Fractional Differential Equations

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.

New exact solutions of Burgers’ type equations with conformable derivative

In this work, we aim to present a new fractional extension of regularized long-wave equation. The regularized long-wave equation is a very important mathematical model in physical sciences, which unfolds the nature of shallow water waves and ion acoustic plasma waves. The existence and uniqueness of the solution of the regularized long-wave equation associated with Atangana–Baleanu fractional derivative having Mittag-Leffler type kernel is verified by implementing the fixed-point theorem. The numerical results are derived with the help of an iterative algorithm. In order to show the effects of various parameters and variables on the displacement, the numerical results are presented in graphical and tabular form.

Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel

The key purpose of the present work is to constitute a numerical scheme based on q-homotopy analysis transform method to examine the fractional model of regularized long-wave equation. The regularized long-wave equation explains the shallow water waves and ion acoustic waves in plasma. The proposed technique is a mixture of q-homotopy analysis method, Laplace transform, and homotopy polynomials. The convergence analysis of the suggested scheme is verified. The scheme provides ℏ and n-curves, which show that the range convergence of series solution is not a local point effects and elucidate that it is superior to homotopy analysis method and other analytical approaches. Copyright © 2017 John Wiley & Sons, Ltd.

A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves

The purpose of this study is to find some novel soliton solutions of Fokas–Lenells (FL) equation where the perturbation terms are taken into account with nonlinearity. The sine-Gordon expansion method (SGEM) and the $$(m+({G'}/{G}))$$
 -expansion method are used in this context. The dark, bright, dark–bright and singular optical soliton solutions are successfully obtained. Moreover, the constraint conditions for guaranteeing the existence of solutions are also given.

Optical soliton solutions to the Fokas-Lenells equation via sine-Gordon expansion method and (m +(G ' /G ) ) -expansion method

In this paper, we investigate the numerical solutions of one dimensional modified Burgers’ equation with the help of Haar wavelet method. In the solution process, the time derivative is discretized by finite difference, the nonlinear term is linearized by a linearization technique and the spatial discretization is made by Haar wavelets. The proposed method has been tested by three test problems. The obtained numerical results are compared with the exact ones and those already exist in the literature. Also, the calculated numerical solutions are drawn graphically. Computer simulations show that the presented method is computationally cheap, fast, reliable and quite good even in the case of small number of grid points.

A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation

In this paper, we are going to investigate numerical solutions of the regularized long wave (RLW) equation by using Haar wavelet (HW), combined with finite difference method. The motion of a single solitary wave, interaction of two solitary waves, Maxwellian initial condition and wave undulation are our test problems for measuring performance of the proposed method. The results of computations are compared with exact solutions and those already published. ${L_{2}}$ and ${L_{\infty}}$ error norms and the numerical conservation laws are computed for discussing the accuracy and efficiency of the proposed method.

Numerical Solutions of Regularized Long Wave Equation By Haar Wavelet Method

In this study, we established a wavelet method, based on Haar wavelets and finite difference scheme for two-dimensional time fractional reaction–subdiffusion equation. First by a finite difference approach, time fractional derivative which is defined in Riemann–Liouville sense is discretized. After time discretization, spatial variables are expanded to truncated Haar wavelet series, by doing so a fully discrete scheme obtained whose solution gives wavelet coefficients in wavelet series. Using these wavelet coefficients approximate solution constructed consecutively. Feasibility and accuracy of the proposed method is shown on three test problems by measuring error in $$L_{\infty }$$
 norm. Further performance of the method is compared with other methods available in literature such as meshless-based methods and compact alternating direction implicit methods.

A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation

A computational method based on Hermite wavelets for two-dimensional Sobolev and regularized long wave equations in fluids

In this paper, we state a fuzzy initial value problem of the second order fuzzy differential equations. Here we investigate problems with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions. We propose an algorithm based on alpha-cut of a fuzzy set. Finally we present some examples by using our proposed algorithm.

Ömer Oruç

Papers

A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation

Numerical Solutions of Regularized Long Wave Equation By Haar Wavelet Method

A haar wavelet approximation for two-dimensional time fractional reaction–subdiffusion equation

A computational method based on Hermite wavelets for two-dimensional Sobolev and regularized long wave equations in fluids

An algorithm for the solution of second order fuzzy initial value problems