Jacobi多项式解变分数阶非线性微积分方程
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摘要
为求解一类变分数阶非线性微积分方程,提出了一种求解该类方程数值解的方法.该方法主要利用移位的Jacobi多项式将方程中的函数逼近,再结合Captuo类型的变分数阶微积分定义,推导出移位Jacobi多项式的微积分算子矩阵,将最初的方程转化为矩阵相乘的形式,然后通过离散变量,将原方程转化为一系列非线性方程组.通过解该非线性方程组得到移位Jacobi多项式的系数,进而可得原方程的数值解.最后,通过数值算例的精确解和数值解的绝对误差验证了该方法的高精度性和有效性.
In order to solve a class of variable fractional order nonlinear differential-integral equations,the numerical solution is proposed.Function approximation based on Shifted Jacobi polynomials,combining Caputo-type variable order fractional derivate definition,which are used to get the operational matrixes of Shifted Jacobi polynomials,are the main characteristic behind this method.With the operational matrix,the original equation is translated into the products of several dependent matrixes,which can be regarded as a system of nonlinear equations after dispersing the variable.By solving the nonlinear system of algebraic equations,the coefficients of Shifted Jacobi polynomials are got,then the numerical solutions of the original equation are acquired.Finally,some numerical examples illustrate the accuracy and effectiveness of the method.
In order to solve a class of variable fractional order nonlinear differential-integral equations,the numerical solution is proposed.Function approximation based on Shifted Jacobi polynomials,combining Caputo-type variable order fractional derivate definition,which are used to get the operational matrixes of Shifted Jacobi polynomials,are the main characteristic behind this method.With the operational matrix,the original equation is translated into the products of several dependent matrixes,which can be regarded as a system of nonlinear equations after dispersing the variable.By solving the nonlinear system of algebraic equations,the coefficients of Shifted Jacobi polynomials are got,then the numerical solutions of the original equation are acquired.Finally,some numerical examples illustrate the accuracy and effectiveness of the method.
引文
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[11]任建娅,尹建华,耿万海.小波方法求一类变系数分数阶微分方程数值解[J].辽宁工程技术大学学报(自然科学版),2012,31(6):925-928.REN Jianya,YIN Jianhua,GENG Wanhai.Wavelet method for numerical solution of a class of fractional differential equation with variable coefficients[J].Journal of Liaoning Technical University(Natural Science),2012,31(6):925-928.
[12]LIN R,LIU F,ANH V,et al.Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation[J].Applied Mathematics and Computation,2009(212):435-445.
[13]ZHUANG P,LIU F,ANH V,et al.Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term[J].SIAM Journal on Numerical Analysis,2009(47):1 760-1 781.
[14]COIMBRA C F M.Mechanics with variable-order differential operators[J].Annals of Physics,2003,12(11-12):692-703.
[15]KAZEM S.An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations[J].Applied Mathematical Modelling,2013(37):1 126-1 136.
[2]ROSSIKHIN Y A,SHITIKOVA M V.Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems[J].Acta Mechanica,1997(120):109-125.
[3]ABBASBANDY S.An Approximation solution of a nonlinear equation with riemann-liouville fractional derivatives by he’s variational iteration method[J].Journal of Computation and Applied Mathematics,2007(207):53-58.
[4]WANG Y X,FAN Q B,CUI Y H,et al.Wavelet method for a class of fractional convection-diffusion equation with variable coefficients[J].Journal of Computational Science,2010,1(3):146-149.
[5]Samko S G,Ross B.Intergation and differentiation to a variable fractional order[J].Integral Transforms and Special Functions,1993,1(4):277-300.
[6]RAMIREZ L E S,COIMBRA C F M.A variable-order constitutive relation for viscoelasticity[J].Annals of Physics,2007,16(7-8):543-552.
[7]SOON C M,COIMBRA C F M,KOBAYASHI M H.The variable viscoelasticity oscillator[J].Annals of Physics,2005,14(6):378-389.
[8]LIN R,LIU F,ANH V,et al.Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation[J].Applied Mathematics and Computation,2009(212):435-445.
[9]INGMAN D,SUZDALNITSKY J,ZEIFMAN M.Constitutive dynamicorder model for nonlinear contact phenomena[J].Journal of Applied Mechanies,2000(67):383-390.
[10]陈一鸣,孙慧,刘丽丽.Legendre小波在非线性分数阶微分方程中的应用[J].辽宁工程技术大学学报(自然科学版),2013,32(4):573-576.CHEN Yiming,SUN Hui,LIU Lili,et al.Application of Legendre wavelet in nonlinear fractional differential equations[J].Journal of Liaoning Technical University(Natural Science),2013,32(4):573-576.
[11]任建娅,尹建华,耿万海.小波方法求一类变系数分数阶微分方程数值解[J].辽宁工程技术大学学报(自然科学版),2012,31(6):925-928.REN Jianya,YIN Jianhua,GENG Wanhai.Wavelet method for numerical solution of a class of fractional differential equation with variable coefficients[J].Journal of Liaoning Technical University(Natural Science),2012,31(6):925-928.
[12]LIN R,LIU F,ANH V,et al.Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation[J].Applied Mathematics and Computation,2009(212):435-445.
[13]ZHUANG P,LIU F,ANH V,et al.Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term[J].SIAM Journal on Numerical Analysis,2009(47):1 760-1 781.
[14]COIMBRA C F M.Mechanics with variable-order differential operators[J].Annals of Physics,2003,12(11-12):692-703.
[15]KAZEM S.An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations[J].Applied Mathematical Modelling,2013(37):1 126-1 136.