TY - JOUR
T1 - Tau numerical solution of the Volterra-Fredholm Hammerstein integro-differential equations with the Bernstein multi-scaling functions
AU - Yadollah Ordokhani Solmaz Moosavi and Mohsen Shahrezaee
JO - Journal of Information and Computing Science
VL - 4
SP - 243
EP - 255
PY - 2024
DA - 2024/01
SN - 8
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jics/22600.html
KW - Bernstein multi-scaling functions , Operational Tau method , Hammerstein integro-differential 
equation, Algebraic equation,  Fredholm, Volterra. 

AB -  This  paper  involves  the  development  of  the  Tau  method  with  Bernstein  multi-scaling  (BMS) 
functions  basis  for  the  numerical  solution  of  the  Volterra-Fredholm  Hammerstein  integro-differential 
equations (VFHIDEs). For this purpose at the beginning we define BMS functions and express briefly some 
properties  of  BMS  functions  and  after  function  approximation  by  using  BMS  functions,  will  be  presented. 
Then, the operator matrix representation for the differential and integral parts seeming in the equation using 
the  operational  Tau  method  base  on  BMS  functions  basis,  will  be  displaced.  The  operational  Tau  method 
transforms the differential and integration parts of the desired VFHIDEs to some operational matrices. In fact, 
this  method  reduces  VFHIDEs  to  a  system  of  algebraic  equations.    Numerical  examples  demonstrate  the 
validity and applicability of the proposed method with BMS functions basis.