.

Fractional-order Bernoulli functions and their applications in solving fractional Fredholem–Volterra integro-differential equations

This study introduces an indirect approach to find the solutions of the fractional Bessel differential equation using fractional Bernoulli functions and Caputo’s fractional derivative. This method transforms the problem into a nonlinear system of algebraic equations. The existence and uniqueness of the solution for the problem are proved. Moreover, operational matrices for fractional derivatives and time‐fractional integration are constructed. These operational matrices, together with the least square approximation technique, are used to reduce the problem to a nonlinear system of algebraic equations. Newton’s iterative method is applied to solve the nonlinear algebraic equations. The convergence analysis and the error estimate of the proposed method are also provided. This study demonstrates the effectiveness and the significance of the proposed method for obtaining approximate fractional solutions and for researchers in related fields or specific areas. Four examples are given to illustrate the accuracy and the performance of the proposed method.

In this work, we introduce a method based on the Müntz–Legendre polynomials (M‐LPs) for solving fractal‐fractional 2D optimal control problems that the fractal‐fractional derivative is described in Atangana‐Riemann‐Liouville's sense. First, we obtain operational matrices of fractal‐fractional‐order derivative, integer‐order integration, and derivative of the M‐LPs. Second, the under study problem is converted into an equivalent variational problem. Then, by applying the M‐LPs, their operational matrices and Gauss–Legendre integration, the mentioned problem is converted to a system of algebraic equations. Finally, this system is solved by Newton's iterative method. Also, we introduce an error bound for the described method. Two examples are included to test the applicability and validity of the present scheme.

New one- and two-level Newton iterative mixed finite element methods for stationary conduction–convection problems

In this paper, a similarity transformation is used to reduce the three-dimensional steady state condensation film on an inclined rotating disk by a set of nonlinear boundary value problems. This problem is solved using a new hybrid technique based on differential transform method (DTM) and Iterative Newton's Method (INM). The differential equations and its boundary conditions are transformed to a set of algebraic equations, and the Taylor series of solution is calculated. After finding Jacobian matrix, the unknown parameters computed using Multi-Variable Iterative Newton's Method. These techniques are used to obtain an approximate solution of the problem. In this solution, there is no need to restrictive assumptions or linearization. The results compared with the numerical solution of the problem, and a good accuracy of the proposed hybrid method observed. Finally, the velocity and temperature profiles demonstrated for different values of problem parameters. Key words: Condensation film, rotating disk, nonlinear boundary value problem, differential transform method, iterative Newton's method, Jacobian matrix.

In this article, a variable-order operational matrix of Gegenbauer wavelet method based on Gegenbauer wavelet is applied to solve a space–time fractional variable-order non-linear reaction–diffusion equation and non-linear Galilei invariant advection diffusion equation for different particular cases. Operational matrices for integer-order differentiation and variable-order differentiation have been derived. Applying collocation method and using the said matrices, fractional-order non-linear partial differential equation is reduced to a system of non-linear algebraic equations, which have been solved using Newton iteration method. The salient feature of the article is the stability analysis of the proposed method. The efficiency, accuracy and reliability of the proposed method have been validated through a comparison between the numerical results of six illustrative examples with their existing analytical results obtained from literature. The beauty of the article is the physical interpretation of the numerical solution of the concerned variable-order reaction–diffusion equation for different particular cases to show the effect of reaction term on the pollution concentration profile.

This paper intends to present a novel fractional basis function derived from Lagrange polynomials and to establish a new interpolation formula for approximating solutions of nonlinear fractional weakly singular Volterra integro–differential systems (VIDS) with Caputo fractional derivative. These systems are solved using a comprehensive approach that includes product integration, Nyström, and spectral collocation methods founded on sixth–kind Chebyshev polynomials. Utilizing a helpful connection between the Riemann–Lioville fractional integral operator and the Caputo derivative, weakly singular integro–differential equations and initial conditions are converted into an equivalent system of nonlinear algebraic equations. Approximate solutions and numerical derivatives of solutions are determined together using Newton's iterative method. The proposed scheme's convergence is examined, and an upper error bound in the L ∞ –norm and weighted L 2 –norm is acquired. Finally, the suggested approach is thoroughly evaluated for its simplicity and accuracy by including numerical examples.

Several noniterative procedures for solving the nonlinear Richards equation are introduced and compared to the conventional Newton and Picard iteration methods. Noniterative strategies for the numerical solution of transient, nonlinear equations arise from explicit or linear time discretizations, or they can be obtained by linearizing an implicit differencing scheme. We present two first order accurate linearization methods, a second order accurate two‐level “implicit factored” scheme, and a second order accurate three‐level “Lees” method. The accuracy and computational efficiency of these four schemes and of the Newton and Picard methods are evaluated for a series of test problems simulating one‐dimensional flow processes in unsaturated porous media. The results indicate that first order accurate schemes are inefficient compared to second order accurate methods; that second order accurate noniterative schemes can be quite competitive with the iterative Newton and Picard methods; and that the Newton scheme is no less efficient than the Picard method, and for strongly nonlinear problems can outperform the Picard scheme. The two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods, although there are concerns regarding the stability behavior of the three‐level scheme which need to be resolved. We conclude that of the four noniterative strategies presented, the implicit factored scheme is the most promising, and we suggest improved formulations of the method.