Very often because no analytical solution is available.
Some partial differential equations do not have analytical solutions. These can only be solved numerically.
It happens when the solution for the equation is periodic and contains oscillatory functions such as cos, sin and their combinations.
Euler's Method (see related link) can diverge from the real solution if the step size is chosen badly, or for certain types of differential equations.
Finite Differential Methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
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.
P. Quittner has written: 'Superlinear parabolic problems' -- subject(s): Differential equations, Elliptic, Differential equations, Parabolic, Differential equations, Partial, Elliptic Differential equations, Parabolic Differential equations, Partial Differential equations
George Francis Denton Duff has written: 'Partial differential equations' -- subject(s): Differential equations, Partial, Partial Differential equations 'Differential equations of applied mathematics' -- subject(s): Differential equations, Differential equations, Partial, Mathematical physics, Partial Differential equations
J. L Blue has written: 'B2DE' -- subject(s): Computer software, Differential equations, Elliptic, Differential equations, Nonlinear, Differential equations, Partial, Elliptic Differential equations, Nonlinear Differential equations, Partial Differential equations
Jerrold Stephen Rosenbaum has written: 'Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits' -- subject(s): Differential equations, Electronic circuits, Numerical solutions, Stiff computation (Differential equations)
Leon Lapidus has written: 'Numerical solution of ordinary differential equations' -- subject(s): Differential equations, Electronic data processing, Numerical analysis, Mathematics
Laurent Veron has written: 'Singularities of solutions of second order quasilinear equations' -- subject(s): Differential equations, Elliptic, Differential equations, Nonlinear, Differential equations, Parabolic, Elliptic Differential equations, Nonlinear Differential equations, Numerical solutions, Parabolic Differential equations, Singularities (Mathematics)
It happens when the solution for the equation is periodic and contains oscillatory functions such as cos, sin and their combinations.
Monge's method, also known as the method of characteristics, is a mathematical technique used to solve certain types of partial differential equations. It involves transforming a partial differential equation into a system of ordinary differential equations by introducing characteristic curves. By solving these ordinary differential equations, one can find a solution to the original partial differential equation.
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)