فهرست مطالب

Computational Methods for Differential Equations
Volume:11 Issue: 1, Winter 2023

  • تاریخ انتشار: 1401/12/16
  • تعداد عناوین: 15
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  • Rooholah Abedian * Pages 1-11
    In this paper, the 2D incompressible Navier-Stokes (INS) equations in terms of vorticity and stream function are considered. These equations describe the physics of many phenomena of scientific and engineering. By combining monotone upwind methods and weighted essentially non-oscillatory (WENO) procedures, a new numerical algorithm is proposed to approximate the solution of INS equations. To design this algorithm, after obtaining an optimal polynomial, it is rewritten as a convex combination of second-order modified ENO polynomials. Following the methodology of the traditional WENO procedure, the new non-linear weights are calculated. The performance of the new scheme on a number of numerical examples is illustrated.
    Keywords: Navier-Stokes equation, WENO, UNO limiter
  • Youssri Youssri *, Shahenda Sayed, Amany Mohamed, Emad Aboeldahab, Waleed Abd-Elhameed Pages 12-31
    This paper is devoted to the construction of certain polynomials related to Lucas polynomials, namely, modified Lucas polynomials. The constructed modified Lucas polynomials are utilized as basis functions for the numerical treatment of the linear and non-linear second-order boundary value problems (BVPs) involving some specific important problems such as singular and Bratu-type equations. To derive our proposed algorithms, the operational matrix of derivatives of the modified Lucas polynomials is established by expressing the first-order derivative of these polynomials in terms of their original ones. The convergence analysis of the modified Lucas polynomials is deeply discussed by establishing some inequalities concerned with these modified polynomials. Some numerical experiments accompanied by comparisons with some other articles in the literature are presented to demonstrate the applicability and accuracy of the presented algorithms.
    Keywords: Lucas polynomials, Boundary value problems, Bratu equations, Singular initial value problems, Spectral methods, Operational matrix, Convergence analysis
  • Hossein Fazli *, Fariba Bahrami, Sedaghat Shahmorad Pages 32-41
    This paper is devoted to prove the existence of extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputo type and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. As an application, some examples are presented to illustrate the main results.
    Keywords: Caputo fractional derivative, Extremal solutions, existence, Approximation, Nonlinear boundary conditions
  • Narahari Raji Reddy, Jugal Mohapatra * Pages 42-51
    This article proposes a parameter uniform numerical method for solving a singularly perturbed delay parabolic initial boundary-value problem involving mixed space shifts. The model also involves a large delay in time. Taylor’s series expansion is applied to approximate the retarded terms in the spatial direction. For the time discretization, the implicit trapezoidal scheme is applied on uniform mesh, and for the spatial discretization, we use a proper combination of the mid-point upwind and the central difference scheme on Shishkin mesh. The proposed scheme provides a second-order convergence rate uniformly with respect to the perturbation parameter. Some comparison results are presented by using the proposed method to support our claim.
    Keywords: Time delay parabolic problem, mixed shifts, Singular perturbation, Boundary layer, uniform convergence
  • Maryam Alipour, Samaneh Soradi Zeid * Pages 52-64
    In this paper, we introduce a new direct scheme based on Dickson polynomials and collocation points to solve a class of optimal control problems (OCPs) governed by Volterra integro-differential equations namely Volterra integro-OCPs (VI-OCPs). This topic requires to calculating the corresponding operational matrices for expanding the solution of this problem in terms of Dickson polynomials. Further, the highlighted method allows us to transform the VI-OCP into a system of algebraic equations for choosing the coefficients and control parameters optimally. The error estimation of this technique is also investigated which given the high efficiency of the Dickson polynomials to deal with these problems. Finally, some examples are brought to confirm the validity and applicability of this approach in comparison with those obtained from other methods.
    Keywords: Dickson polynomials, Optimal control problem, Volterra integro-differential equation, Algebraic equations, Collocation points, Error estimation
  • Chafai Imzegouan, Aziz Zouine, Hassane Bouzahir, Cemil Tunç * Pages 65-80
    This paper focuses on existence, uniqueness, and stability analysis of solutions for a new kind of delayed integro-differential neural networks with Markovian switches in delays and noises. The studied system combines many types of integro-differential neural network treatises in the literature. After having presented the studied system, the existence and uniqueness of solutions are shown under Lipschitz condition. By using the Lyapunov-Krasovskii functional, some stochastic analysis techniques and the M-matrix approach, stochastic stability, and general decay stability are established. Finally, a numerical example is given to validate the main established theoretical results.
    Keywords: Neural Networks, Markovian jumps systems, Levy noise, Gaussian noise, Neutral-type systems, Time-varying delays, General decay stability
  • Samira Bonyadi, Yaghoub Mahmoudi *, Mehrdad Lakestani, Mohammad Jahangiri Rad Pages 81-94
    The paper reports a spectral method for generating an approximate solution for the space-time fractional PDEs with variable coefficients based on the spectral shifted Jacobi collocation method in conjunction with the shifted Jacobi operational matrix of fractional derivatives. The spectral collocation method investigates both temporal and spatial discretizations. By applying the shifted Jacobi collocation method, the problem reduces to a system of algebraic equations, which greatly simplifies the problem. Numerical results are given to establish the validity and accuracy of the presented procedure for space-time fractional PDE.
    Keywords: Jacobi polynomials, Operational matrices, space-time PDEs, Collocation method
  • Melike Karta * Pages 95-107
    In this paper, numerical computation of the modified equal width equation (MEW), which is one of the equations used to model nonlinear events, will be carried out. For this equation, numerical computations have been obtained by many researchers using different methods. The goal of the new approach is to check how well it performs with respect to the numerical calculations the researchers found. For this, the proposed study presents a Lie-Trotter splitting algorithm in accordance with the time-splitting technical rules combined with Lumped Galerkin FEM based on the basis function of the cubic B-spline. Two valid test examples are given to determine the validity and effectiveness of the current technique. The results obtained in a new way with the Matlab computational software are compared with the studies of other authors in the literature and are shown graphically. Based on these new results, it can clearly be stated that the benefit of the proposed approach is to demonstrate that reliability is achieved in obtaining approximate computations.
    Keywords: The MEW equation, B-Spline, Collocation method, Lumped Galerkin method, Lie-Trotter splitting
  • MohammadReza Yaghouti *, Farnaz Farshadmoghadam Pages 108-129

    The radial basis functions (RBFs) meshless method has high accuracy for the interpolation of scattered data in high dimensions. Most of the RBFs depend on a parameter, called shape parameter which plays a significant role to specify the accuracy of the RBF method. In this paper, we present three algorithms to choose the optimal value of the shape parameter. These are based on Rippa’s theory to remove data points from the data set and results obtained by Fasshauer and Zhang for the iterative approximate moving least square (AMLS) method. Numerical experiments confirm stable solutions with high accuracy compared to other methods.

    Keywords: Radial basis functions, Shape Parameter, Leave-One-Out Cross Validation, Leave-Two-Out Cross Validation, Approximate Moving Least Squares
  • Yesim Cicek, Nurcan Gucuyenen Kaymak *, Ersin Bahar, Gurhan Gurarslan, Gamze Tangolu Pages 130-142
    In this article, the coupled Burger’s equation which is one of the known systems of the nonlinear parabolic partial differential equations is studied. The method presented here is based on a combination of the quintic B-spline and a high order time integration scheme known as adaptive Runge-Kutta method. First of all, the application of the new algorithm on the coupled Burger’s equation is presented. Then, the convergence of the algorithm is studied in a theorem. Finally, to test the efficiency of the new method, coupled Burger’s equations in literature are studied. We observed that the presented method has better accuracy and efficiency compared to the other methods in the literature.
    Keywords: Quintic B-spline, Adaptive Runge-Kutta Method, Coupled Burger’s equation, Non-linear parabolic partial differential equation
  • Parastoo Reihani *, Hossein Esmailpour, Fahimeh Soltanian Pages 143-160
    In the present study, we consider an important mathematical model of the spread of two competing species in an ecological system with two species considering the interactions between these species. This model is derived from a system of nonlinear reaction-diffusion equations. We investigate this model as an inverse problem. Using appropriate initial and boundary conditions, the finite difference method in the time variable and the Quartic Bspline collocation method in the spatial variable are used to develop a numerical method. The proposed numerical approach results in an ill-posed linear system of equations and to overcome the ill-posedness, the Tikhonov regularization method is implemented. An effective approach based on the ABC algorithm is established to determine the regularization parameter. To show the robustness and ability of the present approach, for a test case, the results are compared with the results of the L-curve and GCV methods.
    Keywords: Inverse problem, Reaction-Diffusion Problem, Regularization, ABS Algorithm
  • Gemeda Tolessa Lubo *, Gemechis File Duressa Pages 161-174
    This paper is concerned with the numerical treatment of delay reaction-diffusion with the Dirichlet boundary condition. The finite element method with linear B-spline basis functions is utilized to discretize the space variable. The Crank-Nicolson method is used for the processes of time discretization. Sufficient and necessary conditions for the numerical method to be asymptotically stable are investigated. The convergence of the numerical method is studied. Some numerical experiments are performed to verify the applicability of the numerical method.
    Keywords: Delay reaction diffusion equation, Crank Nicolson, Linear B-spline, finite element method, Asymtotic stability, Convergence
  • Mehdi Jafari *, Razie Darvazebanzade Pages 175-182
    In this paper, we apply the approximate symmetry transformation group to obtain the approximate symmetry group of the perturbed mKdV-KS equation which is a modified Korteweg-de Vries (mKdV) equation with a higher singularity perturbed term as the Kuramoto-Sivashinsky (KS) equation. Also, an optimal system of one-dimensional subalgebras of symmetry algebra is constructed and the corresponding differential invariants and some approximately invariant solutions of the equation are computed.
    Keywords: Perturbed mKdV-KS equation, Approximate symmetry, Approximately invariant solution, Optimal system
  • Narges Mahmoodi Darani * Pages 183-196
    In the current study, a fast, accurate, and reliable numerical scheme for approximating second-kind nonlinear Fredholm, Volterra, and Fredholm-Volterra integral equations with a weakly singular kernel and invertible nonlinearity is presented. The computational approach is based upon function, especially the hybrid one. Hybrid functions give us the opportunity to achieve an appropriate solution by adjusting a suitable order for polynomials’ degrees and block-pulse functions. The basic idea of this method is based on using the invertibility of the nonlinear function as a benefit to reduce the total error and simplify the procedure. The scheme reduces these types of equations to nonlinear systems of algebraic equations. Convergence analysis of the method under the infinity norm is wellstudied. Numerical results indicate the superiority of the present method compared with another existing method in the literature
    Keywords: Hybrid functions, collocation scheme, Weakly singular kernel, Nonlinear Fredholm-Volterra integral equation, Convergence analysis
  • Tahir Gadjiev, Sardar Aliev *, Aybeniz ‎Yagnalieva Pages 197-206
    Holder estimates of solutions of initial-boundary problem degenerate nonlinear parabolic equations are obtained. Estimates for solutions and parabolic Harnack inequality are proved. Also, one variant of weighted Poincare inequality is shown.
    Keywords: degenerate, parabolic equation, Nonlinear, Holder estimates