Chemical Engineering Hand Written_Notes_Part_67

Chemical Engineering Hand Written_Notes_Part_67 - 136 4....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

Chemical Engineering Hand Written_Notes_Part_67 - 136 4....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online