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A representation formula for second-order nonhomogeneous nonlinear ordinary differential equations (ODEs) has been recently constructed by M. Frasca using its Green’s function, i.e. the solution of the corresponding nonlinear differential equation with a Dirac delta function instead of its nonhomogeneity. It has been shown that the first-order term–the convolution of the nonlinear Green’s function and the right-hand side, analogous to the Green’s representation formula for linear equations — provides a numerically efficient solution of the original equation, while the higher order terms add complementary corrections. The cases of square and sine nonlinearities have been studied by Frasca. Some new cases of explicit determination of nonlinear Green’s function have been studied previously by us. Here, we gather nonlinear equations and their explicitly determined Green’s functions from existing references, as well as investigate new nonlinearities leading to implicit determination of nonlinear Green’s function. Some transformations allowing to reduce second-order nonlinear partial differential equations (PDEs) to nonlinear ODEs are considered, meaning that Frasca’s method can be applied to second-order PDEs as well. We perform a numerical error analysis for a generalized Burgers’ equation and a nonlinear wave equation with a damping term in comparison with the method of lines.

The well-known Green’s function method has been recently generalized to nonlinear second-order differential equations. In this paper, we study possibilities of exact Green’s function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Green’s function can be represented in terms of the homogeneous solution. Specific examples and a numerical analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg–de Vries equations, we are forced to introduce higher order Green’s functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.

In this work, two numerical schemes were developed to overcome the problem of shock waves that appear in the solutions of one/two-layer shallow water models. The proposed numerical schemes were based on the method of lines and artificial viscosity concept. The robustness and efficiency of the proposed schemes are validated on many applications such as dam-break problem and the problem of interface propagation of two-layer shallow water model. The von Neumann stability of proposed schemes is studied and hence, the sufficient condition for stability is deduced. The results were presented graphically. The verification of the obtained results is achieved by comparing them with exact solutions or another numerical solutions founded in literature. The results are satisfactory and in much have a close agreement with existing results.

This paper introduces a model of growth and dispersion of the marine phytoplankton, focusing on the effects of the currents (3D) and vertical mixing. Our method consists of describing these effects as the product of the horizontal current, which is solved along the characteristic lines, and the coupled action of vertical current and vertical diffusion, restricted on each characteristic line of the horizontal current. We show that the trivial steady state loses its stability and a nontrivial (non-constant in space) steady state is created.

The high level of human morbidity caused by E. coli O157:H7 necessitates an improved understanding of the infection dynamics of this bacterium within the bovine reservoir. Until recently, a degree of uncertainty surrounded the issue of whether these bacteria colonize the bovine gut and as yet, only incomplete in-vivo datasets are available. Such data typically consist of bacterial counts from fecal samples. The development of a deterministic model, which has been devised to make good use of such data, is presented. A partial differential equation, which includes advection, diffusion and growth terms, is used to model the (unobserved) passage of bacteria through the bovine gut. A set of experimentally-obtained fecal count data is used to parameterize the model. Between-animal variability is found to be greater than between-strain variability, with some results adding further weight to the hypothesis that E. coli O157:H7 can colonize the bovine gastrointestinal tract.

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer [26] for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen and Toivanen [21]. The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

This paper is devoted to possibilities of a semi-analytical approximation of nonlinear wave equations with nonlinearities depending on the absolute value of the unknown function, which arise in different areas of physics and mechanics. The main difficulty of analysis of such equations is that the derivation of their rigorous solution is highly sophisticated, while their numerical solution requires burdensome computational costs. Using the traveling wave ansatz, we first reduce the wave equation to a nonlinear ordinary differential equation. Then, applying Frasca’s method, we construct its general solution in terms of the nonlinear Green’s function. For particular nonlinearities, it is shown that the first-order approximation of the Green’s function solution is numerically comparable with the solution obtained by the well-known numerical method of lines. The contribution of the higher order terms is studied for a particular nonlinearity.

We consider process of generation of difference frequency in a GaAs crystal with periodic domain structure during propagation of a few-cycle laser pulse in the crystal in the regime when chromatic dispersion is expressed weakly. Method of lines is used to obtain numerical solution to the system of coupled nonlinear differential equations in partial derivatives describing the evolution of the electric field of a few-cycle laser pulse in GaAs crystal both with periodic and aperiodic domain structure. Time-frequency distribution is obtained, with use of Wigner transformation, for linearly-polarized femtosecond laser pulse and for orthogonally polarized difference-frequency pulse generated via filtration of the orthogonally-polarized pulse at the crystal output.