Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

In this paper, we present a novel discrete scheme based on Genocchi polynomials and fractional Laguerre functions to solve multiterm variable‐order time‐fractional partial differential equations (M‐V‐TFPDEs) in the large interval. In this purpose, the accurate modified operational matrices are constructed to reduce the problems into a system of algebraic equations. Also, the computational algorithm based on the method and modified operational matrices in the large interval is easily implemented. Furthermore, we discuss the error estimation of the proposed method. Ultimately, to confirm our theoretical analysis and accuracy of numerical approach, several examples are presented.     

The dual reciprocity boundary elements method for the linear and nonlinear two‐dimensional time‐fractional partial differential equations

In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two‐dimensional linear and nonlinear time‐fractional modified anomalous subdiffusion equations and time‐fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiquadric quasi‐interpolation methods for solving partial differential algebraic equations

In this article, we propose two meshless collocation approaches for solving time dependent partial differential algebraic equations (PDAEs) in terms of the multiquadric quasi‐interpolation schemes. In presenting the process of the solution, the error is estimated. Furthermore, the comparisons on condition numbers of the collocation matrices using different methods and the sensitivity of the shape parameter c are given. With the use of the appropriate collocation points, the method for PDAEs with index‐2 is improved. The results show that the methods have some advantages over some known methods, such as the smaller condition numbers or more accurate solutions for PDAEs which has an modal index‐2 or an impulse solution with index‐2. Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 95–119, 2014     

B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations

In this work, we discuss two methods for solving a fourth order parabolic partial differential equation. In Method-I, we decompose the given equation into a system of second order equations and solve them by using cubic B-spline method with redefined basis functions. In Method-II, the equation is solved directly by applying quintic B-spline method with redefined basis functions. Stability of these methods have been discussed. Both methods are unconditionally stable. These methods are tested on four examples. The computed results are compared wherever possible with those already available in literature. We have developed Method-I for fourth order non homogeneous parabolic partial differential equation from which we can obtain displacement and bending moment both simultaneously, while Method-II gives only displacement. The results show that the derived methods are easily implemented and approximate the exact solution very well.     

Critical dimension for a system of partial differential equations with time‐dependent generators

We give sufficient conditions for blows up of positive mild solutions for the weakly coupled system: where is a fractional Laplacian, 0 < α_i≤2,β_i≥1,ρ_i>0,σ_i>?1 are constants, and the initial condition ?_i are positive, bounded and integrable functions. We also discuss the critical dimension. Copyright © 2014 John Wiley & Sons, Ltd.     

Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014     

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations

In this paper, a new computational scheme based on operational matrices (OMs) of two‐dimensional wavelets is proposed for the solution of variable‐order (VO) fractional partial integro‐differential equations (PIDEs). To accomplish this method, first OMs of integration and VO fractional derivative (FD) have been derived using two‐dimensional Legendre wavelets. By implementing two‐dimensional wavelets approximations and the OMs of integration and variable‐order fractional derivative (VO‐FD) along with collocation points, the VO fractional partial PIDEs are reduced into the system of algebraic equations. In addition to this, some useful theorems are discussed to establish the convergence analysis and error estimate of the proposed numerical technique. Furthermore, computational efficiency and applicability are examined through some illustrative examples.     

Numerical solution of nonlinear partial quadratic integro‐differential equations of fractional order via hybrid of block‐pulse and parabolic functions

In this paper, an effective numerical approach based on a new two‐dimensional hybrid of parabolic and block‐pulse functions (2D‐PBPFs) is presented for solving nonlinear partial quadratic integro‐differential equations of fractional order. Our approach is based on 2D‐PBPFs operational matrix method together with the fractional integral operator, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro‐differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h³) . The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

Identification of a time‐dependent coefficient in a partial differential equation subject to an extra measurement

The problem of recovering a time‐dependent coefficient in a parabolic partial differential equation has attracted considerable recent attention. Several finite difference schemes are presented for identifying the function u(x, t) and the unknown coefficient a(t) in a one‐dimensional partial differential equation. These schemes are developed to determine the unknown properties in a region by measuring only data on the boundary. Our goal has been focused on coefficients that presents physical quantities, for example, the conductivity of a medium. For the convenience of discussion, we will present the results of numerical experiment on several test problems. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Finite integration method for partial differential equations

A finite integration method is proposed in this paper to deal with partial differential equations in which the finite integration matrices of the first order are constructed by using both standard integral algorithm and radial basis functions interpolation respectively. These matrices of first order can directly be used to obtain finite integration matrices of higher order. Combining with the Laplace transform technique, the finite integration method is extended to solve time dependent partial differential equations. The accuracy of both the finite integration method and finite difference method are demonstrated with several examples. It has been observed that the finite integration method using either radial basis function or simple linear approximation gives a much higher degree of accuracy than the traditional finite difference method.     

Generalized fractional‐order Jacobi functions for solving a nonlinear systems of fractional partial differential equations numerically

In this paper, a collocation spectral numerical algorithm is presented for solving nonlinear systems of fractional partial differential equations subject to different types of conditions. A proposed error analysis investigates the convergence of the mentioned algorithm. Some numerical examples confirm the efficiency and accuracy of the method.     

Chebyshev polynomial approximation for high‐order partial differential equations with complicated conditions

In this article, a new method is presented for the solution of high‐order linear partial differential equations (PDEs) with variable coefficients under the most general conditions. The method is based on the approximation by the truncated double Chebyshev series. PDE and conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown Chebyshev coefficients, via Chebyshev collocation points. Combining these matrix equations and then solving the system yields the Chebyshev coefficients of the solution function. Some numerical results are included to demonstrate the validity and applicability of the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A meshless method based on the dual reciprocity method for one‐dimensional stochastic partial differential equations

This article describes a new meshless method based on the dual reciprocity method (DRM) for the numerical solution of one‐dimensional stochastic heat and advection–diffusion equations. First, the time derivative is approximated by the time–stepping method to transforming the original stochastic partial differential equations (SPDEs) into elliptic SPDEs. The resulting elliptic SPDEs have been approximated with the new method, which is a combination of radial basis functions (RBFs) method and the DRM method. We have used inverse multiquadrics (IMQ) and generalized IMQ (GIMQ) RBFs, to approximate functions in the presented method. The noise term has been approximated at the source points, at each time step. The developed formulation is verified in two test problems with investigating the convergence and accuracy of numerical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 292–306, 2016     

Quintic spline technique for time fractional fourth‐order partial differential equation

Higher order non‐Fickian diffusion theories involve fourth‐order linear partial differential equations and their solutions. A quintic polynomial spline technique is used for the numerical solutions of fourth‐order partial differential equations with Caputo time fractional derivative on a finite domain. These equations occur in many applications in real life problems such as modeling of plates and thin beams, strain gradient elasticity, and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical, and aerospace engineering. The quintic polynomial spline technique is used for space discretization and the time‐stepping is done using a backward Euler method based on the L1 approximation to the Caputo derivative. The stability and convergence analysis are also discussed. The numerical results are given, which demonstrate the effectiveness and accuracy of the numerical method. The numerical results obtained in this article are also compared favorably well with the results of (S. S. Siddiqi and S. Arshed, Int. J. Comput. Math. 92 (2015), 1496–1518). © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 445–466, 2017     

Fractional‐order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro‐differential equations

In this paper, a new two‐dimensional fractional polynomials based on the orthonormal Bernstein polynomials has been introduced to provide an approximate solution of nonlinear fractional partial Volterra integro‐differential equations. For this aim, the fractional‐order orthogonal Bernstein polynomials (FOBPs) are constructed, and its operational matrices of integration, fractional‐order integration, and derivative in the Caputo sense and product operational matrix are derived. These operational matrices are utilized to reduce the under study problem to a nonlinear system of algebraic equations. Using the approximation of FOBPs, the convergence analysis and error estimate associated to the proposed problem have been investigated. Finally, several examples are included to clarify the validity, efficiency, and applicability of the proposed technique via FOBPs approximation.     

Finite‐time synchronization of a class of N‐coupled complex partial differential systems with time‐varying delay

This study examines finite‐time synchronization for a class of N‐coupled complex partial differential systems (PDSs) with time‐varying delay. The problem of finite‐time synchronization for coupled drive‐response PDSs with time‐varying delay is similarly considered. The synchronization error dynamic of the PDSs is defined in the q‐dimensional spatial domain. We construct a feedback controller to achieve finite‐time synchronization. Sufficient conditions are derived by using the Lyapunov‐Krasoviskii stability approach and inequalities technology to ensure that the proposed networks achieve synchronization in finite time. The proposed systems demonstrate extensive application. Finally, an example is used to verify the theoretical results.     

Oscillation of second‐order perturbed differential equations

We give constructive proof of the existence of vanishing at infinity oscillatory solutions for a second‐order perturbed nonlinear differential equation. In contrast to most results reported in the literature, we do not require oscillatory character of the associated unperturbed equation, monotonicity of nonlinearity, and we establish global existence of oscillatory solutions without assuming it a priori. Furthermore, as our example demonstrates, existence of bounded oscillatory solutions does not exclude existence of unbounded nonoscillatory solutions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

High‐order compact schemes for fractional differential equations with mixed derivatives

In this article, we consider two‐dimensional fractional subdiffusion equations with mixed derivatives. A high‐order compact scheme is proposed to solve the problem. We establish a sufficient condition and show that the scheme converges with fourth order in space and second order in time under this condition.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2141–2158, 2017