Iterative methods for solving inverse problems for nonlinear partial differential equations

Inverse coefficient problems are considered for the mathematical models of sorption dynamics and heat conduction. Iterative methods proposed for solving these inverse problems transform a supplementary condition into an integral relationship containing the unknown coefficient. Combined with the original boundary-value problem, this integral relationship makes it possible to construct an iterative process. A priori representation of the unknown nonlinear coefficients in parametric form is not required. Results of computational experiments are reported.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 142–149.     

Three-step iterations for nonlinear accretive operator equations

In this paper, we suggest and analyze a three-step iterative scheme for solving nonlinear strongly accretive operator equation Tx=f without continuous condition in a uniformly smooth Banach space. Our results include the Ishikawa, Mann and Noor iterations as special cases. The results presented in this paper improve and extend almost all the current results in the more general setting.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.     

On certain nonlinear systems of partial differential equations of the second order

We consider nonlinear systems of five partial differential equations of the second order with one and two unknown functions. We obtain explicit conditions of compatibility which provide the unique solvability of problems with initial data.     

On solving certain differential equations with variable coefficients

We show how to solve certain types of linear ordinary differential equations with variable coefficients by using Appell polynomials.     

Some decomposition method for analytic solving of certain nonlinear partial differential equations in physics with applications

Certain nonlinear partial differential equations (NPDEs) can be decomposed into several more simple equations, which can possess enough general analytic solutions. This approach and some interesting kinds of solutions (obtained by using this method) of some NPDEs in physics will be presented. The presented approach is somewhat similar to the homogeneous balance method, however they are different.     

On a continuous method for solving nonlinear operator equations

We suggest a continuous method for solving nonlinear operator equations in Banach spaces. The proof of the convergence of the method is based on stability criteria for solutions of differential equations. The implementation of the method does not require the construction of inverse operators. Criteria for the global convergence are derived.     

One method for solving systems of nonlinear partial differential equations

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).     

Auxiliary equation method for solving nonlinear Wick-type partial differential equations

Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally.     

Modified extended tanh-function method for solving nonlinear partial differential equations

Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers–Fisher equation and Drinfeld–Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.     

Iterative methods for solving nonlinear irregular operator equations in banach space

We present an approach to constructing stable methods for solving nonlinear operator equations in Banach space without anyassumptions on the regularity of the operator. The approach is based on the linearization of the equation and the use of aregularization algorithm to find an approximate solution of the linearized equation at each iteration. The local convergence of proposed methods is proved and the estimations of the rate of convergence are established, provided that solution satisfies a sourcewise representation condition. The case of noisy data is also analysed.     

Implicit ad hoc methods for nonlinear partial differential equations

Several special methods including implicit separation of variables, explicit and implicit generalized traveling waves are introduced and employed to obtain solutions for nonlinear equations. Certain nonlinear wave propagation problems are shown to yield to implicit separation while generalized traveling wave concepts are applied in diffusion, fluid mechanics and wave propagation.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

A new method for solving nonlinear second order partial differential equations

In this paper, a new method for finding the approximate solution of a second order nonlinear partial differential equation is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a distributed parameter control system, the theory of measure is used for obtaining the approximate solution of the original problem.     

The method of approximate particular solutions for solving certain partial differential equations

A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 506–522, 2012