G. Adomian

Solving frontier problems of physics the decomposition method

An electric watt-hour meter socket which is provided with means for by-passing the meter-socket contact prior to removal of the meter, thus enabling current to be supplied when necessary, and when the interruption of the current supply by removal of the meter, would otherwise occur. Also, provision is made to loosen the socket contacts when the current by-pass is activated, thus facilitating disengagement of the meter contact and the removal of the meter, and to tighten the socket contacts when the meter is restored and the by-pass deactivated. The invention resides in the simplicity of the means provided for accomplishing these operations.

A review of the decomposition method in applied mathematics

Abstract The decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions, closure approximations, perturbation theory, or restrictive assumptions on stochasticitiy.

Nonlinear stochastic systems theory and applications to physics

I: A Summary of the Decomposition Method.- 1: The Decomposition Method.- 1.1 Introduction.- 1.2 Summary of the Decomposition Method.- 1.3 Generation of the An Polynomials.- 1.4 The An for Differential Nonlinear Operators.- 1.5 Convenient Computational Forms for the An Polynomials.- 1.6 Calculation of the An Polynomials for Composite Nonlinearities.- 1.7 New Generating Schemes - the Accelerated Polynomials.- 1.8 Convergence of the An Polynomials.- 1.9 Eulers Transformation.- 1.9.1 Solution of a Differential Equation by Decomposition.- 1.9.2 Application of Euler Transform to Decomposition Solution.- 1.9.3 Numerical Comparison.- 1.9.4 Solution of Linearized Equation.- 1.10 On the Validity of the Decomposition Solution.- 2: Effects of Nonlinearity and Linearization.- 2.1 Introduction.- 2.2 Effects on Simple Systems.- 2.3 Effects on SOlution for the General Case.- 3: Research on Initial and Boundary Conditions for Differential and Partial Differential Equations.- II: Applications to the Equations of Physics.- 4: The Burgers Equation.- 5: Heat Flow and Diffusion.- 5.1 One-Dimensional Case.- 5.2 Two-Dimensional Case.- 5.3 Three-Dimensional Case.- 5.4 Some Examples.- 5.5 Heat Conduction in an Inhomogeneous Rod.- 5.6 Nonlinear Heat Conduction.- 5.7 Heat Conduction Equation with Discontinuous Coefficients.- 5.8 Nonlinear Boundary Conditions.- 5.9 Comparisons.- 5.10 Uncoupled Equations with Coupled Conditions.- 6: Nonlinear Oscillations in Physical Systems.- 6.1 Oscillatory Motion.- 6.2 Pendulum Problem.- 6.3 The Duffing and Van der Pol Oscillators.- 7: The KdV Equation.- 8: The Benjamin-Ono Equation.- 9: The Sine-Gordon Equation.- 10: The Nonlinear Schrodinger Equation and the Generalized Schrodinger Equation.- 10.1 Nonlinear Schrodinger Equation.- 10.2 Generalized Schrodinger Equation.- 10.3 Schrodingers Equation with a Quartic Potential.- 11: Nonlinear Plasmas.- 12: The Tricomi Problem.- 13: The Initial-Value Problem for the Wave Equation.- ChaDter 14: Nonlinear Dispersive or Dissipative Waves.- 14.1 Wave Propagation in Nonlinear Media.- 14.2 Dissipative Wave Equations.- 15: The Nonlinear Klein-Gordon Equation.- 16: Analysis of Model Equations of Gas Dynamics.- 17: A New Approach to the Efinger Model for a Nonlinear Quantum Theory for Gravitating Particles.- 18: The Navier-Stokes Equations.- Epilogue.

A review of the decomposition method and some recent results for nonlinear equations

Abstract The decomposition method can be an effective procedure for solution of nonlinear and/or stochastic continous-time dynamical systems without usual restrictive assumptions. This paper is intended as a convenient tutorial review of the method. 1

A new approach to nonlinear partial differential equations

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

Inversion of nonlinear stochastic operators

Abstract The operator-theoretic method (Adomian and Malakian, J. Math. Anal. Appl. 76(1), (1980), 183–201) recently extended Adomians solutions of nonlinear stochastic differential equations (G. Adomian, Stochastic Systems Analysis, in “Applied Stochastic Processes,” Nonlinear Stochastic Differential Equations, J. Math. Anal. Appl. 55(1) (1976), 441–452; On the modeling and analysis of nonlinear stochastic systems, in “Proceeding, International Conf. on Mathematical Modeling.” Vol. 1, pp. 29–40) to provide an efficient computational procedure for differential equations containing polynomial, exponential, and trigonometric nonlinear terms N(y). The procedure depends on the calculation of certain quantities An and Bn. This paper generalizes the calculation of the An and Bn to much wider classes of nonlinearities of the form N(y, y′,…). Essentially, the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance. This procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities An and Bn for any order n in a convenient form. This paper also demonstrates the significance of the iterative series decomposition proposed by Adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators. Since both the nonlinear and stochastic behavior is quite general, the results are extremely significant for applications. Processes need not, for example, be limited to either Gaussian processes, white noise, or small fluctuations.

On the solution of algebraic equations by the decomposition method

Abstract The decomposition method ( G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983 ) developed to solve nonlinear stochastic differential equations has recently been generalized to nonlinear (and/or) stochastic partial differential equations, systems of equations, and delay equations and applied to diverse applications. As pointed out previously (see reference above) the methodology is an operator method which can be used for nondifferential operators as well. Extension has also been made to algebraic equations involving real or complex coefficients. This paper deals specifically with quadratic, cubic, and general higher-order polynomial equations and negative, or nonintegral powers, and random algebraic equations. Further work on this general subject appears elsewhere (G. Adomian, “Stochastic Systems II,” Academic Press, New York, in press).

Solution of physical problems by decomposition

Recent generalizations are discussed and results are presented for the theory and applications of the decomposition method. Application is made to the Duffing equation with an error of 0.0001% in only four terms and less than 10−16 in thirteen terms of the decomposition series. Application is also made to a dissipative wave equation, a matrix Riccati equation, and advection-diffusion nonlinear transport.

Convergent series solution of nonlinear equations

Abstract The authors decomposition method [1] provides a new, efficient computational procedure for solving large classes of nonlinear (and/or stochastic) equations. These include differential equations containing polynomial, exponential, and trigonometric terms, negative or irrational powers, and product nonlinearities [2]. Also included are partial differential equations [3], delay-differential equations [4], algebraic equations [5], and matrix equations [6] which describe physical systems. Essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance. The procedure depends on calculation of the authors A n , a finite set of polynomials [1,13] in terms of which the nonlinearities can be expressed. This paper shows important properties of the A n which ensure an accurate and computable convergent solution by the authors decomposition method [1]. Since the nonlinearities and/or stochasticity which can be handled are quite general, the results are potentially extremely useful for applications and make a number of common approximations such as linearization, unnecessary.

Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations

Abstract We consider the solution of partial differential equations for initial/boundary conditions using the decomposition method. The partial solutions obtained from the seperate equations for the highest-ordered linear operator terms are shown to be identical when the boundary conditions are general, and asymptotically equal when the boundary conditions in one independent variable are independet of other variables.