Solving linear differential equations as a minimum norm least squares problem with error-bounds

Linear ordinary/partial differential equations (DEs) with linear boundary c onditions (BCs) are posed as an error minimization problem. This problem ha s a linear objective function and a system of linear algebraic (constraint) equations and inequalities derived using both the forward and the backward Taylor series expansion. The DEs along with the BCs are approximated as li near equations/inequalities in terms of the dependent variables and their d erivatives so that the total error due to discretization and truncation is minimized. The total error along with the rounding errors render the equati ons and inequalities inconsistent to an extent or, equivalently, near-consi stent, in general. The degree of consistency will be reasonably high provid ed the errors are not dominant. When this happens and when the equations/in equalities are compatible with the DEs, the minimum value of the total disc retization and truncation errors is taken as zero. This is because of the f act that these errors could be negative as well as positive with equal prob ability due to the use of both the backward and forward series. The inequal ities are written as equations since the minimum value of the error (implyi ng error-bound and written/expressed in terms of a nonnegative quantity) in each equation will be zero. The minimum norm least-squares solution (that always exists) of the resulting over-determined system will provide the req uired solution whenever the system has a reasonably high degree of consiste ncy. A lower error-hound and an upper error-bound of the solution are also included to logically justify the quality/validity of the solution.