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Register a new userThe BLUES function method applied to partial differential equations and analytic approximants for interface growth under shear
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An iteration sequence based on the BLUES (beyond linear use of equation superposition) function method is presented for calculating analytic approximants to solutions of nonlinear partial differential equations. This extends previous work using this method for nonlinear ordinary differential equations with an external source term. Now, the initial condition plays the role of the source. The method is tested on three examples: a reaction-diffusion-convection equation, the porous medium equation with growth or decay, and the nonlinear Black-Scholes equation. A comparison is made with three other methods: the Adomian decomposition method (ADM), the variational iteration method (VIM), and the variational iteration method with Green function (GVIM). As a physical application, a deterministic differential equation is proposed for interface growth under shear, combining Burgers and Kardar- Parisi-Zhang nonlinearities. Thermal noise is neglected. This model is studied with Gaussian and space-periodic initial conditions. A detailed Fourier analysis is performed and the analytic coefficients are compared with those of ADM, VIM, GVIM, and standard perturbation theory. The BLUES method turns out to be a worthwhile alternative to the other methods. The advantages that it offers ensue from the freedom of choosing judiciously the linear part, with associated Green function, and the residual containing the nonlinear part of the differential operator at hand.
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The iteration sequence based on the BLUES (Beyond Linear Use of Equation Superposition) function method for calculating analytic approximants to solutions of nonlinear ordinary differential equations with sources is elaborated upon. Diverse problems in physics are studied and approximate analytic solutions are found. We first treat a damped driven nonlinear oscillator and show that the method can correctly reproduce oscillatory behaviour. Next, a fractional differential equation describing heat transfer in a semi-infinite rod with Stefan-Boltzmann cooling is handled. In this case, a detailed comparison is made with the Adomian decomposition method, the outcome of which is favourable for the BLUES method. As a final problem, the Fisher equation from population biology is dealt with. For all cases, it is shown that the solutions converge exponentially fast to the numerically exact solution, either globally or, for the Fisher problem, locally.
We introduce a computational method in physics that goes beyond linear use of equation superposition (BLUES). A BLUES function is defined as a solution of a nonlinear differential equation (DE) with a delta source that is at the same time a Greens function for a related linear DE. For an arbitrary source, the BLUES function can be used to construct an exact solution to the nonlinear DE with a different, but related source. Alternatively, the BLUES function can be used to construct an approximate piecewise analytical solution to the nonlinear DE with an arbitrary source. For this alternative use the related linear DE need not be known. The method is illustrated in a few examples using analytical calculations and numerical computations. Areas for further applications are suggested.
A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.
The method of characteristics has played a very important role in mathematical physics. Preciously, it was used to solve the initial value problem for partial differential equations of first order. In this paper, we propose a fractional method of characteristics and use it to solve some fractional partial differential equations.
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