A class of strongly nonlinear functional differential equations

Let H be a real Hilbert space, :H [0, + ] a proper l.s.c., convex function with L_k:={u H; u² + (u) k} compact for every k > 0, let > 0 be a given constant and . We prove an existence result for strong solutions to a class of functional differential equations of the form

On the existence of solutions for strongly nonlinear differential equations

The objectives of this paper are twofold. Firstly, to prove the existence of an approximate solution in the mean for some nonlinear differential equations, we also investigate the behavior of the class of solutions which may be associated with the differential equation. Secondly, we aim to implement the homotopy perturbation method (HPM) to find analytic solutions for strongly nonlinear differential equations.     

A one-step optimal homotopy analysis method for nonlinear differential equations

In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

An approach to deducing approximate solutions to the Cauchy problem for nonlinear differential equations

A new technique for the construction of numerical methods based on continued fractions is proposed. A characteristic feature of these algorithms is the fact that for certain values of the parameters it is possible to obtain both novel and traditional (explicit and implicit) numerical methods for the solution of the Cauchy problem for ordinary differential equations. Two-sided formulas are proposed by means of which it is possible to obtain on each integration step not only upper and lower approximations to the exact solution, but also information concerning the magnitude of the leading term of the error without the need for additional calculations of the right-hand side of the initial differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1695–1701, December, 1992.     

The optimal homotopy-analysis method for Kawahara equation

In this paper, the optimal homotopy-analysis method is used to find the travelling-wave solution of the Kawahara equation. The method used here contains three auxiliary convergence-control parameters, which provide us with a simple way to adjust and control the convergence region of the solution. By minimizing the averaged residual error, the optimal convergence-control parameters can be obtained, which give much better approximations than those given by usual homotopy-analysis method.     

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.     

Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

Limiting behavior for strongly damped nonlinear wave equations

In this paper we investigate both the existence and the limiting behavior for the equation $\text{u}{}_{\mathrm{tt}}\text{+ Au}{}_{\mathrm{t}}\text{+ Au = ?(t, u, u}{}_{\mathrm{t}}\text{)}$, where A is a sectorial operator, ? is periodic in t, and ? satisfies certain regularity and growth assumptions. In most results on limiting behavior we will assume A has compact resolvent. We consider the equation as an abstract ODE defined on a paired space X^β × X^α, 0 ? σ ? β < 1. With regard to the limiting behavior, one of our principal results will be to show that if there is a bounded set in one of the spaces considered, for which all points or trajectories enter into and remain, then there is a set J consisting of very “smooth” functions defined on all of the spaces considered, which is the maximum compact invariant set, uniformly asymptotically stable, connected, and having very strong attractivity properties in all these spaces. We will often show it attracts all points in a bounded set uniformly. We will give a few sharper results for the case where A = ?Δ. The work is motivated by recent papers of Webb and Fitzgibbon, and applies techniques found in recent papers by the author.     

An abstract approach to nonlinear boltzmann-type equations

Summary Using a general existence and uniqueness theory for linear time dependent kinetic equations, for general inhomogeneous multidimensional spatial and velocity domains and partially absorbing boundaries, we obtain local in time solutions of a class of nonlinear Boltzmann type equations. For small initial-boundary data we obtain global in time solutions. The ideal norm on certain ideals in the Banach space ofL _p-functions on phase space is used to measure the ?size? of initial-boundary data and solutions. Kaniel-Shinbrot type upper and lower approximation arguments are applied. The combined length of the time interval of existence when applying the method repeatedly is analyzed as a function of the size of the initial-boundary data. Specific applications to the nonlinear Boltzmann equation itself and to the plane Broadwell model are given. Research conducted under the auspices of C.N.R. (Consiglio Nazionale delle Ricerche), Gruppo Fisica-Matematica, and partially supported by M.P.I. (Ministero della Pubblica Intruzione). Research conducted as a visiting professor supported by C.N.R., Gruppo Fisica-Matematica. Permanente address: Dept. of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.     

A probabilistic approach to one class of nonlinear differential equations

Summary We establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes. As an application, we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.Partially supported by National Science Foundation Grant DMS-8802667