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Single soliton and double soliton solutions of the quadratic‐nonlinear Korteweg‐de Vries equation for small and long‐times Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201022
Ali Başhan; Alaattin EsenIn this article, numerical solutions of the seven different forms of the single soliton and double soliton solutions of the Korteweg‐de Vries equation are investigated. Since numerical solution of the six test problems for small‐times do not exist in the literature, the present numerical results firstly are reported with exact solutions. Besides small‐time solutions, long‐time solutions of all test

Haar‐wavelet based approximation for pricing American options under linear complementarity formulations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201021
Devendra Kumar; Komal DeswalIn this manuscript, we present a novel and highly accurate wavelet‐based approximation technique to explore the sensitivities and value of American options diagnosed by linear complementarity problems. For a detailed analysis of such financially relevant problems, we transform the actual final value problem into a dimensionless initial value problem. To avoid the unacceptable large truncation error

Analysis of a novel finance chaotic model via ABC fractional derivative Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201021
Mustafa Ali DokuyucuIn this study, the analysis of the new finance chaotic model was expanded to Atangana–Baleanu–Caputo fractional derivative. The existence solution was investigated using the fixed model theorem of the new model. Then, the uniqueness solution of model was examined by using the Sumudu transformation.

A second‐order isoparametric element method to solve plane linear elastic problem Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201021
Shicang Song; Zhixin LiuConsidering the both effect of boundary approximation and numerical quadrature, a second‐order isoparametric element method is given to solve the homogeneous isotropic plane linear elasticity problem in domain Ω with curved boundary. By using technically analysis, the optimal error estimate with is obtained, where the function is an extension of the true solution to . It yields better accuracy than

Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201021
Haixiang Zhang; Xuehua Yang; Da XuA linearized orthogonal spline collocation (OSC) method with C1 splines of degree ≥3 on a suitable graded mesh is formulated and analyzed for approximate solution of an initial‐boundary‐value problem of semilinear subdiffusion equations with nonsmooth solutions in time. The sharp error estimate in the L2 norm is established without any restriction on the relative temporal and spatial mesh sizes. Such

Numerical implementation of nonlinear system of fractional Volterra integral–differential equations by Legendre wavelet method and error estimation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201019
Lijing Shen; Shuai Zhu; Bingcheng Liu; Zirui Zhang; Yuanda CuiIn the current study, a numerical scheme for solving the nonlinear system of fractional Volterra integro‐differential equations via Legendre wavelet is proposed. The Legendre wavelet operational matrix of fractional integration is derived and utilized to alter the main system to a system of algebraic equations. In addition, the error estimate of the original system is investigated in detail. Lastly

Kernel functions‐based approach for distributed order diffusion equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201019
Fazhan Z. Geng; Xinyuan WuIn this work, we solve distributed order diffusion equations (DODEs) by applying the theory on reproducing kernel functions (RKFs). The classical numerical quadrature formulae is used to approximate the DODE to a multi‐term Caputo fractional order diffusion equation (FDE). The Mittag‐Leffler RKF is introduced to estimate fractional derivatives of Caputo. And a space–time RKFs collocation scheme is

New results concerning to approximate controllability of Hilfer fractional neutral stochastic delay integro‐differential systems Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201016
C. Dineshkumar; R. UdhayakumarThis manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro‐differential equation with infinite delay. We study our primary outcomes by utilizing the theoretical concepts related to the fractional calculus and Krasnoselskii's fixed point theorem. First, we prove the approximate controllability of the fractional evolution system. Then, we extend

A new offer of NTRU cryptosystem with two new key pairs Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201014
Mehmet Sever; Ahmet Şükrü ÖzdemirIn this study, it is aimed to contribute to the NTRU cryptologic system to increase the number of operations of taking module by taking a polynomial module, to increase the number of secret key by adding an isomorphism and also to increase the number of public key by choosing an irreducible polynomial. Since it is calculated module by suitably partitioning a chosen message polynomial, and since it

A key agreement protocol based on group actions Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201014
Abdullah Çağman; Kadirhan Polat; Sait TaşThe key agreement scheme is an important part of the cryptography theory. The first study in this field belongs to Diffie–Hellman and Merkle. We present a new key agreement scheme using a group action of special orthogonal group of 2 × 2 matrices with real entries on the complex projective line.

On the absolute stable difference scheme for the space‐wise dependent source identification problem for elliptic‐telegraph equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201014
Allaberen Ashyralyev; Ahmad Al‐Hammouri; Charyyar AshyralyyevThe present paper is devoted to study the first order of accuracy absolute stable difference scheme for the approximate solution of the space identification problem for the elliptic‐telegraph equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the difference scheme is established. In applications, theorems on the stability of difference

On n‐dimensional quadratic B‐splines Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201012
K. R. Raslan; Khalid K. AliIn this paper, we present new constructions of the n‐dimensional quadratic B‐splines method. The quadratic B‐splines method format is displayed in a single, two, and three‐dimensional format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. Some numerical examples are also presented, through

Symplectic‐preserving Fourier spectral scheme for space fractional Klein–Gordon–Schrödinger equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201009
Junjie WangIn the paper, the symplectic‐preserving Fourier spectral scheme is presented for space fractional Klein–Gordon–Schrödinger equations involving fractional Laplacian. First, we validate space fractional Klein–Gordon–Schrödinger equations that can be expressed as an infinite dimension Hamiltonian system. We apply the Fourier spectral method in space, and the semi‐discrete system preserves the mass and

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201009
Devendra KumarIn this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ϵ leads to a parabolic right boundary layer. A collocation method consisting of cubic B‐spline basis functions on an appropriate piecewise‐uniform mesh is

A stabilizer free weak Galerkin finite element method with supercloseness of order two Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201009
Ahmed Al‐Taweel; Xiaoshen Wang; Xiu Ye; Shangyou ZhangThe weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. A simple WG finite element method is introduced for second‐order elliptic problems. First we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201009
Khosrow Maleknejad; Jalil Rashidinia; Tahereh EftekhariIn this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the

A sixth‐order improved Runge–Kutta direct method for special third‐order ordinary differential equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201009
Mukaddes Ökten Turacı; Merve ÖzdemirIn this paper, we construct a four‐stage explicit improved Runge–Kutta direct (IRKD) method of order six for solving special third‐order ordinary differential equations. The sixth‐order IRKD method is a two‐step method and it requires fewer number of stages compared to the classical Runge–Kutta method of the same order per step. The stability properties of the proposed method are given. Numerical results

On solution of a class of nonlinear variable order fractional reaction–diffusion equation with Mittag–Leffler kernel Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201009
Prashant Pandey; José Francisco Gómez‐AguilarIn this article, an efficient variable‐order Chebyshev collocation method which is based on shifted fifth‐kind Chebyshev polynomials is applied to solve a nonlinear variable‐order fractional reaction–diffusion equation with Mittag–Leffler kernel. The operational matrix of shifted fifth‐kind Chebyshev polynomials is derived for variable‐order ABC derivatives. The Chebyshev operational matrix together

A new study on existence and uniqueness of nonlocal fractional delay differential systems of order 1 < r < 2 in Banach spaces Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201007
W. Kavitha Williams; V. Vijayakumar; R. Udhayakumar; Kottakkaran Sooppy NisarThis article is mainly focusing on the existence and uniqueness of nonlocal fractional delay differential systems of order 1 < r < 2 in Banach spaces. By using the theoretical concepts related to the fractional calculus, cosine, and sine functions of operators and fixed point approach, we prove our main results. By using Kranoselskii's fixed point theorem, we discuss the existence of the mild solution

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201007
Rauf Amirov; Abdullah Ergun; Sevim DurakIn this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and

Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201005
Devendra Kumar; Parvin KumariA numerical scheme is constructed for the problems in which the diffusion and convection parameters (ϵ1 and ϵ2, respectively) both are small, and the convection and source terms have a jump discontinuity in the domain of consideration. Depending on the magnitude of the ratios , and two different cases have been considered separately. Through rigorous analysis, the theoretical error bounds on the singular

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201005
Shallu Gupta; Vijay Kumar KukrejaIn this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201003
Mehdi Dehghan; Nasim Shafieeabyaneh; Mostafa AbbaszadehThis article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has

Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201002
Huan Liu; Xiangcheng Zheng; Hongfei Fu; Hong WangIn this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast

Leverage centrality analysis of infrastructure networks Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201001
Murat Erşen BerberlerLeverage centrality is a novel centrality measure proposed to identify the critical nodes that are highly influential within the network. Leverage centrality considers the extent of connectivity of a node relative to the connectivity of its neighbors. The leverage centrality of a node in a network is determined by the extent to which its direct neighbors rely on that node for information. In this paper

Long time error analysis of the fourth‐order compact finite difference methods for the nonlinear Klein–Gordon equation with weak nonlinearity Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201001
Yue FengWe present the fourth‐order compact finite difference (4cFD) discretizations for the long time dynamics of the nonlinear Klein–Gordon equation (NKGE), while the nonlinearity strength is characterized by ϵp with a constant p ∈ ℕ+ and a dimensionless parameter ϵ ∈ (0, 1]. Based on analytical results of the life‐span of the solution, rigorous error bounds of the 4cFD methods are carried out up to the

A new exploration on existence of Sobolev‐type Hilfer fractional neutral integro‐differential equations with infinite delay Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20201001
V. Vijayakumar; R. UdhayakumarThis article is primarily focusing on the existence of Sobolev‐type Hilfer fractional neutral integro‐differential systems via measure of noncompactness. We study our primary outcomes by employing fractional calculus, measure of noncompactness and fixed point technique. First, we discuss the existence of mild solution for the fractional evolution system. Then, we extend our results to discuss the system

Comparison of efficiency among different techniques to avoid order reduction with Strang splitting Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200930
Isaías Alonso‐Mallo; Begoña Cano; Nuria RegueraIn this article, we offer a comparison in terms of computational efficiency between two techniques to avoid order reduction when using Strang method to integrate nonlinear initial boundary value problems with time‐dependent boundary conditions. We see that it is important to consider an exponential method for the integration of the linear nonhomogeneous and stiff part in the technique by Einkemmer

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200928
Ali Başhan; N. Murat Yağmurlu; Yusuf Uçar; Alaattin EsenThe aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate

Approximation of functions by a new construction of Bernstein‐Chlodowsky operators: Theory and applications Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200928
Fuat UstaThe main motivation of this paper is to provide a generalization of Bernstein‐Chlodowsky type operators which depend on function τ by means of two sequences of functions. The newly defined operators fix the test function set {1, τ, τ2}. Then we present the approximation properties of newly defined operators, such as weighted approximation, degree of approximation and Voronovskaya type theorems. Finally

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200928
Haifeng Wang; Da Xu; Jun Zhou; Jing GuoIn this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L2 norm is established. Numerical experiments are presented to validate the theoretical

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200928
Afshin Babaei; Seddigheh Banihashemi; Carlo CattaniThe 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

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200928
Boniface Nkemzi; Michael JungSolutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ3 with edge singularities and presents, on the one hand, explicit computational

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200928
Qinxu Ding; Patricia J. Y. WongIn this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t). Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method

Investigation of Coriolis effect on oceanic flows and its bifurcation via geophysical Korteweg–de Vries equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200721
Turgut Ak; Asit Saha; Sharanjeet Dhawan; Abdul Hamid KaraIn this work, we have investigated Coriolis effect on oceanic flows in the equatorial region with the help of geophysical Korteweg–de Vries equation (GKdVE). First, Lie symmetries and conservation laws for the GKdVE have been studied. Later, we implement finite element method for numerical simulations. Propagation of nonlinear solitary structures, their interaction and advancement of solitons can be

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200721
Huadong Gao; Weiwei SunNumerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200721
Qifeng Zhang; Lingling Liu; Jiyuan ZhangIn the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two‐level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ2 + h2). For the three‐level linearized scheme

Geometric trigonometrically convexity and better approximations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200801
Mahir KadakalIn this manuscript, we introduce the concept of geometric trigonometrically convex functions. We obtain some refinements of the Hermite–Hadamard inequality for functions whose first derivative in absolute value is geometric trigonometrically convex. In addition, it has been proved that the results obtained with Hölder–İşcan and improved power‐mean integral inequalities give a better approach than Hölder

A second‐order space–time accurate scheme for nonlinear diffusion equation with general capacity term Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200731
Huan Zheng; Guangwei Yuan; Xia CuiA fully implicit finite difference (FIFD) scheme with second‐order space–time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays

Convergence analysis of a hp‐finite element approximation of the time‐harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200817
Serge Nicaise; Jérôme TomezykWe consider a nonconforming hp‐finite element approximation of a variational formulation of the time‐harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H1 as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem

Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200810
Yang Liu; Nan Liu; Hong Li; Jinfeng WangIn this article, a spatial two‐grid finite element (TGFE) algorithm is used to solve a two‐dimensional nonlinear space–time fractional diffusion model and improve the computational efficiency. First, the second‐order backward difference scheme is used to formulate the time approximation, where the time‐fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In

Solution of the linear and nonlinear advection–diffusion problems on a sphere Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200831
Yuri N. Skiba; Roberto C. Cruz‐Rodríguez; Denis M. FilatovVarious linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite‐volume method of second‐order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator

Compact difference scheme for two‐dimensional fourth‐order nonlinear hyperbolic equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200827
Qing Li; Qing Yang; Huanzhen ChenHigh‐order compact finite difference method for solving the two‐dimensional fourth‐order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth‐order equation is written as a system of two second‐order equations by introducing the second‐order spatial derivative as a new variable. The second‐order spatial derivatives

A Haar wavelet collocation approach for solving one and two‐dimensional second‐order linear and nonlinear hyperbolic telegraph equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200804
Muhammad Asif; Nadeem Haider; Qasem Al‐Mdallal; Imran KhanWe have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one‐ and two‐dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one‐ and two‐dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal

On Romanovski–Jacobi polynomials and their related approximation results Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200822
Howayda Abo‐Gabal; Mahmoud A. Zaky; Ramy M. Hafez; Eid H. DohaThe aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F‐distribution over the positive real line. We introduce some basic properties of the Romanovski–Jacobi polynomials, the Romanovski–Jacobi–Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation

High accuracy nonconforming biharmonic element over n‐rectangular meshes Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200810
Xinchen Zhou; Zhaoliang MengThis work gives the high accuracy analysis of a rectangular biharmonic element in arbitrarily high‐dimensional cases. Given an n‐rectangle, we construct the nonconforming finite element and show its explicit standard basis representation. We prove that, if the n‐rectangular mesh is uniform, this element can achieve a second order convergence rate in energy norm when applied to biharmonic problems.

Two‐step discretization method for 2D/3D Allen–Cahn equation based on RBF‐FD scheme Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200815
Lin Yao; Xindong Zhang; Ning LiIn this article, a two‐step discretization method based on multi‐quadrics (MQ) radial basis function (RBF) is presented for solving Allen–Cahn (AC) equation with integer derivative for time and space. In the first step, backward Euler formula with Newton iterative method is used to discrete the time direction of AC equation. And RBF method is applied in space for solving semi‐discrete linearized problem

Controlled fuzzy metric spaces and some related fixed point results Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200918
Müzeyyen Sangurlu SezenIn this paper, we introduce a new extension in the subject of fuzzy metric, called controlled fuzzy metric space. This notion is a generalization of fuzzy b‐metric space. Also, we prove a Banach‐type fixed point theorem and a new fixed point theorem for some self‐mappings satisfying fuzzy ψ‐contraction condition that is more general than existing theorems. Furthermore, we establish some examples about

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200918
Mahboub BaccouchIn this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type uxx + uyy = f(x, y, u, ux, uy), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use

RBF‐ENO/WENO schemes with Lax–Wendroff type time discretizations for Hamilton–Jacobi equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200914
Rooholah Abedian; Mehdi DehghanIn this research, a class of radial basis functions (RBFs) ENO/WENO schemes with a Lax–Wendroff time discretization procedure, named as RENO/RWENO‐LW, for solving Hamilton–Jacobi (H–J) equations is designed. Particularly the multi‐quadratic RBFs are used. These schemes enhance the local accuracy and convergence by locally optimizing the shape parameters. Comparing with the original WENO with Lax–Wendroff

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200914
Raziyeh Erfanifar; Khosro Sayevand; Nasim Ghanbari; Hamid EsmaeiliThis study presents a robust modification of Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200914
Tong Kang; Ran Wang; Huai ZhangWe study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this

Discontinuous Galerkin method for the coupled Stokes‐Biot model Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200912
Jing Wen; Jian Su; Yinnian He; Hongbin ChenIn this paper, a semi‐discrete scheme and a fully discrete scheme of the Stokes‐Biot model are proposed, and we analyze the semi‐discrete scheme in detail. First of all, we prove the existence and uniqueness of the semi‐discrete scheme, and a‐priori error estimates are derived. Then, we present the same conclusions for the fully discrete scheme. Finally, under both matching and non‐matching meshes

Efficient approximation algorithm for the Schrödinger–Possion system Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200907
Xuefei He; Kun WangIn this article, we study an efficient approximation algorithm for the Schrödinger–Possion system arising in the resonant tunneling diode (RTD) structure. By following the classical Gummel iterative procedure, we first decouple this nonlinear system and prove the convergence of the iteration method. Then via introducing a novel spatial discrete method, we solve efficiently the decoupled Schrödinger

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200905
Yun Zhang; Ting Wei; Yuan‐Xiang ZhangThis study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional

Linear barycentric rational collocation method for solving heat conduction equation Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200902
Jin Li; Yongling ChengThe linear barycentric rational collocation method for solving heat conduction equation is presented. The matrix form of discrete heat conduction equation by collocation method is also obtained. With the help of convergence rate of the barycentric interpolation, the convergence rate of linear barycentric rational collocation method for solving heat conduction equation is proved. At last, several numerical

A new approach by two‐dimensional wavelets operational matrix method for solving variable‐order fractional partial integro‐differential equations Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200831
Santanu Saha RayIn 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

The element‐free Galerkin method for a quasistatic contact problem with the Tresca friction in elastic materials Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200829
Rui Ding; Quan Shen; Yuebin HuoThis paper is proposed for the error estimates of the element‐free Galerkin method for a quasistatic contact problem with the Tresca friction. The penalty method is used to impose the clamped boundary conditions. The duality algorithm is also given to deal with the non‐differentiable term in the quasistatic contact problem with the Tresca friction. The error estimates indicate that the convergence

A study over the general formula of regression sum of squares in multiple linear regression Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200828
Mehmet KorkmazIn this study, in addition to the formula of regression sum of squares (SSR) in linear regression, a general formula of SSR in multiple linear regression is given. The derivations of the formula presented are given step by step. This new formula is proposed for estimation of the SSR in multiple linear regression. By using this formula, the researcher can find easily SSR and so the researcher can compose

The unconditional stability and mass‐preserving S‐DDM scheme for solving parabolic equations in three dimensions Numer. Methods Partial Differ. Equ. (IF 2.236) Pub Date : 20200827
Zhongguo Zhou; Jingwen Gu; Lin Li; Hao Pan; Yan WangIn this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior