legendre wavelets technique for special initial-value heat transfer problem in the quarter plane

In this work we have solved the heat transfer equation by means of the volterra integral equation and legendre wavelets. indeed, numerical facts show that resolving the related partial differential equation is difficult which motivates the choice of this approach. the integral equation corresponding to this system is a volterra type of the first kind. these systems are ill posed, therefore an appropriate methods to solve this type of systems is wavelets approach, since wavelets can be generated in the space of solutions. in this work we use legendre wavelets to solve the corresponding integral equation. numerical implementation of the method is illustrated by benchmark problems originated from heat transfer. the time evolution of the initial heat function along with the x-axis is exhibited in a three dimensional plot.