On the Cauchy problem for some abstract nonlinear differential equations

In the present paper, we study the Cauchy problem in a Banach spaceE for an abstract nonlinear differential equation of form $$\frac{{d^2 u}}{{dt^2 }} = - A\frac{{du}}{{dt}} + B(t)u + f(t,W)$$ whereW = (A ₁(t)u,A ₂(t)u,?,A _?(t)u), (A _i (t),i = 1, 2, ?,?), (B(t),t ∈I = [0,b]) are families of closed operators defined on dense sets inE intoE, f is a given abstract nonlinear function onI ×E ^? intoE and ?A is a closed linear operator defined on dense set inE intoE, which generates a semi-group. Further, the existence and uniqueness of the solution of the considered Cauchy problem is studied for a wide class of the families (A _i(t),i = 1, 2, ?,?), (B(t),t ∈I). An application and some properties are also given for the theory of partial diferential equations.     

Line method approximations to the Cauchy problem for nonlinear parabolic differential equations

Summary The Cauchy problemu _t =f(x, t, u, u _x , u _xx ),u(x, o)=(x),xR, is treated with the longitudinal method of lines. Existence, uniqueness, monotonicity and convergence properties of the line method approximations are investigated under the classical assumption that satisfies an inequality |(x)|<=conste ^{Bx 2} . We obtain generalizations of the works of Kamynin [4], who got similar results in the case of the one dimensional heat equation when is allowed to grow likee ^{Bx 2–}, >0, and of Walter [11], who proved convergence in the case of nonlinear parabolic differential equations under the growth condition |(x)|<=conste ^B |x|     

On continuous approximate solutions of nonlinear differential equations

Two recently developed methods for continuous approximate solution of equations involving differential operators—the Adomian (decomposition) method and the Sarafyan method—are compared.     

On the approximate solution of the singular problem of Cauchy for ordinary differential equations

In the present paper an approximate solution of the singular problem of Cauchy for the ordinary differential equation of mth order is constructed and, by the method of finite differences, sufficient conditions are found for the convergence to the exact solution when the mesh width tends to zero.     

On acyclicity of the set of solutions to the Cauchy problem for differential equations

The acyclicity of sets of solutions to the Cauchy problem is considered from the viewpoint of the axiomatic theory of solutions spaces of ordinary differential equations. We prove that the class of acyclic solution spaces is closed with respect to passing to the limit space. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 836–842, December, 1997 Translated by M. A. Shishkova     

Weighted Cauchy problem for nonlinear singular differential equations with deviating arguments

For higher-order nonlinear differential equations with deviating arguments and with nonintegrable singularities with respect to the time variable, we establish sharp sufficient conditions for the Cauchy problem to be solvable and well-posed.     

Verifying approximate solutions to differential equations

It is now standard practice in computational science for large-scale simulations to be implemented and investigated in a problem solving environment (PSE) such as MATLAB or MAPLE. In such an environment, a scientist or engineer will formulate a mathematical model, approximate its solution using an appropriate numerical method, visualize the approximate solution and verify (or validate) the quality of the approximate solution. Traditionally, we have been most concerned with the development of effective numerical software for generating the approximate solution and several efficient and reliable numerical libraries are now available for use within the most widely used PSEs. On the other hand, the visualization and verification tasks have received little attention, even though each often requires as much computational effort as is involved in generating the approximate solution.In this paper, we will investigate the effectiveness of a suite of tools that we have recently introduced in the MATLAB PSE to verify approximate solutions of ordinary differential equations. We will use the notion of ‘effectivity index’, widely used by researchers in the adaptive mesh PDE community, to quantify the credibility of our verification tools. Numerical examples will be presented to illustrate the effectiveness of these tools when applied to a standard numerical method on two model test problems.     

Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

The paper considers the Cauchy problem for linear partial differential equations of non-Kowalevskian type in the complex domain. It is shown that if the Cauchy data are entire functions of a suitable order, the problem has a formal solution which is multisummable. The precise bound of the admissible order of entire functions is described in terms of the Newton polygon of the equation.     

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

This paper is concerned with the Cauchy problem for the nonlinear parabolic equation $${\partial _t}u| = \vartriangle u + F(x,t,u,\nabla u){\text{ in }}{{\text{R}}^N} \times (0,\infty ),{\text{ }}u(x,0) = \varphi (x){\text{ in }}{{\text{R}}^N},$$ , where $$\begin{gathered} N \geqslant 1, \hfill \\ F \in C(R^N \times (0,\infty ) \times R \times R^N ), \hfill \\ \phi \in L^\infty (R^N ) \cap L^1 (R^N ,(1 + |x|^K )dx)forsomeK \geqslant 0 \hfill \\ \end{gathered} $$ . We give a sufficient condition for the solution to behave like a multiple of the Gauss kernel as t → ∞ and obtain the higher order asymptotic expansions of the solution in W ^1,q (R ^N ) with 1 ≤ q ≤ ∞.     

The Cauchy problem for singularly perturbed weakly nonlinear second-order differential equations: An iterative method

For a singularly perturbed weakly nonlinear second-order differential equation, we construct a sequence converging to the Cauchy problem solution. This is an asymptotical sequence because the deviation (in the sense of the norm of the space of continuous functions) of its nth element from the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter. Such a sequence can be used to justify the asymptotics obtained by using boundary functions.     

Uniqueness and stability of solutions for Cauchy problem of nonlinear diffusion equations

The uniqueness of solutions for Cauchy problem of the form $$\frac{{\partial u}}{{\partial t}} = \Delta A(u) + \sum\limits_{i = 1}^N {\frac{{\partial b^i (u)}}{{\partial x_i }} + c(u)} $$ is studied. It is proved that ifu ∈BVx and A(u) is strictly increasing, the solution is unique.     

The Cauchy problem for nonlinear elliptic equations

This paper is devoted to investigation of the Cauchy problem for nonlinear equations with a small parameter. They are actually small perturbations of linear elliptic equations in which case the Cauchy problem is ill-posed. To study the Cauchy problem we invoke purely nonlinear methods, such as successive iterations and L^q$L^q$ Sobolev spaces with large q$q$. We also discuss linearisable problems.     

Analytical and approximate solutions to autonomous,nonlinear, third-order ordinary differential equations

Analytical solutions to autonomous, nonlinear, third-order nonlinear ordinary differential equations invariant under time and space reversals are first provided and illustrated graphically as functions of the coefficients that multiply the term linearly proportional to the velocity and nonlinear terms. These solutions are obtained by means of transformations and include periodic as well as non-periodic behavior. Then, five approximation methods are employed to determine approximate solutions to a nonlinear jerk equation which has an analytical periodic solution. Three of these approximate methods introduce a linear term proportional to the velocity and a book-keeping parameter and employ a Linstedt–Poincaré technique; one of these techniques provides accurate frequencies of oscillation for all the values of the initial velocity, another one only for large initial velocities, and the last one only for initial velocities close to unity. The fourth and fifth techniques are based on the Galerkin procedure and the well-known two-level Picard’s iterative procedure applied in a global manner, respectively, and provide iterative/sequential approximations to both the solution and the frequency of oscillation.     

On the Cauchy problem for higher-order nonlinear dispersive equations

We study the higher-order nonlinear dispersive equation

     

Cauchy problem for nonlinear p-evolution equations

The local solvability of the Cauchy problem in Sobolev spaces is studied for a class of nonlinear partial differential equations incorporating weakly hyperbolic and Schrödinger equations.