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Finite Differential Methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
Partial differential equations are great in calculus for making multi-variable equations simpler to solve. Some problems do not have known derivatives or at least in certain levels in your studies, you don't possess the tools needed to find the derivative. So, using partial differential equations, you can break the problem up, and find the partial derivatives and integrals.
Differential equations were invented separately by Isaac Newton and Gottfried Leibniz. This debate on who was the first one to invent it was argued by both Isaac and Gottfried until their death.
Calc 2, then Calc 3, then usually Differential Equations
Some partial differential equations do not have analytical solutions. These can only be solved numerically.