Construction Of Fractal Models And Analytical Solutions For Several Classes Of Local Fractional Order Nonlinear Partial Differential Equations

Recently,with regard to the latest progress and development direction of fractional calculus theory,the applications of local fractional derivatives and fractal derivatives in nanohydrodynamics and nanothermodynamics are particularly emphasized.Local fractional derivatives and calculus theory are introduced.This exists in fractal geometry,which is the best way to describe the non-differential functions defined on Cantor sets.Physical explanations of local fractional derivatives can be found in the relevant literature listed.In fractal domain,a large number of research results have been reported on the use of local fractional derivatives for non-differentiable phenomena.For example,the new analytical solutions of Klein-Gordon and Helmholtz equations in fractal space are discussed,and a new method for calculating nonlinear local fractional partial differential equations is proposed.In addition,the exact traveling wave solutions,separable local fractional differential equations,local fractional Korteweg-de Vries equation and local fractions of local fractional Boussinesq equation in fractal domain are reported.Second-order two-dimensional Burgers equation,fractal interpolation function and its fractional calculus,non-differentiable exact solutions of non-linear ordinary differential equations in fractal concentration,etc.Fractional partial differential equations can be divided into linear and non-linear ones.For example,many classical nonlinear partial differential equations were studied under smooth conditions in the past.In practice,under a large number of non-differentiable conditions,we must use local fractional derivatives to study fractal dimension and Cantor set.New models and fractal models are established,and the research is carried out by means of fractional complex transformation,various numerical methods and new methods.Chapter 1 introduces the background and significance of the research,the research status of the problem and the main work of this paper.Chapter 2: Analytical solutions of(1+1)and(n+1)dimensional nonlinear local fractional Harry-Dym equations.(1+1)dimensional and(n+1)dimensional nonlinear local fractional Harry-Dym equation(HDE)new fractal model was derived for the first time,with the help of local fractional derivative(LFD)and local fractionalsimplified differential transformation(LFRDTM).)The coupled fractional complex becomes the analytic approximate solution of the two new models.The fractional complex transformation of(n+1)dimensional variable functions is generalized,and the theorem of(n+1)dimensional LFRDTM is supplemented and generalized.The traveling wave solutions of fractal HDE show that the method is effective and simple for solving approximate solutions of nonlinear partial fractional differential equations.In Chapter 3,we propose a new method,called LFYLTDGJM,which couples local fractional Young Laplace transform with Daftardar-Gejji-Jafaris method.This method has been successfully applied to the analytical solution of the time fractional nonlinear modified Korteweg-de-vries(TFNMKDV)equation.The approximate solution shows that the new method is more efficient and accurate in solving local fractional order nonlinear partial differential equations.In chapter 4,six new fractal models of(2+1)dimension and(2n+1)dimension local fractional order non-linear biological population model(LFNBPM)on Cantor set are established.Analytical approximate solutions of these six models are obtained by coupling local fractional derivative and local fractional order simplified differential transformation(LFRDTM)with multidimensional fractional order complex transformation(MDFCT).The fractional complex transformation of(n+1)dimensional variable functions is generalized,and the theorem of(n+1)dimensional LFRDTM is supplemented and generalized.The analytical solution of fractal LFNBPM is obtained,which proves that the approximate solution of local fractional nonlinear partial differential equation is effective and simple.In the fifth chapter,two-dimensional and three-dimensional fractional order heat models with variable coefficients are solved.Fractional derivatives are described in Capto's sense.By using fractional power series method(FPSM),many analytical approximate solutions and exact solutions are obtained,including two-dimensional and three-dimensional fractional heat models with variable coefficients.The results show that the method used provides a very effective,convenient and powerful theoretical tool for solving many other fractional differential equations in mathematical physics.In Chapter 6,two-dimensional and three-dimensional fractional wave models with variable coefficients are solved.Fractional derivatives are described in Caputo sense.We obtain many analytical approximations and exact solutions for two-dimensional and three-dimensional fractional wave models with variablecoefficients.The results show that FPSM is an effective,convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics.Chapter 7,conclusions and prospects.