【分数阶微积分理论与应用研究10000字（论文）】

附录%G-L型分数阶微积分运算函数程序functiondx=GL(x,s,pp)ifstrcmp(class(x),'function_handle'),x=x(s);endo=s(2)-s(1);z=1;x=x(':');s=s(':');fork=2:length(x),z(k)=z(k-1)*(1-(pp+1)/(k-1));endforl=1:length(s),dx(l)=z(1:l)*[x(l:-1:1)]/o^pp;end%G-L型分数阶微积分的求解主程序s=0:0.01:pi;x=exp(-s).*sin(3*s+1);pp=0:0.25:1.25;forp=pp，x1=GL(x,s,p);plot(s,x1);holdon;endaxis([0pi,-33])%R-L型分数阶微积分运算函数程序function[dx,s]=RL(x,s,q)f=s(2)-s(1);m=length(s);dx1=zeros(1,m);x=x(:);s=s(:);ifq>-1，n=ceil(q)+1;p=n-q;x3=s.^(p-1);elseifq==-1m=length(s);xx=0.5*(x(1:m-1)+x(2:m)).*diff(s);dx(1)=0;forj=2:m，dx(j)=dx(j-1)+xx(j-1);end，returnelse，n=-q;x3=s.^(n-1);endforj=1:m，dx1(j)=x(j:-1:1).'*(x3(1:j));endifq>-1，dx=diff(dx1,n)/(f^(n-1))/gamma(p);s=s(1:m-n);else，dx=dx1*k/gamma(n);end%R-L型分数阶微积分的求解主程序f=0.01;s1=0:f:6;x=sin(s1);pp=-1:0.25:0ford=pp，[x2,s2]=RL(x,s1,d);plot(s2,x2);holdon;ylim([-1.52.5]);end%R-L型分数阶微积分与G-L型分数阶微积分比较f=0.01;t1=0:f:6;x=sin(s1);pp=-1:0.25:0forp=pp，[x2,s2]=RL(x,s1,p);x1=GL(x,s1,p);plot(s1,x1,'--',s2,x2,':');holdon;ylim([-1.52.5]);end%R-L型分数阶积分几何解释图绘制函数程序functiongeometry(s,u,pp,w)，x=s;z=u(x);s=x(end);switchwcase1，y=(s^pp-(s-x).^pp)/gamma(1+pp);axis([min(x),max(x),min(y),max(y),min(z),max(z)])，holdonforj=1:length(x)-1,x0=[x(j)x(j)x(j+1)x(j+1)];z0=[0z(j)z(j+1)0];y0=[0000];fill3(x0,y0,z0,'c'),x0=[x(j)x(j)x(j+1)x(j+1)];y0=[y(j)y(j)y(j+1)y(j+1)];fill3(x0,y0,z0,'g')，x0=[0000];fill3(x0,y0,z0,'r')；endcase{2,3}，axis([min(x)max(x)min(z)max(z)])，holdonforj=1:length(x)-1，x1=[0:0.01:x(j+1)];s=x1(end);y=(s^pp-(s-x1).^pp)/gamma(1+pp);ifw==2，plot(x1,y);else，z=u(x1);plot(y,z,[y(end),y(end)],[z(end),0]);end，end，end，holdoff%R-L型分数阶积分几何解释图绘制主程序s=0:0.5:10;u=@(s)s.^2+cos(s);geometry(s,u,0.8,1)grid，set(gca,'xdir','reverse'),set(gca,'ydir','reverse')geometry(s,u,0.8,2),set(gca,'xdir','reverse','Ylim',[0,2*pi]),ylim([0,2*pi])figure，geometry(s,u,0.8,2),xlim([0,2*pi]),ayis([02*pi])%Caputo型分数阶导数运算函数程序functiondy=caputo9(y,s,deta,y0,T)dy=GL(y,s,deta);ifnargin<=4，T=10;endifdeta>0，q=ceil(deta);ifdeta<=1，y0=y(1);endfori=0:q-1，dy=dy-y0(i+1)*s.^(i-deta)./gamma(i+1-deta);endyy1=interp1(s(T+1:end),dy(T+1:end),s(1:T),'spline');dy(1:T)=yy1;end%Caputo型分数阶导数求解主程序symss;y=sin(3*s+1);y10=double(subs(diff(y,s),s,0));y20=double(subs(diff(y,s,2),s,0));y00=sin(1);s=0:0.01:3;y=sin(3*s+1);y1=caputo9(y,s,1.3,[y00y10],10);y2=caputo9(y,s,2.3,[y00,y10,y20],30);y3=caputo9(y,s,1.6,[y00,y10,y20],30);y4=caputo9(y,s,2,[y00,y10,y20],30);plot(s,y1,s,y2,s,y3,s,y4)%图像处理函数程序functionww=tv(f,A)N=20;delta=5;dt=0.1;mmax=50;[m,n]=size(f);q=floor(mmax/2);fl=zeros(m+2*q,n+2*q);foraa=1:qfl(aa,q+1:n+q)=f(mmax-aa,:);fl(m+q+aa,q+1:n+q)=f(m+1-aa,:);fl(q+1:n+q,aa)=f(:,mmax-aa);fl(q+1:n+q,m+q+aa)=f(:,n+1-aa);endfl(q+1:m+q,q+1:n+q)=f;fl(1:q,1:q)=f(1,1);fl(m+q+1:m+2*q,n+q+1:n+2*q)=f(m,n);fl(1:q,n+q+1:n+2*q)=f(1,n);fl(m+q+1:m+2*q,1:q)=f(m,1);v1=0.9;v=v1*ones(m,n);%%ÀûÓ÷½³Ì½øÐеü´úff=fl;ww=ff;fornn=1:Nfork=q+1:m+qforr=1+q:n+qb1(k-q,r-q)=delta*dt./sqrt(((0.5*v(k-q,r-q)*(1-v(k-q,r-q))*ww(k,r+1))+v(k-q,r-q)*ww(k,r+1)-ww(k,r+1)).^2+...0+0.25*v(k-q,r-q)*(ww(k+1,r)-ww(k-1,r)).^2+eps);b2(k-q,r-q)=delta*dt./sqrt((ww(k,r)-v(k-q,r-q)*ww(k,r-1)-(0.5*v(k-q,r-q)*(1-v(k-q,r-q))*ww(k,r-2))).^2+...+0.25*v(k-q,r-q)*(ww(k+1,r-1)-ww(k-1,r-1)).^2+eps);b3(k-q,r-q)=delta*dt./sqrt(((0.5*v(k-q,r-q)*(1-v(k-q,r-q))*ww(k+2,r))+v(k-q,r-q)*ww(k+2,r)-ww(k+2,r)).^2+...0+0.25*v(k-q,r-q)*(ww(k,r+1)-ww(k,r-1)).^2+eps);b4(k-q,r-q)=delta*dt./sqrt((ww(k,r)-v(k-q,r-q)*ww(k-1,r)-(0.5*v(k-q,r-q)*(1-v(k-q,r-q))*ww(k-2,r))).^2+...+0.25*v(k-q,r-q)*(ww(k-1,r+1)-ww(k-1,r-1)).^2+eps);ww(k,r)=(ww(k,r)+dt*A*(ff(k,r)-ww(k,r))+...(b1(k-q,r-q).*ww(k,r+1)+b2(k-q,r-q).*ww(k,r-1)+...b3(k-q,r-q).*ww(k+1,r)+b4(k-q,r-q).*ww(k-1,r)))./...(1+(b1(k-q,r-q)+b2(k-q,r-q)+b3(k-q,r-q)+b4(k-q,r-q)));endendend%图像处理主程序I=imread('512.jfif');I=double(I);[row,col]=size(I);figure;imshow(I,[]);In=I;sigma=20In=I+sigma*randn(row,col);figure;imshow(In,[])A=1;It=tv(In,A);figure;imshow(It,[]);%求分数阶微分方程数值解的函数程序functionoo_delta=o(P,delta)oo_delta=zeros(P+1,1);oo_delta(1,1)=1;forj=1:Poo_delta(j+1,1)=gamma(j-delta)/(gamma(-delta)*gamma(j+1));endend%求分数阶微分方程数值解的主程序W=1;P=100;Q=P;h=1/P;pp=W/Q;x=0:h:1;s=0:pp:W;delta=1.5;VE=zeros(P+1,Q+1);V=VE;E=zeros(P-1,1);c=E;f=@(x,s)-2.*exp(-s).*x.^(4+delta);%右端函数initial=@(x)x.^(4+delta);left=@(s)0;right=@(s)exp(-s);exact=@(x,s)exp(-s).*x.^(4+delta);d=@(x)24*x.^delta/gamma(5+delta);forr=1:Q+1VE(1:end,r)=exact(x(1:end),s(r));%真解end%初边值条件V(1:end,1)=initial(x);V(1,1:end)=left(s);V(end,1:end)=right(s);%矩阵的构造B=zeros(P-1,P-1);forj=1:P-1E(j,1)=d(x(j+1));endc=pp*E/(2*h^delta);oo=o(P,delta);forj=1:P-1forr=1:Q-1ifr<=j-1B(j,r)=c(j,1)*oo(j-r+2,1);elseifr==jB(j,r)=c(j,1)*oo(2,1);elseifr==j+1B(j,r)=c(j,1)*oo(1,1);elseB(j,r)=0;endendendforr=1:Qd=(eye(P-1)+B)*V(2:end-1,r)+pp*f(x(2:end-1),s(r)+pp/2)'+...c.*(oo(3:end)