Jacobi spectral Galerkin method for the integrated forms of second-order differential equations

Efficiently direct solvers based on the Jacobi-Galerkin method for the integrated forms of second-order elliptic equations in one and two space variables are presented. They are based on appropriate base functions for the Galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted. The homogeneous Dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the Jacobi polynomials. The direct solution algorithm here developed for the homogeneous Dirichlet problem in two-dimensions relies upon a diagonalization process. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.     

A Petrov–Galerkin spectral method for fourth‐order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth‐order elliptic problems on rectangular domains subject to non‐homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher‐order problems. By applying these results to a fourth‐order problem, we establish the H²‐error and L²‐error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Jacobi spectral Galerkin method for elliptic Neumann problems

This paper is concerned with fast spectral-Galerkin Jacobi algorithms for solving one- and two-dimensional elliptic equations with homogeneous and nonhomogeneous Neumann boundary conditions. The paper extends the algorithms proposed by Shen (SIAM J Sci Comput 15:1489–1505, 1994) and Auteri et al. (J Comput Phys 185:427–444, 2003), based on Legendre polynomials, to Jacobi polynomials with arbitrary α and β. The key to the efficiency of our algorithms is to construct appropriate basis functions with zero slope at the endpoints, which lead to systems with sparse matrices for the discrete variational formulations. The direct solution algorithm developed for the homogeneous Neumann problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Neumann data are accounted for by means of a lifting. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.     

Oscillation theorems for fourth‐order delay differential equations with a negative middle term

This paper deals with the oscillation of the fourth‐order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third‐order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.     

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     

A C0‐weak Galerkin finite element method for fourth‐order elliptic problems

This article proposes and analyzes a C⁰‐weak Galerkin (WG) finite element method for solving the biharmonic equation in two‐dimensional and three‐dimensional. The new WG method uses continuous piecewise‐polynomial approximations of degree for the unknown u and discontinuous piecewise‐polynomial approximations of degree k for the trace of on the interelement boundaries. Optimal error estimates are obtained in H², H¹, and L² norms. Numerical experiments illustrate and confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1090–1104, 2016     

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.     

Jacobi rational approximation and spectral method for differential equations of degenerate type

We introduce an orthogonal system on the half line, induced by Jacobi polynomials. Some results on the Jacobi rational approximation are established, which play important roles in designing and analyzing the Jacobi rational spectral method for various differential equations, with the coefficients degenerating at certain points and growing up at infinity. The Jacobi rational spectral method is proposed for a model problem appearing frequently in finance. Its convergence is proved. Numerical results demonstrate the efficiency of this new approach.

     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Oscillation criteria for fourth‐order differential equations with middle term

Oscillation criteria for self‐adjoint fourth‐order differential equations were established for various conditions on the coefficients r(x) > 0, q(x) and p(x). However, most of these results deal with the case when lim_{x → ∞}∫^x₁q(s) ds < +∞. In this note we give a new oscillation criterion in the case when this condition is not fulfilled, in particular when q(x)↗ + ∞ (even with exponential growth).     

Bernstein modal basis: Application to the spectral Petrov‐Galerkin method for fractional partial differential equations

In the spectral Petrov‐Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein‐spectral Petrov‐Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.     

The general Jacobi matrix method for solving some nonlinear ordinary differential equations

In this paper, we obtain the approximate solutions for some nonlinear ordinary differential equations by using the general Jacobi matrix method. Explicit formulae which express the Jacobi expansion coefficients for the powers of derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. Three test problems are discussed to illustrate the efficiency of the proposed method.     

A Hermite collocation method for the approximate solutions of high‐order linear Fredholm integro‐differential equations

In this study, a Hermite matrix method is presented to solve high‐order linear Fredholm integro‐differential equations with variable coefficients under the mixed conditions in terms of the Hermite polynomials. The proposed method converts the equation and its conditions to matrix equations, which correspond to a system of linear algebraic equations with unknown Hermite coefficients, by means of collocation points on a finite interval. Then, by solving the matrix equation, the Hermite coefficients and the polynomial approach are obtained. Also, examples that illustrate the pertinent features of the method are presented; the accuracy of the solutions and the error analysis are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1707–1721, 2011     

Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition

In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equations where w is a constant, and . By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well‐known (AR)‐condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth‐order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.     

Time evolution of discrete fourth‐order elliptic operators

The evolution equation is considered. A discrete parabolic methodology is developed, based on a discrete elliptic (fourth‐order) calculus. The main ingredient of this calculus is a discrete biharmonic operator (DBO). In the general case, it is shown that the approximate solutions converge to the continuous one. An “almost optimal” convergence result (O(h^{4 ? ?})) is established in the case of constant coefficients, in particular in the pure biharmonic case. Several numerical test cases are presented that not only corroborate the theoretical accuracy result, but also demonstrate high‐order accuracy of the method in nonlinear cases. The nonlinear equations include the well‐studied Kuramoto–Sivashinsky equation. Numerical solutions for this equation are shown to approximate remarkably well the exact solutions. The numerical examples demonstrate the great improvement achieved by using the DBO instead of the standard (five‐point) discrete bilaplacian.     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

Highlights are the following:

• For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
• To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
• To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.

Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.