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

Mittal, R. C.; Jain, Reema

doi:10.1016/j.amc.2011.04.061

Cited by 23 publications

(20 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These PDEs describe various physical phenomenon including deformation of beams, viscoelastic and inelastic flows, transverse vibrations of a homogeneous beam, plate deflection theory, engineering and applied sciences [1][2][3][4][5][6][7][8][9][10][11][12]. In recent years, various methods have been proposed for solving the fourth-order parabolic PDEs, such that adomian decomposition method (ADM) [13,14], variational iteration method (VIM) [15,16], B-spline methods [17][18][19], homotopy perturbation method (HPM) [20] and homotopy analysis method (HAM) [21].…”

Section: Introductionmentioning

confidence: 99%

Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

Ibis¹

2014

J. Math. Computer Sci.

View full text Add to dashboard Cite

The purpose of this paper is to obtain the approximate solution of fourth-order parabolic partial differential equations by the reduced differential transform method (RDTM).This method provides the solution in the form of a convergent series with easily calculable terms. Comparing RDTM with some other methods in the literature shows present approach is very simple, effective, powerful and can be easily applied to other linear or nonlinear PDEs in science and engineering.Keywords: Reduced differential transform method (RDTM), differential transform method (DTM) fourthorder parabolic partial differential equations, initial value problems.

show abstract

Section: Introductionmentioning

confidence: 99%

Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

Ibis¹

2014

J. Math. Computer Sci.

View full text Add to dashboard Cite

show abstract

“…, m − 3. Substituting the above two expressions in (51) (with d = 2), we arrive at the semi-discrete equation (47) with the amplification factor…”

Section: Qbsqi Methodsmentioning

confidence: 99%

“…It is known that quadratic splines are C 1 -functions and therefore are not suitable for (1). In such circumstances, it is common to rewrite (1) into a system of first order equations (see for instance [46]) v = u x u t + f (u) x − δv x − βv xt = g(x, t)  (6) and then use the quadratic B-spline quasi-interpolation (also see [47] for a fourth order equation). We also propose to use the cubic B-spline quasi-interpolation for the same system, and show that this method is more accurate and achieves higher rate of convergences when compared to the cubic B-spline quasi-interpolation applied to (1) directly.…”

Section: Introductionmentioning

confidence: 99%

B-spline quasi-interpolation based numerical methods for some Sobolev type equations

Kumar

Baskar

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tIn this article, we intend to use quadratic and cubic B-spline quasi-interpolants to develop higher order numerical methods for some Sobolev type equations in one space dimension. Our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence. We also discuss another approach to the cubic B-spline quasi-interpolation based method, where we achieve fourth order of accuracy in space. We theoretically establish the order of accuracy for the three proposed methods and also establish the L 2 -stability in the linear case using von Neumann analysis. As a particular case of the Sobolev type equations, we take the equal width and the Benjamin-Bona-Mahony-Burgers equations, and perform several numerical experiments to support our theoretical results.

show abstract

“…The differential equation is converted to its corresponding weak form. Then, it is approximated using B-spline functions thereby constructing the eigenvalue problem (Hopkins and Wait, 1979;Mittal and Jain, 2011;Gu et al, 2011;Yanan et al, 2011;Mohammadi, 2014;Xiaoyun and Weiyin, 2014). Finally, after applying essential boundary conditions, natural frequencies and corresponding mode shapes are calculated and compared with those obtained from finite element (FE) analysis.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of combined system of framed tube and shear walls by Galerkin method using B‐spline functions

Rahgozar

Mahmoudzadeh

Malekinejad

et al. 2014

Structural Design Tall Build

View full text Add to dashboard Cite

SUMMARYIn this article, dynamic parameters (natural frequencies and mode shapes) of tall buildings that consist of framed tube and shear walls are obtained using a simple approximate method. The three-dimensional structure is replaced by an equivalent cantilever beam, considering both bending and shear deformations. On the basis of dynamic equilibrium, the governing differential equation of motion is obtained and converted to its corresponding weak form. B-spline functions are then utilized to approximate the weak form and to obtain the final matrix form of the problem. Finally, by applying essential boundary conditions, the natural frequencies and corresponding mode shapes are calculated. To demonstrate the accuracy of the proposed method, numerical examples are solved, and the results are compared with those obtained from SAP2000 computer analysis. The results show that the proposed method is efficient and accurate enough to be used in preliminary design.

show abstract

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

Cited by 23 publications

References 22 publications

Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

Application Of Reduced Differential Transformation Method For Solving Fourth-order Parabolic Partial Differential Equations

B-spline quasi-interpolation based numerical methods for some Sobolev type equations

Dynamic analysis of combined system of framed tube and shear walls by Galerkin method using B‐spline functions

Contact Info

Product

Resources

About