Zehra Pinar

Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology

In this work we consider nonlinear reaction-diffusion equations arising in mathematical biology. We use the exp-function method in order to obtain conventional solitons and periodic solutions. The proposed scheme can be applied to a wide class of nonlinear equations.

A note on the fractional hyperbolic differential and difference equations

Abstract The initial boundary value problem for the fractional differential equation. d 2 u ( t ) dt 2 + D t 1 2 u ( t ) + Au ( t ) = f ( t ) , 0 t 1 , u ( 0 ) = 0 , u ′ ( 0 ) = ψ , in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem and its first and second order derivatives are established. The first order of accuracy difference scheme for the approximate solution of this problem is presented. The stability estimates for the solution of this difference scheme and its first and second order difference derivatives are established. In practice, the stability estimates for the solution of difference schemes for one dimensional fractional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional fractional hyperbolic equation with Dirichlet condition in space variables are obtained.

On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations

The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation is presented. Stability estimates for the solution of this difference scheme and for the first and second orders difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

The Periodic Solutions to Kawahara Equation by Means of the Auxiliary Equation with a Sixth-Degree Nonlinear Term

It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the Kawahara equation. By the aid of the solutions of the original auxiliary equation, some other physically important nonlinear equations can be solved to construct novel exact solutions.

Solutions of Modified Equal Width Equation by Means of the Auxiliary Equation with a Sixth-Degree Nonlinear Term

In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with sixth-degree nonlinear term are presented to obtain novel exact solutions of the modified equal width equation. By the aid of the solutions of the original auxiliary equation; some other physically important nonlinear equations can be solved to construct novel exact solutions.

An Improved Analytical Solution of Population Balance Equation Involving Aggregation and Breakage via Fibonacci and Lucas Approximation Method

Abstract This study presents a novel analytical solution for the Population Balance Equation (PBE) involving particulate aggregation and breakage by making use of the appropriate solution(s) of the associated complementary equation of a nonlinear PBE via Fibonacci and Lucas Approximation Method (FLAM). In a previously related study, travelling wave solutions of the complementary equation of the PBE using Auxiliary Equation Method (AEM) with sixth order nonlinearity was taken to be analogous to the description of the dynamic behavior of the particulate processes. However, in this study, the class of auxiliary equations is extended to Fibonacci and Lucas type equations with given transformations to solve the PBE. As a proof-of-concept for the novel approach, the general case when the number of particles varies with respect to time is chosen. Three cases i. e. balanced aggregation and breakage and when either aggregation or breakage can dominate are selected and solved for their corresponding analytical solution and compared with the available analytical approaches. The solution obtained using FLAM is found to be closer to the exact solution and requiring lesser parameters compared to the AEM and thereby being a more robust and reliable framework.

Population Balances Involving Aggregation and Breakage Through Homotopy Approaches

Abstract Homotopy techniques in nonlinear problems are getting increasingly popular in engineering practice. The main reason is because the homotopy method deforms continuously a difficult problem under study into a simple problem, which then can be easy to solve. This study explores several homotopy approaches to obtain semi- or approximate analytical solutions for various cases involving mechanistic phenomena such as aggregation and breakage. The well-established approximate analytical methods namely, the Homotopy Perturbation Method (HPM), the Homotopy Analysis Method (HAM), and the more recent forms of homotopy approaches such as the Optimal Homotopy Asymptotic Method (OHAM) and the Homotopy Analysis Transform Method (HATM) have been used to solve using a general mathematical framework based on population balances. In this study, several test cases have been discussed such as conditions in which the aggregation kernel is not only constant, but also sum or product dependent. Furthermore cases involving pure breakage, pure aggregation and a combined aggregation-breakage have been studied to understand the sensitivity of these homotopy-based methods in solving PBM. In all these cases, the solutions have been analytically studied and compared with literature. Using symbolic computation and carefully chosen perturbation parameters, the approximate analytical solutions are compared with each other and with the available analytical solution. A convergence analysis of the solution methods is made in comparison to the available solution. The case studies indicate that OHAM performs slightly better than both HATM and HPM in solving nonlinear equations such as the PBEs.

Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities

The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time– space modulated nonlinearities of coupled Schrodinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.

The Periodic Solutions of the Model of Drop Formation on a Coated Vertical Fibre

Nonlinear differential equations and its systems are used to describe various processes in physics, biology, economics etc. There are a lot of methods to look for exact solutions of nonlinear differential equations: the inverse scattering transform, Hirota method, the Backlund transform, the truncated Painlev’e expansion. Here, we present a well known auxiliary equation method that produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with sixth-degree nonlinear term are presented to obtain novel exact solutions of the leading-order evolution equation which is the model of drop formation on a coated vertical fibre.

The Exact Solutions to Analytical Model of Tsunami Generation by Sub-Marine Landslides

Nonlinear differential equations and its systems are used to describe various processes in physics, biology, economics etc. There are a lot of methods to look for exact solutions of nonlinear differential equations: the inverse scattering transform, Hirota method, the Backlund transform, the truncated Painleve expansion. It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the analytical model of Tsunami generation by sub-marine landslides.