Numerical integration and numerical solution of integral and differential equations using PL transform

A new class of numerical integral operator for indefinite integral is first derived using piecewise-linear transform. The operator is a matrix by which the piecewise-linear transforms of functions are converted into the transforms of their integrals. The operator is then modified to cover definite integrals. The definite integral operator is a simple row vector. Application of the new operators to integral equations yields simple algebraic solutions for both Volterra’s and Fredholm’s equations. Initial value and boundary value problems of linear differential equations can also be solved by the use of the new integral operators through the solution of integral equations. The new operators provide wider range of applications with high accuracy than the existing methods using Walsh transform.