There are many numerical methods adopted to solve mathematical problems. Early researchers focused on the methods to reduce computational costs. In recent years, reduction in computational costs makes many numerical methods available which were not tried for this reason. The use of higher order Legendre polynomials for more than 5-7 orders is usually not common. The efficient and quick numerica...

The optimal control problem of a linear distributed parameter system is studied via shifted Legendre polynomials (SLPs) in this paper. The partial differential equation, representing the linear distributed parameter system, is decomposed into an n set of ordinary differential equations, the optimal control problem is transformed into a two-point boundary value problem, and the twopoint boundary...

In this report, we present two mathematical results which can be useful in a variety of settings. First, we present an analysis of Legendre polynomials triple product integral. Such integrals arise whenever two functions are multiplied, with both the operands and the result represented in the Legendre polynomial basis. We derive a recurrence relation to calculate these integrals analytically. W...