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CH E374 (5)

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Chemical Engineering
CH E374
Joe Mmbaga

CHE 374 Computational Methods in Engineering MODULE 6 Ordinary differential equations ODEs initial value problems IVP BasicsEquations involving derivatives are called differential equations If there is one independent variable the equations are known ordinary differential equations ODEs if there is more than one independent variable and the equations involve partial derivatives we speak of partial differential equations to be discussed in Lecture Module 8 Order of an ODE order of the highest derivative thyx General form of an n order ODE with 2ndydydy0Fxy61 2ndxdxdx th General form of a linear n order ODE 1nndydydy0axaxaxaxygx62 110nn1nndxdxdx0gxIn a homogeneous linear ODE the term1nndydydy0axaxaxaxy63 110nn1nndxdxdx Important feature of homogeneous linear ODEs yxyxyxyxSupposeandare solutions to a linear ODE then alsois a 1122solution of the same ODE withandconstantsNonlinear ODEs eg contain products of y and its derivatives eg 2dydysin0bycx64 2dxdx th An n order ODE needs n boundary conditions If these are known at one point eg x0 we call the problem an initial value problem An ODE with boundary ndconditions at different locations or moments in time eg a 2 order ODE with two boundary conditions one at x0 one at xL is called a boundary value problem Boundary value problems are discussed in Lecture Module 7 1 ndIn the context of numerically solving ODEs it is common practice to write 2 and ndhigher order ODEs as systems of first order ODEs The most common 2 order ODE comes from applying Newtons second law of motion remember the falling object problem 221dxdxmamFxtORFxt65 22dtdtm with time t the independent variable x the position of mass m in space and F a force that in general depends on x and t stWe can write the second order equation as a system of two 1 order ODEs by introducing the velocity vdxvanddt66 dvmFxtdtWe write this in a vector form as dy1fyyt112dt67 dy2fyyt212dt ordyfyt68 dt with yx1 yyv21Giving our two first order differential equations asdy1y2dx69 dy1dv2Ftxdxdxm Eulers method Take the single firstorder ODE2
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