SOLUTIONS OF A CLASS OF N-TH ORDER ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS VIA FRACTIONAL CALCULUS

Abstract

Solutions of the n-th order linear ordinary differential equations \begin{eqnarray*} \begin{array}{l} (z+b)^l \prod^{n-l}_{k=1}(z+a_k)\varphi_n+ \sum^n_{k=1}\varphi_{n-k} \{C^\lambda_{k}\{Q(z)\}_{k}+C^\lambda_{k-1}\{G(z)\}_{k-1}\}=f \\[] (z\neq -a_{k}\ (k=1,2,\ldots , n-l)\ z\neq -b;~a_i\neq a_j\neq b\ {\rm if}\ i\neq j;\ n\gt l,\ l\geq 2) \end{array} \end{eqnarray*} and the partial differential equations \begin{eqnarray*} \begin{array}{ll} (z+b)^l \prod^{n-l}_{k=1}(z+a_k)\cdot \frac{\partial^{n}\mu}{\partial z^{n}} & +\sum^{n-1}_{k=1} \frac{\partial^{n-k}\mu}{\partial z^{n-k}} \{C^\lambda_{k}\{Q(z)\}_{k}+C^\lambda_{k-1}\{G(z)\}_{k-1}\} \\[] &\hspace{-0.3cm} +\alpha\mu(z,t)=M \frac{\partial^2\mu}{\partial t^2}+N \frac{\partial\mu}{\partial t} \end{array} \end{eqnarray*} $$ (z\neq -a_{k}\ (k=1,2,\ldots , n-l)\ z\neq -b;\ a_i\neq a_j\neq b\ {\rm if}\ i\neq j;\ n\gt l,\ l\geq 2) $$ are discussed.