Exact solutions for the ZK-MEW equation by using the tanh and sine–cosine methods

The tanh and sine–cosine methods are used to handle the two-dimensional ZK-modified equal-width equation (ZK-MEW). The two methods work well to obtain exact solutions of different physical structures; solitary wave solutions and periodic solutions are also obtained. The framework presented here reveals a number of useful features of the methods applied.     

The evolution of periodic waves of the coupled nonlinear Schrödinger equations

Systems of coupled nonlinear Schrödinger equations (CNLS) arise in several branches of physics, e.g., hydrodynamics and nonlinear optics. The Hopscotch method is applied to solve CNLS numerically. The algorithm is basically a finite difference method but with a special procedure for marching forward in time. The accuracy of the scheme is ensured as the system is proved to satisfy certain conserved quantities. Physically, the goal is to study the effects of an initial phase difference on the evolution of periodic, plane waves. The outcome will depend on the precise nature of the cubic nonlinearity, or in physical terms, the nature of polarization in optical applications.     

Numerical solution of linear Fredholm integral equations using sine–cosine wavelets

If we divide the interval [0,1] into N sub-intervals, then sine–cosine wavelets on each sub-interval can approximate any function. This ability helps us to obtain a more accurate approximation of piecewise continuous functions, and, hence, we can obtain more accurate solutions of integral equations. In this article we use a combination of sine–cosine wavelets on the interval [0,1] to solve linear integral equations. We convert the integral equation into a system of linear equations. Numerical examples are given to demonstrate the applicability of the proposed method.     

A Numerical method for computing the periodic solutions of the ring-spinning balloon equations ∗

The ring-spinning process is the last phase of the manufacturing process of textile yarn, where a loop of yarn rotates rapidly about a fixed axis and twisting occurs while the yarn is wound onto the bobbin. The surface generated by the rotating loop of the yarn is called a balloon. In this paper a numerical method is developed for the calculation of periodic solutions of a non-stationary mathematical model of the ring-spinning process recently proposed in the technical literature. Some results are given, from which the method seems to be very efficient and reliable in the simulation of realistic process.     

The numerical solution of the nonlinear Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method

In this work, we study the numerical solutions of one-dimensional Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method based on the integration–differentiation (CTMMID). First, we apply CTMMID to discretize both space and time variables. The initial and boundary conditions could be incorporated efficiently with full CTMMID. Furthermore, we introduce the Domain Decomposition Method (DDM) in space and the block-marching technique in time for problems defined in large interval and long time computing. The numerical results are more accurate and with less computational effort than some existing studies.     

High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations

In this paper, we develop a new kind of multisymplectic integrator for the coupled nonlinear Schrödinger (CNLS) equations. The CNLS equations are cast into multisymplectic formulation. Then it is split into a linear multisymplectic formulation and a nonlinear Hamiltonian system. The space of the linear subproblem is approximated by a high-order compact (HOC) method which is new in multisymplectic context. The nonlinear subproblem is integrated exactly. For splitting and approximation, we utilize an HOC–SMS integrator. Its stability and conservation laws are investigated in theory. Numerical results are presented to demonstrate the accuracy, conservation laws, and to simulate various solitons as well, for the HOC–SMS integrator. They are consistent with our theoretical analysis.     

Solving nonlinear Volterra–Fredholm integro-differential equations using He's variational iteration method

In this paper, a nonlinear Volterra–Fredholm integro-differential equation is solved by using He's variational iteration method. The approximate solution of this equation is calculated in the form of a sequence where its components are computed easily. The accuracy of the proposed numerical scheme is examined by comparing with the modified Adomian decomposition method. The existence and uniqueness of the solution and convergence of the proposed method are proved.     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.     

Numerical solutions of the Burgers–Huxley equation by the IDQ method

In this paper, the Burgers–Huxley equation has been solved by a generalized version of the Iterative Differential Quadrature (IDQ) method for the first time. The IDQ method is a method based on the quadrature rules. It has been proposed by the author applying to a certain class of non-linear problems. Stability and error analysis are performed, showing the efficiency of the method. Besides, an error bound is tried. In the discussion about the numerical examples, the generalized Burgers–Huxley equation is involved too.     

Special types of the nonlinear dispersive Zakharov–Kuznetsov equation with compactons,solitons, and periodic solutions

In this article, we study nonlinear dispersive special types of the Zakharov–Kuznetsov equation with positive and negative exponents. The approach depends mainly on the sine–cosine algorithm. Compactons, solitary patterns, solitons, and periodic solutions are formally derived.     

Stabilization of geometrically nonlinear topology optimization by the Levenberg–Marquardt method

This paper deals with the design of compliant mechanisms in a continuum-based finite-element representation. Because the displacements of mechanisms are intrinsically large, the geometric nonlinearity is essential for designing such mechanisms. However, the consideration of the geometric nonlinearity may cause some instability in topology optimization. The problem is in the analysis part but not in the optimization part. To alleviate the analysis problem and eventually stabilize the optimization process, this paper proposes to apply the Levenberg–Marquardt method to the nonlinear analysis of compliant mechanisms.     

An analytical approach to the Fornberg–Whitham type equations by using the variational iteration method

In this paper, He’s variational iteration method (VIM) is applied to solve the Fornberg–Whitham type equations. The VIM provides fast converged approximate solutions to nonlinear equations without linearization, discretization, perturbation, or the Adomian polynomials. This makes the method attractive and reliable for solving the Fornberg–Whitham type equations. Numerical examples related to two initial value problems are presented to show the efficiency of the VIM.     

Solitary-wave solutions of the Degasperis–Procesi equation by means of the homotopy analysis method

The homotopy analysis method (HAM) is applied to the Degasperis–Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the HAM is a powerful tool for finding excellent approximations to nonlinear solitary waves.     

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

The hyperbolic Lindstedt-Poincaré method is applied to determine the homoclinic and heteroclinic solutions of cubic strongly nonlinear oscillators of the form x + c1 x + c3 x 3= ε f (μ,x,x).In the method,the hyperbolic functions are employed instead of the periodic functions in the Lindstedt-Poincaré procedure.Critical value of parameter μ under which there exists homoclinic or heteroclinic orbit can be determined by the perturbation procedure.Typical applications are studied in detail.To illustrate the acc...     

Numerical solution of a coupled modified Korteweg–de Vries equation by the Galerkin method using quadratic B-splines

In this paper, numerical solutions of a coupled modified Korteweg–de Vries equation have been obtained by the quadratic B-spline Galerkin finite element method. The accuracy of the method has been demonstrated by five test problems. The obtained numerical results are found to be in good agreement with the exact solutions. A Fourier stability analysis of the method is also investigated.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Sinc Nyström method

In this paper, a double-exponential (DE) Sinc Nyström method is utilized to solve nonlinear two-dimensional Fredholm integral equations of the second kind. Using the DE transformation, the Sinc quadrature rule for a definite integral is extended to double integral over a rectangular region. Therefore, a nonlinear Fredholm integral equation is reduced to a system of nonlinear algebraic equations, which is solved using the Newton iteration method. Convergence analysis shows that the proposed method can converge exponentially. Several numerical examples are provided to demonstrate the high efficiency and accuracy of the proposed method.     

Comparing the Adomian decomposition method and the Runge–Kutta method for solutions of the Stefan problem

The solution of the one-phase Stefan problem is presented. A Stefan's task is first approximated with a system of ordinary differential equations. A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving this system is then presented.