Solutions Of Second-order Nonlinear Equations Research Articles

New exact solutions for a generalised Burgers-Fisher equation

New travelling wave solutions for a generalised Burgers-Fisher (GBF) equation are obtained. They arise from the solutions of nonlinear second-order equations that can be linearised by a generalised Sundman transformation. The reconstruction problem involves a one-parameter family of first-order equations of Chini type. Firstly we obtain a unified expression of a one-parameter family of exact solutions, few of which have been reported in the recent literature by using hitherto not interrelated procedures, such as the tanh method, the modified tanh-coth method, the Exp-function method, the first integral method, or the improved (G′/G)−expansion method. Upon certain condition on the coefficients of the GBF equation, the procedure successes in finding all the possible travelling wave solutions, given through a single expression depending on two arbitrary parameters, and expressed in terms of the Lerch Transcendent function. Finally, the case n=1 is completely solved, classifying all the admitted travelling wave solutions into either a one-parameter family of exponential solutions, or into a two-parameter family of solutions that involve Bessel functions and modified Bessel functions. For particular subclasses of the GBF equation new families of solutions, depending on one or two arbitrary parameters and given in terms of the exponential, trigonometric, and hyperbolic functions, are also reported.

A generalized Cole–Hopf transformation for nonlinear ODEs

We introduce a ‘hybrid’ Cole–Hopf–Darboux transformation to relate the solutions of nonlinear and linear second-order differential equations and derive classification and sufficient condition for this correspondence. We explore physical applications of this correspondence to nonlinear oscillations, the Duffing equation and a nonlinear form of the Schrödinger equation for the nonrelativistic hydrogen atom. In addition, we show that solutions of some nonlinear second-order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the ‘class of special nonlinear equations’ which correspond to the linear differential equations which define the special functions of mathematical physics.

Global existence and boundedness for a class of second-order nonlinear differential equations

In this paper we obtain new conditions for the global existence and boundedness of solutions for nonlinear second-order equations of the form ( r ( t ) | u ′ | p − 2 u ′ ) ′ + g ( t , u , u ′ ) u ′ + a ( t ) f ( u ) = e ( t ) , where p > 1 is a real constant. The results are applicable to well-known Emden–Fowler and Lienard type equations. An illustrative example is also provided.

Behavior at infinity of solutions of second-order nonlinear equations of a particular class

Let Ω be an arbitrary, possibly unbounded, open subset of ℝn, and letL be an elliptic operator of the form $$L = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{\partial }{{\partial x_j }}} \right)} $$ . The behavior at infinity of the solutions of the equationLu=ƒ(¦u¦)signu in Ω is studied, whereƒ is a measurable function. In particular, given certain conditions at infinity, the uniqueness theorem for the solution of the first boundary value problem is proved.

Uniqueness of Positive Solutions of Nonlinear Second-Order Equations

Uniqueness of Positive Solutions of Nonlinear Second-Order Equations

Uniqueness of positive solutions of nonlinear second-order equations

In this paper we study the uniqueness question of positive solutions of the two-point boundary value problem: u ( t ) + f ( | t | , u ( t ) ) = 0 , − R > t > R , u ( ± R ) = 0 u(t) + f(|t|,u(t)) = 0, - R > t > R,u( \pm R) = 0 where R > 0 R > 0 is fixed and f : [ 0 , R ] × [ 0 , ∞ ) → R f:[0,R] \times [0,\infty ) \to \mathbb {R} is in C 1 ( [ 0 , R ] × [ 0 , ∞ ) ) {C^1}([0,R] \times [0,\infty )) . A uniqueness result is proved when f satisfies some appropriate conditions. Some examples illustrating our theorem are also given.

Periodic solutions of nonlinear second-order equations which are not solved for the highest derivative

Periodic solutions of nonlinear second-order equations which are not solved for the highest derivative

Rational approximations to the solution of the second order Riccati equation

Continued fraction rational approximations to solution of second-order nonlinear equation, including Ricatti equations treated by Merkes, Scott and Fair