The method of finding solutions of partial dynamic equations on time scales
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RESEARCH
Open Access
The method of finding solutions of partial dynamic equations on time scales Hsuan-Ku Liu* *
Correspondence: [email protected] Department of Mathematics and Information Education, National Taipei University of Education, Taipei, Taiwan
Abstract On time scales, one area lacking of development is the method of finding solutions on partial dynamic equations. This paper proposes a method for finding the exact solution of linear partial dynamic equations on arbitrage time scales. We modify the variational iteration method on R to find an approximation of the nonlinear partial dynamic equation on qN . As an example, the modified variational iteration method is applied to q-Berger equations and to q-Fisher equations. Their numerical results reveal that the proposed method is very effective. Keywords: partial dynamic equations on time scales; nonlinear q-difference equation; variational iterative method; approximate solutions
1 Introduction A time scale is a nonempty closed subset of real numbers. On time scale calculus, notations and theorems have been well established for the univariate case []. Solutions of ordinary differential equations, such as initial value problems and boundary value problems, have been studied and published during the past two decades on time scales. In recent years, Hoffacker [] and Ahlbrandt and Morian [] demonstrated the related ideas to the multivariate case and studied partial dynamic equations on time scales. Notations and definitions on multivariate time scales calculus can be found in Bohner and Guseinov [, ]. Jackson [] extended the existing ideas of the time scales calculus [] to the multivariate case. The method of generalized Laplace transform on time scales is applied to find solutions of the homogeneous and nonhomogeneous heat and wave equations. Recent developments in the method of finding solutions have aroused further interest in the discussion of partial dynamic equations on time scales. For the nonlinear cases, methods of finding solutions are not mentioned for partial dynamic equations on time scales. One of the difficulties for developing a theory of series solutions to nonlinear equations on time scales is that the formula for multiplications of two generalized polynomials is not easily found. If a time scale has constant graininess, Haile and Hall [] provided an exact formula for the multiplication of two generalized polynomials. Using the obtained results, the series solutions for linear dynamic equations are proposed on the time scales R and T = hZ (difference equations with step size h). On generalized time scales, Mozyrska and Pawtuszewicz [] presented a formula for the multiplication of generalized polynomials of degree one and degree n ∈ N. Liu [] provided a product rule of two generalized polynomials on the time scale qZ = {qn | n ∈ N} ∪ {}. © 2013 Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted us
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