Accurate numerical approximation of nonlinear fourth order Emden-Fowler type equations: A Haar based wavelet-collocation approach

2017 ◽  
Vol 3 (1) ◽  
pp. 75-83
Author(s):  
Najeeb Alam Khan ◽  
Amber Shaikh ◽  
Muhammad Ayaz
Keyword(s):  
Fourth Order ◽  
Numerical Approximation ◽  
Collocation Approach

scholarly journals Block Backward Differentiation Formulas for Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
T. A. Biala ◽  
S. N. Jator
Keyword(s):  
Numerical Approximation ◽  
Initial Value ◽  
One Dimensional ◽  
Numerical Tests ◽  
Numerical Integrators ◽  
Backward Differentiation Formulas ◽  
Collocation Approach ◽  
Fractional Differential ◽  
Large Systems ◽  
Differentiation Formulas

This paper concerns the numerical approximation of Fractional Initial Value Problems (FIVPs). This is achieved by constructing k-step continuous BDFs. These continuous schemes are developed via the interpolation and collocation approach and are used to obtain the discrete k-step BDF and (k-1) additional methods which are applied as numerical integrators in a block-by-block mode for the integration of FIVP. The properties of the methods are established and regions of absolute stability of the methods are plotted in the complex plane. Numerical tests including large systems arising form the semidiscretization of one-dimensional fractional Burger’s equation show that the methods are highly accurate and efficient.

NUMERICAL SOLUTION FOR THE NONLINEAR EMDEN–FOWLER TYPE EQUATIONS BY A FOURTH-ORDER ADAPTIVE METHOD

2013 ◽  
Vol 11 (01) ◽  
pp. 1350052 ◽  
Author(s):  
S. A. KHURI ◽  
A. SAYFY
Keyword(s):  
Finite Element ◽  
Numerical Solution ◽  
Rate Of Convergence ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Adaptive Method ◽  
B Splines ◽  
Special Cases ◽  
Collocation Approach

A finite element collocation approach, based on cubic B-splines, is manipulated for obtaining numerical solutions of a generalized form of the Emden–Fowler type equations. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order by using the double-mesh principle. The efficiency of the scheme is tested on a number of examples which represent special cases of the problem under consideration. The results are compared with analytical and other numerical solutions that are available in the literature. The proposed method reveals that the outcomes are reliable and very accurate when contrasted with other existing methods.

scholarly journals Comparison of Proposed and Existing Fourth Order Schemes for Solving Non-linear Equations

2019 ◽  
pp. 1-7
Author(s):  
Khushbu Rajput ◽  
Asif Ali Shaikh ◽  
Sania Qureshi
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Numerical Approximation ◽  
Real Root ◽  
Graphical Representation ◽  
Linear Equations ◽  
Order Convergence ◽  
Derivative Free ◽  
Non Linear Equations ◽  
Number Of Iterations

This paper, investigates the comparison of the convergence behavior of the proposed scheme and existing schemes in literature. While all schemes having fourth-order convergence and derivative-free nature. Numerical approximation demonstrates that the proposed schemes are able to attain up to better accuracy than some classical methods, while still significantly reducing the total number of iterations. This study has considered some nonlinear equations (transcendental, algebraic and exponential) along with two complex mathematical models. For better analysis graphical representation of numerical methods for finding the real root of nonlinear equations with varying parameters has been included. The proposed scheme is better in reducing error rapidly, hence converges faster as compared to the existing schemes.

scholarly journals Fourth-Order Compact Difference Schemes for the Riemann-Liouville and Riesz Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yuxin Zhang ◽  
Hengfei Ding ◽  
Jincai Luo
Keyword(s):  
Fourth Order ◽  
Numerical Approximation ◽  
Difference Schemes ◽  
Order Convergence ◽  
Transform Method ◽  
Numerical Examples ◽  
Compact Difference Schemes ◽  
Compact Difference ◽  
The Fourier Transform ◽  
Convergence Orders

We propose two new compact difference schemes for numerical approximation of the Riemann-Liouville and Riesz derivatives, respectively. It is shown that these formulas have fourth-order convergence order by means of the Fourier transform method. Finally, some numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence orders.

scholarly journals NUMERICAL SOLUTION OF A CLASS OF NONLINEAR SYSTEM OF SECOND-ORDER BOUNDARY-VALUE PROBLEMS: A FOURTH-ORDER CUBIC SPLINE APPROACH

2015 ◽  
Vol 20 (5) ◽  
pp. 681-700 ◽  
Author(s):  
Suheil A. Khuri ◽  
Ali M. Sayfy
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Spline Collocation ◽  
Extended System ◽  
Collocation Approach ◽  
Linear And Nonlinear

A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions.

scholarly journals The boundary layer problem: A fourth-order adaptive collocation approach

2012 ◽  
Vol 64 (6) ◽  
pp. 2089-2099 ◽  
Author(s):  
S.A. Khuri ◽  
A. Sayfy
Keyword(s):  
Boundary Layer ◽  
Fourth Order ◽  
Boundary Layer Problem ◽  
Collocation Approach ◽  
Layer Problem

scholarly journals Implicit Two-Point Block Method for Solving Fourth-Order Initial Value Problem Directly with Application

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Reem Allogmany ◽  
Fudziah Ismail ◽  
Zanariah Abdul Majid ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Approximation ◽  
Stability And Convergence ◽  
Block Method ◽  
Initial Value ◽  
Order Zero ◽  
First Derivative ◽  
Hermite Interpolating Polynomial

This paper proposes an implicit block method with two-point to directly solve the fourth-order Initial Value Problems (IVPs). The implicit block method is derived by adopting Hermite interpolating polynomial as the basis function, incorporating the first derivative of ft,y,y′,y′′,y′′′ to enhance the solution’s accuracy. A block formulation is presented to acquire the numerical approximation at two points simultaneously. The introduced method’s basic properties, including order, zero stability, and convergence, are presented. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed method compared with those of the several existing methods. Application in ship dynamics is also presented which yield impressive results for the proposed two-point block method.

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