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Very often because no analytical solution is available.
Tarek P. A. Mathew has written: 'Domain decomposition methods for the numerical solution of partial differential equations' -- subject(s): Decomposition method, Differential equations, Partial, Numerical solutions, Partial Differential equations
Some partial differential equations do not have analytical solutions. These can only be solved numerically.
Granville Sewell has written: 'The numerical solution of ordinary and partial differential equations' -- subject(s): Data processing, Differential equations, Mathematics, Nonfiction, Numerical solutions, OverDrive, Partial Differential equations 'Computational Methods of Linear Algebra' -- subject(s): OverDrive, Mathematics, Nonfiction
David L. Colton has written: 'Analytic theory of partial differential equations' -- subject(s): Differential equations, Partial, Numerical solutions, Partial Differential equations 'Partial differential equations' -- subject(s): Differential equations, Partial, Partial Differential equations
S. H. Lui has written: 'Numerical analysis of partial differential equations' -- subject(s): Partial Differential equations, Numerical solutions
Elemer E. Rosinger has written: 'Generalized solutions of nonlinear partial differential equations' -- subject(s): Differential equations, Nonlinear, Differential equations, Partial, Nonlinear Differential equations, Numerical solutions, Partial Differential equations 'Distributions and nonlinear partial differential equations' -- subject(s): Differential equations, Partial, Partial Differential equations, Theory of distributions (Functional analysis)
Stephen F Wornom has written: 'Critical study of higher order numerical methods for solving the boundary-layer equations' -- subject(s): Boundary layer, Differential equations, Partial, Numerical solutions, Partial Differential equations
A. J. Werthen has written: 'Optimal moving grids for time-dependent partial differential equations' -- subject(s): Differential equations, Numerical grid generation (Numerical analysis)
The answer will depend very much on the nature of the equation. The steps required for a one-step equation are very different from the steps required for a partial differential equation. For some equations there are no straightforward analytical methods of solution: only numerical methods.
Finite element analysis is a numerical technique that will help find solutions to partial differential equations (PDE) as well as of integral equations.
G. Evans has written: 'Analytic methods for partial differential equations' -- subject(s): Differential equations, Partial, Numerical solutions, Partial Differential equations