Chemical Engineering Hand Written_Notes_Part_67

# Chemical Engineering Hand Written_Notes_Part_67 - 136 4...

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136 4. ODE-IVPS AND RELATED NUMERICAL SCHEMES 2.2. Vector case. Now consider system of equations given by equation (2.1). Taking clues from the scalar case, let us investigate a candidate solution of the form (2.8) x ( t )= e λt v ; v R m where v is a constant vector. The above candidate solution must satisfy the ODE, i.e., (2.9) d dt ( e λt v )= A ( e λt v ) λ v e λt = A v e λt Cancelling e λt from both the sides, as it is a non-zero scalar, we get an equation that vector v must satisfy, (2.10) λ v = A v This fundamental equation has two unknowns λ and v and the resulting problem is the well known eigenvalue problem in linear algebra. The number λ is called the eigenvalue of the matrix A and v is called the eigenvector. Now, λ v = A v is a non-linear equation as λ multiplies v . if we discover λ then the equation for v would be linear. This fundamental equation can be rewritten as (2.11) ( A λI ) v =0 This implies that vector v should be to the row space of

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Chemical Engineering Hand Written_Notes_Part_67 - 136 4...

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