The document discusses numerical solutions of differential equations using B-splines. It introduces B-splines and some of their key properties like non-negativity, local support, and smoothness. It then presents the explicit expression for cubic B-spline basis functions. The document describes the B-spline collocation and Galerkin methods for solving differential equations, which involve approximating the solution as a linear combination of B-spline basis functions and discretizing the equations at collocation points or using weak formulations respectively.
1. Numerical Solutions of Differential Equations using
B-Splines
R. C. Mittal
Department of Mathematics,
Indian Institute of Technology Roorkee
2. Why B-Splines ?
• Splines are piece wise smooth functions
having continuity and differentiability at the
nodes.
• All splines of the same degree form a vector
space defined on an interval.
• B-Splines form basis of vector space of
Spline functions.
• A B-spline function is defined as a spline
function that has minimal support with
respect to a given degree, smoothness, and
domain partition.
3. Properties of B- Splines
Non-negativity
Local support
Partition of unity, Smoothness, Shape
preserving
Boundary conditions of various types can
be easily incorporated
4. Cubic B-Spline
The explicit expression for the cubic B-spline
basis functions is given as follows:
otherwise
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0
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(
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4
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1
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2
1
1
1
1
2
3
2
3
1
3
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1
3
2
3
2
3
3
,
j
j
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x
x
x
x
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x
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x
x
h
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B
7. Coefficient of cubic B-spline and its derivatives at knots
0
2
6
2
12
2
6
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(
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0
3
0
3
0
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8. B-Splines Based Collocation Method
The approximate solution of a PDE using cubic B-
spline functions can be written as
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(
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(
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(
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,
( 1
1
1
1 j
j
j
j
j
j
j
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j x
B
c
x
B
c
x
B
c
t
x
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1
2
1
(
6
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1
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3
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2
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c
j
c
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c
j
U
h
j
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j
c
j
hU
j
c
j
c
j
c
j
U
9. Description of the Method
Step-1: Consider numerical approximate solution
by using cubic B-spline basis functions as
Step-2: Discretizing the Neumann’s boundary
conditions as
1
1
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(
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(
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,
(
N
j
x
j
B
t
j
c
t
x
U
1
1
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(
1
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(
'
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,
(
1
1
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(
0
)
0
(
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)
,
0
(
N
N
j
t
g
N
x
j
B
j
c
t
N
x
x
U
j
t
g
x
j
B
j
c
t
x
x
U
10. Description of the Method
• So for Neumann’s Conditions
Step-3: Put the values of u(x), u’(x) and u”(x) in
terms of B-Spline functions in the differential
equations as given the table.
Step-4 : By putting collocation points x0 , x1,….,
xN, obtain a system of equations.
)
(
1
3
1
1
)
(
0
3
1
1
t
g
h
N
c
N
c
t
g
h
c
c
11. Step-5 : Substitute the approximation for boundary
conditions (Dirichlet’s or Neumann’s) in the obtained
equations.
Step-6 : Solve this system to get the solutions.
Note 1: If the given differential equation is linear then
the corresponding obtained system of equations will be
linear. This system can easily be solved by Thomas
algorithm.
If the given differential equations is non-linear then the
obtained system of equations is also non-linear. This
can be solved by Newton-Raphson’s method.
12. Note – 2 : If given differential equation is a time
dependant partial differential equation , then the
B- Spline approximation coefficients C’s will be
time dependant.
These problems can be solved in two ways :
Method – 1: Replace time derivative by a finite
difference approximation and form system of
equations.
Method – 2 : Form a system of ordinary
differential equations in coefficients C’s. Solve this
system by Runge-Kutta method.
Note – 3 : For time dependant problem, the initial
values of C’s are to be computed from the initial
conditions of the differential equation.
13. B- Spline Based Galerkin’s Method
Consider a two point boundary value problem
- y” + p(x)y = r(x) in (a, b) with boundary
conditions y(a) = α and y(b) = β.
The weak formulation of this problem is
14. Suppose we now select B- Splines as basis
functions in [a, b] and are used for local
approximation.
• Cubic Splines-
A typical finite element [xm, xm+1] has
nodes xm, xm+1. Since each cubic B-spline
covers four elements, each element is covered
by four cubic B-splines. In each element, using
the following transformation
x = xm + η dx,
15. where 0⩽η⩽1, cubic B-spline shape functions
having representations over the element
[xm,xm+1] can be defined as
Dag AMM
