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Burger and Fisher diffusion transfer properties and reactions from the characteristics are studied using a non-linear equation. The nonlinear fractional Burgers–Fisher equation (NFB-FE) appears in realistic physical situations such as ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of ribonucleic acid and deoxyribonucleic acid, continuous random movement, and formation of wave patterns. The present study focuses on the collocation scheme based on the shifted Chebyshev basis (SCB) and the compact finite difference method to obtain the numerical scheme of the NFB-FE. The simulation model is created in the two steps: Initially, a semi-discrete is formed in a temporal sense, applying a linear approximation with an accuracy order of two. Next, we examine the unconditional stability and the convergence order. In the second stage, the collocation approach based on the SCB of the fourth type is used to discretize the spatial derivative parts and generate the full-discrete scheme.

In this paper, the space-time fractional advection-diffusion equation (STFADE) is considered in the finite domain that the time and space derivatives are the Caputo fractional derivative. At first, a quadratic interpolation with convergence order O ( τ 3 - α ) is applied to obtain the semi-discrete in time variable. Then, the Chebyshev collocation method of the fourth kind has been used to approximate the spatial fractional derivative. In addition, the energy method has been employed to show the unconditional stability and gained convergence order of the time-discrete scheme. Finally, the accuracy of the numerical method is analyzed and showed that our method is much more accurate than existing techniques.

The application of Levy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Levy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.

The time-fractional Black-Scholes model governing European options in which the temporal derivative is focused on the Caputo fractional derivative with $$0

Purpose The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients. Design/methodology/approach The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well. Findings Several test problems are provided to confirm the validity and efficiently of the proposed method. Originality/value For the first time, some famous examples are solved by using the proposed high-order technique.

This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the$$(2-\alpha)$$(2-α)-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

In this paper, the improved wavelet transform domain least mean squares (IWTDLMS) adaptive algorithm is established. The IWTDLMS algorithm has a faster convergence speed than the conventional WTDLMS for colored input signals. Since the performances of WTDLMS and IWTDLMS are degraded in impulsive noise interference, the IWTDLMS sign algorithm (IWTDLMS-SA) is proposed. In comparison with IWTDLMS, the IWTDLMS-SA has lower computational complexity. In order to improve the performance of IWTDLMS-SA, the variable step-size IWTDLMS-SA (VSS-IWTDLMS-SA) is introduced. The VSS-IWTDLMS-SA is derived by minimizing the $$\ell _1$$ -norm of the a posteriori error vector. To increase the tracking ability of the VSS-IWTDLMS-SA, the modified VSS-IWTDLMS-SA (MVSS-IWTDLMS-SA)is presented. The simulation results demonstrate that the proposed algorithms have a faster convergence rate and lower misadjustment than the conventional WTDLMS. The robustness feature of the IWTDLMS-SA, VSS-IWTDLMS-SA, and MVSS-IWTDLMS-SA against impulsive noises is also verified through several experiments in a system identification setup.

The computation of the nonlinear fractional Burgers–Fisher problem stated in the Caputo sense is the topic of this paper. The model depicts the issue of biological invasion and can be found in a variety of fields, including ecology, physiology, and basic stage transition situations. To produce the time discretization, the suggested methodology employs a one-order correct expression in the first process. To generate the full-discretization in the second level, the spectral collocation method approach that relies on the Legendre basis is presented. The theoretical investigation confirms the temporal discretized formulation’s stability and convergence, which are examined in relation to the associated norm. Three test examples demonstrate the computing capability and efficiency of the approach. We can use the provided approach to resolve more engineering and physics models and can also increase the convergence order of the method.

We consider Fredholm integral equation of the first kind with noisy data and use Landweber-type iterative methods as an iterative solver. We compare regularization property of Tikhonov, truncated singular value decomposition and the iterative methods. Furthermore, we present a necessary and sufficient condition for the convergence analysis of the iterative method. The performance of the iterative method is shown and compared with modulus-based iterative methods for the constrained Tikhonov regularization.

The present study introduces a robust variable step-size affine projection sign adaptive algorithm (RVSS-APSA) in impulsive noise environments. In the proposed RVSS-APA, the weight coefficients are updated based on the minimization of $$\ell _2$$-norm of the a posteriori error and the step size changes according to the minimization of $$\ell _1$$-norm of the a posteriori error. This algorithm reduces the steady-state misalignment and increases the convergence rate for colored input signal as well as with or without impulsive noise interference. Also, a new simple reset algorithm is proposed to improve the tracking ability of the introduced algorithm. The simulation results demonstrate a good performance for the proposed algorithm in different situations.

Many researchers have expanded a new form of fractional diffusion models named the temporal-space fractional Bloch–Torrey equation (TSF-BTE) to evaluate the diffusion structure of human brain tissues as well as prepare additional insight into other studies of cells and tissues and the microenvironment. The main objective of this paper is to present an effective computational method of solving such models in two dimensions. The temporal and spatial directions are based on the Caputo and the Riemann–Liouville fractional derivative, respectively. The presented numerical scheme is derived from the following manners: at first, the semi-discrete is constructed in the temporal based on a quadratic interpolation with accuracy order $$\mathcal {O}(\tau ^{2-\alpha })$$ and secondly, the unconditional stability and convergence order are analyzed. For the constructed full-discrete scheme, the spatial derivative terms approximated with the helping of the collocation method based on the Legendre basis. Finally, to illustrate the high precision of the proposed design, we use some test problems. Furthermore, the obtained results are compared with some other techniques under which the suggested methodology is highly accurate and feasible.

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature.

The present paper is primarily aimed at obtaining the numerical solution of space fractional advection–diffusion equation including two fractional space derivatives of order. At the first stage, a difference approach with the second-order accuracy is formulated to obtain a semi-discrete scheme. Unconditional stability and convergence analysis has been analyzed. At the second stage, the spatial discretization is accomplished by means of the second kind shifted Chebyshev polynomials. Two examples are investigated, and numerical results are reported to confirm the theoretical results.

This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order $${\mathcal {O}}(\delta \tau ^{2})$$ is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.

In this paper, the WTDLMS adaptive algorithm is established based on the multiple-constraint optimization criterion. Furthermore, the WTDLMS with dynamic subband-coefficients update (WTDLMS-DU) is introduced. In this algorithm, the coefficients belonging to a certain subbands are dynamically selected for the update. The optimum selection of the subband-coefficients is derived by the largest decrease of the mean-square deviation. The WTDLMS-DU has a fast convergence speed and a low steady-state error similar to the WTDLMS. In addition, the proposed algorithm has lower computational complexity in comparison to WTDLMS algorithm. The good performance of WTDLMS-DU is demonstrated in various applications such as system identification, linear prediction, and acoustic echo cancellation. Also, a general formalism for the establishment and the theoretical mean-square performance analysis of the family of WTDLMS adaptive algorithms such as WTDLMS, WTDLMS with partial update (WTDLMS-PU), and the proposed WTDLMS-DU are presented. The transient, the steady-state, and the stability bounds of these algorithms are studied in a unified way. This analysis is based on energy conservation arguments and does not need to assume a Gaussian or white distribution for the regressors. It is demonstrated through simulations that the results are useful in predicting the performance of the family of WTDLMS adaptive filter algorithms.

In this paper, the wavelet transform domain least mean squares (WTDLMS) adaptive algorithm with variablestep-size (VSS) is established. The step-size changes according to the largest decrease in mean square deviation. To keep the computational complexity low, the Haar wavelet transform (HWT) is utilized as a transform. In addition, the mean square performance analysis of the VSS-WTDLMS is studied in the stationary and nonstationary environments and the theoretical relations for transient and steady-state performances are established. The simulation results show that the proposed VSS-WTDLMS has faster convergence rate and lower misadjustment than conventional WTDLMS. The theoretical relations are also verified by presenting various experimental results.

In this article, numerical solution of a neutral differential equation with deviating argument by means of the Sinc scheme and fixed point theorem is considered. Properties of the DE-Sinc and SE-Sinc quadratures are utilized to reduce the computation of the neutral differential equations to an iterative technique. Then convergence of this technique is discussed by preparing a theorem. To guarantee the analytical results and show the efficiency and accuracy of the present method, some examples are presented.

In this paper, a new nonmonotone trust region method with adaptive radius is proposed for solving system of equalities and inequalities. This method combines a new nonmonotone technique with a new adaptive strategy based on the Shi and Guo’s adaptive technique in (J Comput Appl Math 213:509–520, 2008), which makes full use of the current point information. Under some standard assumptions, the global convergence property as well as the superlinear convergence rate are established for the new method. Numerical results on some nonlinear systems of equalities and inequalities indicate the efficiency and robustness of the proposed method in practice.

In this paper, we present a new nonmonotone trust region method with adaptive radius which is equipped by a nonmonotone line search technique for solving unconstrained optimization problems. The proposed method combines a modified version of the Li and Fukushima's nonmonotone technique in D.H. Li and M. Fukushima [A derivative-free line search and global convergence of Broyden like method for nonlinear equations, Optim. Methods Softw.13 2000, pp. 181–201] for solving nonlinear systems with a new variant of Shi and Guo's adaptive strategy in Z.J. Shi and J.H. Guo [A new trust region methods for unconstrained optimization, J. Comput. Appl. Math. 213 2008, pp. 509–520] for updating the trust region radius. The method performs a nonmonotone Armijo-type line search whenever the trial step is rejected. Under some standard assumptions, we provide the global convergence property as well as the superlinear and quadratic convergence rates for the new method. Numerical results show the efficiency and effectiveness of the new proposed method in practice.

In this article, we employ trigonometric wavelet bases to numerical solution of Fredholm integral equations of first kind in Holder space. Employment of Galerkin method for trigonometric wavelets in Fredholm integral equations of first kind has resulted in occurrence of two-dimensional trigonometric wavelets. Here, we present the convergence of two-dimensional trigonometric wavelets in numerical solution in Holder space C α([a, b]).

We study a boundary integral equation method for solving Laplace’s equation Δu = 0 with nonlinear boundary conditions. This nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought. The integral equation is solved numerically by using the Wavelet–Galerkin method, with Legendre wavelet functions used as approximations to u . numerical example are given for problem.

PurposeAims to present a boundary integral equation method for solving Laplace's equation Δu=0 with nonlinear boundary conditions.Design/methodology/approachThe nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought. The integral equation is solved numerically by using the collocation method on smooth or nonsmooth boundary; the singularities of solution degrade the rates of convergence.FindingsVariants of the methods for finding numerical solutions are suggested. So these methods have been compared with respect to number of iterations.Practical implicationsNumerical experiments show the efficiency of the proposed methods.Originality/valueProvides new methods to solve nonlinear weakly singular integral equations and discusses difficulties that arise in particular cases.

In this study, we propose one of the new techniques used in solving numerical problems involving integral equations known as the Sinc-collocation method. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this article, a mixed Volterra-Fredholm integral equation which has been appeared in many science an engineering phenomena is discredited by using some properties of the Sinc-collocation method and Sinc quadrature rule to reduce integral equation to some algebraic equations. Then exponential convergence rate of this numerical technique is discussed by preparing a theorem. Finally, some numerical examples are included to demonstrate the validity and applicability of the convergence theorem and numerical scheme.

In this paper, we apply Aitken extrapolation and epsilon algorithm as acceleration technique for the solution of a weakly singular nonlinear Volterra integral equation of the second kind. In this paper, based on Tao and Yong (2006 J. Math. Anal. Appl. 324 225–37.) the integral equation is solved by Navot's quadrature formula. Also, Tao and Yong (2006) for the first time applied Richardson extrapolation to accelerating convergence for the weakly singular nonlinear Volterra integral equations of the second kind. To our knowledge, this paper may be the first attempt to apply Aitken extrapolation and epsilon algorithm for the weakly singular nonlinear Volterra integral equations of the second kind.