For large-k sin(h) 20k = 2 cos(h) – (9.302) 4h k3 and this gives for (Ao, A1, A2) the values sin(h) Ao = cos(h), A1 = 0, A2 (9.303) 8h Substitution of these values into equations (9.292), (9.293), (9.294), and (9.295) gives 0 = 0, Bo = ih, B1 (9.304a) 8h 1 sin(h) 16h sin(h) cos(h) + 8h B2 = (9.304b)
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Show me the steps of determine red and all the equation are there
![and that the corresponding asymptotic form for
Yk
is
B1
+
k
B2
k2
Вз
+0
k3
Yk = k°eBok
(9.283)
whoro +he noromotona (0 R.
R.) mor ba oomnlov vol,und and ora
to
These equations may be solved, in turn, to give (Bo, 0, B1, B2). Doing this
gives
Bo = (+) cosh(Ao) = In ( Ao + V A3 – 1
(9.292)
%3D
A1
(9.293)
sinh(Bo)'
[0(0 – 1)
+
sinh(Bo)
A2
B1 =
tanh(Bo),
(9.294)
2
496
Difference Equations
[(1 – 0) cosh(Bo) + 0(0 – 1)/2 – A2]B1
2 sinh(Bo)
0(е — 1)(ө — 2)
B2 =
Аз
(9.295)
2 sinh(Bo)
12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf606203-a5b8-4153-85ec-20a78972fb86%2F633f5067-14ea-48d2-9574-237860fe1320%2Fc65opw8_processed.png&w=3840&q=75)
![9.6.3
An Example
To illustrate the use of the technique given in the previous section, consider
the STDE
y"(») + 1+v(e) = 0,
(9.297)
472
(see equation (9.263)). For this case,
1
f(x) = 1+
4x2
(9.298)
and its discretization is
1
fk = f(xk) = f(hk) = 1+
(9.299)
4h?k2 °
Using the Mickens-Ramadhani finite difference discretization, we obtain
1/27
1
Yk+1+ Yk-1 = 2 cos (hVfk)| Yk = 2{ cos h (1+
4h?k2
Yk,
(9.300)
and this gives for
the expression
Ok
1/27
1
= cos |h | 1+
(9.301)
Ok = cos
4h2k2
ADVANCED APPLICATIONS
497
For large-k
20%-2c0()-의 ()+0().
sin(h)
(9.302)
4h
and this gives for (Ao, A1, A2) the values
sin(h)
Ao = cos(h), A1 = 0, A2 = -
8h
(9.303)
Substitution of these values into equations (9.292), (9.293), (9.294), and
(9.295) gives
Co =0,
Bo = ih, Bị = -
8h
(9.304a)
sin(h)
cos(h) +
1
B2
(9.304b)
16h sin(h)
8h
where i = V-1.
If these values are now placed in the yk of equation (9.283) and appropriate
linear combinations of yk and (Yk)* are made, then the following expression
is obtained for the asymptotic solution
cos(Tk)
9 sin(2k)]
2(8k)2
(MR)
A sin(xk)
- a(h) ·
%3D
8xk
9 cos(Tk)]
a(h) ·
2(8®k)?
sin(2k)
+B cos(xk) +
8xk
'(주)아기
+0
(9.305)
where
8h
sin(h)
9 sin(h).
cos(h)
8h
(9.306)
Note that
Ca(0) = Lim a(h):
(9.307)
= 1.
h-0
Also, we have replaced (hk) by Xk.
A similar kind of calculation can be done for the other three finite difference
schemes given in Section 9.6.1, and the results for the asymptotic solutions
are as follows:
cos(o*k)]
8xk
(S)
(Standard scheme) y
= A sin(ørk) – ß.
sin(øæk)]
+ B cos(pxk)+B.
8xk
o(4).
(9.308)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf606203-a5b8-4153-85ec-20a78972fb86%2F633f5067-14ea-48d2-9574-237860fe1320%2Fnmq5aya_processed.jpeg&w=3840&q=75)
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