A new approach to nonlinear partial differential equations

In this paper, a novel method called variational iteration method is proposed to solve nonlinear partial differential equations without linearization or small perturbations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. An analytical solution can be obtained from its trial-function with possible unknown constants, which can be identified by imposing the boundary conditions, by successively iteration.     

A new approach to nonlinear partial differential equations

The author's decomposition method for the solution of operator equations which may be nonlinear and/or stochastic is generalized to multidimensional cases.     

Solution of nonlinear differential equation and special functions

In this paper, we find the approximate solution of a nonlinear differential equations pertaining to M-series, $\overline{H}$-function, and I-function of two variables by making use of the Hermite, Legendre, and Jacobi polynomials. The nonlinear differential equations play a key role in distinct branches of engineering and physics and many vibration problems. The results obtained in the present study are general in nature and can be used to obtain the solution of various problems of physics and engineering.     

Iterative solution of some nonlinear differential equations

We propose an iterative method to solve some non-linear ordinary differential equations. Comparing on the Mathieu, van der Pol and Hill equation of fourth order, we see that this method is much more efficient than the well known methods by Lyapunov or Picard.     

Stability of some nonlinear functional differential equations

The procedure to construct Liapunov functionals for some nonlinear functional differential equations (FDEs) is proposed. Stability conditions for some nonlinear FDEs are obtained.Work partially supported by the Italian M.U.R.S.T. National Project Problemi non lineari ...     

Discrete exact solutions of modified Volterra and Volterra lattice equations via the new discrete sine-Gordon expansion algorithm

Recently, we have presented a sine-Gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations. In this paper we further develop the method to be the discrete sine-Gordon expansion method in nonlinear differential-difference equations, in particular, discrete soliton equations. We choose the modified Volterra lattice and Volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained. Moreover some figures display the profiles of the obtained solutions. Our method can be also applied to other discrete soliton equations.     

A differential equation approach to nonlinear programming

A new method is presented for finding a local optimum of the equality constrained nonlinear programming problem. A nonlinear autonomous system is introduced as the base of the theory instead of usual approaches. The relation between critical points and local optima of the original optimization problem is proved. Asymptotic stability of the critical points is also proved. A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points. Some numerical results are given to show the efficiency of the method.     

A new approach to some nonlinear geometric equations in dimension two

We give new arguments for several Liouville type results related to the equation −Δ u = Ke^2u. The new approach is based on the holomorphic function associated with any solution, which plays a similar role as the Hopf differential for harmonic maps from a surface.     

Tension spline solution of nonlinear sine-Gordon equation

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders O(k²+k²h²+h²)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2}) and O(k²+k²h²+h⁴)\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4}). At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

Periodic solutions of some fourth-order nonlinear differential equations

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence of T-periodic solutions for fourth-order nonlinear differential equations. Two illustrative examples are provided to demonstrate that the results in this paper hold under weaker conditions than the existing ones, and are more effective.     

A class of new special solution of nonlinear diffusion equation

One of the most interesting problems of nonlinear differential equations is the construction of partial solutions. A new method is presented in this paper to seek special solutions of nonlinear diffusion equations. This method is based on seeking suitable function to satisfy Bernolli equation. Many new special solutions are obtained.     

A numerical method for some nonlinear differential equation models in biology

In this paper, the authors present a full discretization of nonlinear generalisations of the Fischer and Burgers equations with the zero flux on the boundary. Efficiency of the method is derived via a numerical comparison between their numerical solution and the exact solution.     

A nonlinear differential delay equation

A procedure reported elsewhere for solution of linear and nonlinear, deterministic or stochastic, delay differential equations developed by the authors as an extension of the first author's methods for nonlinear stochastic differential equations is now applied to a nonlinear delay-differential equation arising in population problems and studied by Kakutani and Markus. Examples involving time-dependent constants and even stochastic coefficients and delays can also be done.     

On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form $$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$ , where Φ is a coercive continuous operator from L _p to L _q , and $$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$ ; first- and second-order partial differential equations $$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$ ; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type. We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.     

Blow up oscillating solutions to some nonlinear fourth order differential equations

We give strong theoretical and numerical evidence that solutions to some nonlinear fourth order ordinary differential equations blow up in finite time with infinitely many wild oscillations. We exhibit an explicit example where this phenomenon occurs. We discuss possible applications to biharmonic partial differential equations and to the suspension bridges model. In particular, we give a possible new explanation of the collapse of bridges.     

The first integral method to some complex nonlinear partial differential equations

In this paper, the first integral method is used to construct exact solutions of the Hamiltonian amplitude equation and coupled Higgs field equation. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones.