Mathematical Modelling in One Dimension: An Introduction Via Difference and Differential Equations
Mathematical Modelling in One Dimension demonstrates the universality of mathematical techniques through a wide variety of applications. Learn how the same mathematical idea governs loan repayments, drug accumulation in tissues or growth of a population, or how the same argument can be used to find the trajectory of a dog pursuing a hare, the trajectory of a self-guided missile or the shape of a satellite dish. The author places equal importance on difference and differential equations, showing how they complement and intertwine in describing natural phenomena.
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Basic difference equations models and their
Basic differential equations models
Qualitative theory for a single equation
From discrete to continuous models and back
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2-cycle air resistance assume asymptotically stable equilibrium behaviour breeding season Cauchy problem chain rule coefficients concentration constant continuous function conversion period crystals curve decreases deﬁned deﬁnition denote derivative describe difference equations differential equations discrete model dynamics Elaydi equa equilibrium point escape velocity Example Exercise exponential growth fact ﬁnal ﬁnd ﬁnding ﬁnite ﬁrst ﬁrst-order equation ﬁsh ﬁshing ﬁxed formula given gives graph of f growth rate Hence I N(to increases inhomogeneous initial condition initial value injection integrating interest rate interval of existence introduce iterations Lemma Let us consider linear equation Lipschitz continuous logistic equation mathematical modelling maximal interval modelling process monotonic Nk+1 nonlinear observe obtain parameter Picard theorem population models Ré/b reﬂect repayment right hand side satisﬁes the assumptions semelparous separable equation sin2 solved speed stationary solution substitution tent map terminal velocity Theorem 1.6 theory tion unstable variables xn+1 yields yn+1