2nd order, linear, homogeneous, non-constant coefficient ODEs
Example: Cauchy-Euler (Euler) O.D.E.
ax 2
d2y
dy
+ bx + cy = 0
2
dx
dx
here P(x)= ax 2 , Q(x)=bx, R(x)=c a nd a , b, c = constants
This is one of the few cases where an exact solution can be fo
2.8 Set of Simultaneous Nonlinear Equations
All methods involve iteration. Problems with multiple roots and finding a scheme that converges are
common. In applied problems, we usually have an idea of the answer i.e. some idea of what the
root(s) should be
2.8 Set of Simultaneous Nonlinear Equations
All methods involve iteration. Problems with multiple roots and finding a scheme that converges are
common. In applied problems, we usually have an idea of the answer i.e. some idea of what the
root(s) should be
2.8 Set of Simultaneous Nonlinear Equations
All methods involve iteration. Problems with multiple roots and finding a scheme that converges are
common. In applied problems, we usually have an idea of the answer i.e. some idea of what the
root(s) should be
3.1.4 Runge-Kutta Formulas
dy
= F ( x, y )
dx
To solve a first-order O.D.E.
yi +1 = yi + xFi where Fi = F ( xi , yi ) is the slope of y ( x ) at xi
(Explicit) Euler method:
- uses y i and the slope of y ( x ) at xi (both known) to calculate y i +1
- this
3.1.4 Runge-Kutta Formulas
dy
= F ( x, y )
dx
To solve a first-order O.D.E.
yi +1 = yi + xFi where Fi = F ( xi , yi ) is the slope of y ( x ) at xi
(Explicit) Euler method:
- uses y i and the slope of y ( x ) at xi (both known) to calculate y i +1
- this
3.1.4 Runge-Kutta Formulas
dy
= F ( x, y )
dx
To solve a first-order O.D.E.
yi +1 = yi + xFi where Fi = F ( xi , yi ) is the slope of y ( x ) at xi
(Explicit) Euler method:
- uses y i and the slope of y ( x ) at xi (both known) to calculate y i +1
- this
3.2 Second-Order I.V.P.
ex.
y ' '+ y '2 y = 2 x
y (0 ) = 0 , y ' (0 ) = 1
dy
= F where
dx
dv
=G
dx
y (0 ) = 0
F =v
G = 2x v + 2 y
v(0 ) = 1
Explicit Euler Scheme
yi +1 = yi + xFi
(Fi = vi )
vi +1 = vi + xGi
(Gi = 2 xi vi + 2 yi )
Start at x0 = 0 where ICs
3.2 Second-Order I.V.P.
ex.
y ' '+ y '2 y = 2 x
y (0 ) = 0 , y ' (0 ) = 1
dy
= F where
dx
dv
=G
dx
y (0 ) = 0
F =v
G = 2x v + 2 y
v(0 ) = 1
Explicit Euler Scheme
yi +1 = yi + xFi
(Fi = vi )
vi +1 = vi + xGi
(Gi = 2 xi vi + 2 yi )
Start at x0 = 0 where ICs
3.2 Second-Order I.V.P.
ex.
y ' '+ y '2 y = 2 x
y (0 ) = 0 , y ' (0 ) = 1
dy
= F where
dx
dv
=G
dx
y (0 ) = 0
F =v
G = 2x v + 2 y
v(0 ) = 1
Explicit Euler Scheme
yi +1 = yi + xFi
(Fi = vi )
vi +1 = vi + xGi
(Gi = 2 xi vi + 2 yi )
Start at x0 = 0 where ICs
3.4 Example: Finite Difference Method for Higher Order IVPs
y (0 ) = 1 ,
y ' '+ xy '+ y = e x
ex.
dy
(0) = 0
dx
Find y ( x ) for 0 x 10
Choose x = 0.5 to give 21 x-locations x0 = 0, x1 = 0.5,K , x 20 = 10
The ODE must hold at every x . Apply at xi
()
y '
3.4 Example: Finite Difference Method for Higher Order IVPs
y (0 ) = 1 ,
y ' '+ xy '+ y = e x
ex.
dy
(0) = 0
dx
Find y ( x ) for 0 x 10
Choose x = 0.5 to give 21 x-locations x0 = 0, x1 = 0.5,K , x 20 = 10
The ODE must hold at every x . Apply at xi
()
y '
2.7 Solution of systems of linear equations
Our goal in this section is to develop methods for solving a system of n linear equations with n
unknowns. We will use these methods later in the course since solving ODEs/PDEs often requires
solving systems of
2.7 Solution of systems of linear equations
Our goal in this section is to develop methods for solving a system of n linear equations with n
unknowns. We will use these methods later in the course since solving ODEs/PDEs often requires
solving systems of
2.7 Solution of systems of linear equations
Our goal in this section is to develop methods for solving a system of n linear equations with n
unknowns. We will use these methods later in the course since solving ODEs/PDEs often requires
solving systems of
2nd order, linear, homogeneous, non-constant coefficient ODEs
Example: Cauchy-Euler (Euler) O.D.E.
ax 2
d2y
dy
+ bx + cy = 0
2
dx
dx
here P(x)= ax 2 , Q(x)=bx, R(x)=c a nd a , b, c = constants
This is one of the few cases where an exact solution can be fo
2nd order, linear, homogeneous, non-constant coefficient ODEs
Example: Cauchy-Euler (Euler) O.D.E.
ax 2
d2y
dy
+ bx + cy = 0
2
dx
dx
here P(x)= ax 2 , Q(x)=bx, R(x)=c a nd a , b, c = constants
This is one of the few cases where an exact solution can be fo
2.2.4 Unequally spaced points
All finite difference formulas (i.e., forward, backward, and central difference formulas)
developed in this section and provided in the previous handout assume equal x spacing
(uniform grid). There are some applications which
2.2.4 Unequally spaced points
All finite difference formulas (i.e., forward, backward, and central difference formulas)
developed in this section and provided in the previous handout assume equal x spacing
(uniform grid). There are some applications which
2.2.4 Unequally spaced points
All finite difference formulas (i.e., forward, backward, and central difference formulas)
developed in this section and provided in the previous handout assume equal x spacing
(uniform grid). There are some applications which
2.3 Approximations for Partial Derivatives
Extension of the ideas developed in section 2.2 for partial derivatives is straightforward. Just
keep in mind the physical/geometrical meaning of partial derivatives. Consider a function of two
variables: z = f (
2.3 Approximations for Partial Derivatives
Extension of the ideas developed in section 2.2 for partial derivatives is straightforward. Just
keep in mind the physical/geometrical meaning of partial derivatives. Consider a function of two
variables: z = f (
2.3 Approximations for Partial Derivatives
Extension of the ideas developed in section 2.2 for partial derivatives is straightforward. Just
keep in mind the physical/geometrical meaning of partial derivatives. Consider a function of two
variables: z = f (
Convergence of the direct iteration method
The x = g ( x) rearrangement that we choose can have a strong effect on convergence of the
iteration. We can show graphically that the iteration will converge only if the function g ( x) on
dg
the right hand side