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The main aim of the current paper is to propose a new numerical scheme based on the Adams-Bashforth method for solving two component time fractional differential equations involving the Caputo-Prabhakar fractional derivative of order μ. The existence and uniqueness of the solutions based on the Adams-Bashforth method reported for two component differential equations are discussed. Also, in this paper stability and convergence are demonstrated. Moreover, two numerical example are demonstrated in order to show the validity and reliability of the proposed methods. [ABSTRACT FROM AUTHOR]

Since real-world data are often inaccurate and working with fuzzy data and Z-numbers are very important and necessary, and in the real world, we need to rank and compare data, in this article, we introduce a new method for ranking Z-numbers. This ranking algorithm is based on centroid point. We evaluate the distance between the centroid points, and based on this distance, we rank the Z-numbers. The algorithm of this method has simple calculations and it provides satisfactory results for data in different conditions; we use this method in two practical examples. First, by ranking the return on assets of Tehran stock exchanges, with this method, we can choose the most risky stock and the least risky stock among the available stocks. The advantage of this method over conventional fuzzy methods is considering uncertainty and allocating credit in the opinion of experts to estimate fuzzy parameters, and in the second example, we ranked the factors which affect the productivity of tourism security. Then, by using the proposed algorithm, we introduce the most important and the least important factors for tourist security budgeting. [ABSTRACT FROM AUTHOR]

This paper concerns a nonlinear inverse stochastic parabolic partial differential equation. The source term of this problem consists branches of science and engineering, of known multiplicative noise. We study the existence of the quasi solution for this problem. Our suggested method to this approach is the minimization method based on the stochastic variational formulation. Moreover, we prove a stability estimation and continuity of minimization functional. In the proof procedure, we show that there is a compact subset of the admissible functions set. These results prove the existence of a quasi solution for the proposed problem. [ABSTRACT FROM AUTHOR]

In this article, we use the Haar wavelets (HWs) method to numerically solve the nonlinear Drinfel’d–Sokolov (DS) system. For this purpose, we use an approximation of functions with the help of HWs, and we approximate spatial derivatives using this method. In this regard, to linearize the nonlinear terms of the equations, we use the quasilinearization technique. At the end, to show the effectiveness and accuracy of the method in solving this system one numerical example is provided. [ABSTRACT FROM AUTHOR]

In this paper, for the first time, the distributed‐order time‐fractional forced Korteweg–de Vries equation is studied. We use a numerical method based on the shifted Legendre operational matrix of distributed‐order fractional derivative with Tau method to find approximate solution of distributed‐order forced Korteweg–de Vries equation. This shifted Legendre operational matrix of distributed‐order fractional derivative with Tau method is used to reduce the solution of the distributed‐order time‐fractional forced Korteweg–de Vries equations to a system of algebraic equations. An error analysis and convergence are obtained. Finally, to display the applicability and validity of the numerical method, some examples are implemented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical scheme based on the iterative method for solving the fractional diffusion‐wave equation involving the Caputo–Weyl fractional derivative of order 0<α≤2. The convergence of the approximate solutions of the fractional diffusion‐wave equation is rigourously established. Moreover, the approximate solutions of the fractional diffusion‐wave equation have been presented graphically. [ABSTRACT FROM AUTHOR]

In this article, we survey the Lyapunov direct method for distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives. We provide various ways to determine the stability or asymptotic stability for these types of fractional differential systems. Some examples are applied to determine the stability of certain distributed-order systems. [ABSTRACT FROM AUTHOR]

In this article, an accurate and robust numerical method is proposed to solve a class of nonlinear variable order fractional differential equations (FDEs) in the Caputo‐Prabhakar sense by using Bernstein polynomials (BPs). Here, we use the BPs to obtain operational matrices. This operational matrices of BPs are utilized to transform the studied variable order FDEs into a system of algebraic equations after dispersing the variable which greatly simplifies the solution process. By solving the system of equations, the numerical solutions are acquired. Some examples are provided in order to demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we use Chebyshev polynomials to seek the numerical solution of a class of multi-variable order fractional differential equation (MVODEs) that the fractional derivative is described in the Caputo-Prabhakar sense. Using operational matrices, the original equations are transferred to a system of algebraic equations. By solving the system of equations, the numerical solutions are acquired that this system may be solved numerically using an iterative algorithm. The effectiveness and convergence analysis of the numerical scheme is illustrated through four numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number R0. In particular, when R0 ⩽ 1, the infection free equilibrium is globally asymptotically stable, and conversely if R0 > 1, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for solving a nonlinear third-order Korteweg–de Vries equation. This method is based on the multiquadric (MQ) quasi-interpolation operator L W 2 and an integrated radial basis function networks scheme. In the present scheme, the second-order central divided difference of the spatial derivative is used to approximate the third-order spatial derivative, and the Taylors series expansion to discretize the temporal derivative. Then, the spatial derivative is approximated by the MQ quasi-interpolation operator L W 2 . This method is applied on some test experiments and the numerical results have been compared with the exact solutions and the solutions of other numerical methods. The L ∞ , L 2 and root-mean-square errors of the solutions show the efficiency and the accuracy of the method. Furthermore, the stability analysis of the method is surveyed. [ABSTRACT FROM AUTHOR]

The nonlinear Burgers' equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers' equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers' equation. These methods are based on the multiquadric (MQ) quasi-interpolation operator LW2 and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers' equation. [ABSTRACT FROM AUTHOR]

We have presented a numerical scheme for solving one-dimensional nonlinear sine-Gordon equation. We apply the spectral method with a basis of a new orthogonal polynomial which is orthogonal over the interval [0, 1]with weighting function one. The results show the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

During the last two decades, there has been a considerable interest in developing efficient radial basis functions (RBFs) algorithms for solving partial differential equations (PDEs). In this paper, we introduce the Petrov-Galerkin method for the numerical solution of the one-dimensional nonlinear Burger equation. In this method, the trial space is generated by the multiquadric (MQ) RBF and the test space is generated by the compactly supported RBF. In the time discretization of the equation, the Taylor series expansion is used. Thismethod is applied on some test experiments, and the numerical results have been compared with the exact solutions. The L2, L8, and root-meansquare (RMS) errors in the solutions show the efficiency and the accuracy of the method. [ABSTRACT FROM AUTHOR]

This paper presents numerical solution of elliptic partial differential equations (Poisson's equation) using a combination of logarithmic and multiquadric radial basis function networks. This method uses a special combination between logarithmic and multiquadric radial basis functions with a parameter r. Further, the condition number which arises in the process is discussed, and a comparison is made between them with our earlier studies and previously known ones. It is shown that the system is stable. [ABSTRACT FROM AUTHOR]

We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of (Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002). Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials. [ABSTRACT FROM AUTHOR]

We consider a Cauchy problem of unidimensional nonlinear diffusion equation on finite interval. This problem is ill-posed and its approximate solution is unstable. We apply the He's homotopy perturbation methodHPM and obtain the third-order asymptotic expansion. We show that if the conductivity term in diffusion equation has a specified condition, the above solution can be estimated. Finally, a numerical experiment is provided to illustrate the method. [ABSTRACT FROM AUTHOR]