R EFERENCES FOR C HAPTER 11 451TABLE 11.6 Areas of application of integral transforms first in the last column come references to appropriate sections of the current book Area of applica
Trang 1R EFERENCES FOR C HAPTER 11 451
TABLE 11.6 Areas of application of integral transforms (first in the last column come references to appropriate sections of the current book) Area of application Integral transforms References
Evaluation of improper integrals Laplace, Mellin Paragraph 7.2.8-5;
Ditkin & Prudnikov (1965) Summation of series (direct and inverse)Laplace Paragraphs 8.1.5-2 and 8.4.4-1 Computation of coefficients
of asymptotic expansions Laplace, Mellin Paragraph 7.2.9-1
Linear constant- and variable-coefficient
ordinary differential equations
Laplace, Mellin, Euler, and others
Paragraphs 12.4.1-3 and 12.4.2-6; Ditkin & Prudnikov (1965); Doetsch (1974); E Kamke (1977); Sveshnikov & Tikhonov (1970); LePage (1980) Systems of linear constant-coefficient
ordinary differential equations Laplace
Paragraph 12.6.1-4; Doetsch (1974); Ditkin & Prudnikov (1965)
Linear equations
of mathematical physics
Laplace, Fourier, Fourier sine, Hankel, Kontorovich–Lebedev, and others
Section 14.5; Doetsch (1974); Ditkin & Prudnikov (1965); Antimirov (1993); Sneddon (1995); Zwillinger (1997); Bracewell (1999); Polyanin (2002); Duffy (2004)
Linear integral equations
Laplace, Mellin, Fourier, Meler–Fock, Euler, and others
Subsections 16.1.3, 16.2.3, 16.3.2, 16.4.6; Krasnov, Kiselev, & Makarenko (1971); Ditkin & Prudnikov (1965); Samko, Kilbas, & Marichev (1993); Polyanin & Manzhirov (1998) Nonlinear integral equations Laplace, Mellin,
Fourier
Paragraphs 16.5.2-1 and 16.5.3-2; Krasnov, Kiselev, & Makarenko (1971); Polyanin & Manzhirov (1998) Linear difference equations Laplace Ditkin & Prudnikov (1965) Linear differential-difference equations Laplace Bellman & Roth (1984)
Linear integro-differential equations Laplace, Fourier Paragraph 11.5.2-2; LePage (1980)
References for Chapter 11
Antimirov, M Ya., Applied Integral Transforms, American Mathematical Society, Providence, Rhode Island,
1993.
Bateman, H and Erd´elyi, A., Tables of Integral Transforms Vols 1 and 2, McGraw-Hill, New York, 1954 Beerends, R J., ter Morschem, H G., and van den Berg, J C., Fourier and Laplace Transforms, Cambridge
University Press, Cambridge, 2003.
Bellman, R and Roth, R., The Laplace Transform, World Scientific Publishing Co., Singapore, 1984 Bracewell, R., The Fourier Transform and Its Applications, 3rd Edition, McGraw-Hill, New York, 1999 Brychkov, Yu A and Prudnikov, A P., Integral Transforms of Generalized Functions, Gordon & Breach,
New York, 1989.
Davis, B., Integral Transforms and Their Applications, Springer-Verlag, New York, 1978.
Ditkin, V A and Prudnikov, A P., Integral Transforms and Operational Calculus, Pergamon Press, New
York, 1965.
Doetsch, G., Handbuch der Laplace-Transformation Theorie der Laplace-Transformation, Birkh¨auser, Basel–
Stuttgart, 1950.
Doetsch, G., Handbuch der Laplace-Transformation Anwendungen der Laplace-Transformation, Birkh¨auser,
Basel–Stuttgart, 1956.
Doetsch, G., Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag,
Berlin, 1974.
Duffy, D G., Transform Methods for Solving Partial Differential Equations, 2nd Edition, Chapman & Hall/CRC
Press, Boca Raton, 2004.
Trang 2452 INTEGRALTRANSFORMS
Hirschman, I I and Widder, D V., The Convolution Transform, Princeton University Press, Princeton, New
Jersey, 1955.
Kamke, E., Differentialgleichungen: L¨osungsmethoden und L ¨osungen, I, Gew¨ohnliche Differentialgleichungen,
B G Teubner, Leipzig, 1977.
Krantz, S G., Handbook of Complex Variables, Birkh¨auser, Boston, 1999.
Krasnov, M L., Kiselev, A I., and Makarenko, G I., Problems and Exercises in Integral Equations, Mir
Publishers, Moscow, 1971.
LePage, W R., Complex Variables and the Laplace Transform for Engineers, Dover Publications, New York,
1980.
Miles, J W., Integral Transforms in Applied Mathematics, Cambridge University Press, Cambridge, 1971 Oberhettinger, F., Tables of Bessel Transforms, Springer-Verlag, New York, 1972.
Oberhettinger, F., Tables of Fourier Transforms and Fourier Transforms of Distributions, Springer-Verlag,
Berlin, 1980.
Oberhettinger, F., Tables of Mellin Transforms, Springer-Verlag, New York, 1974.
Oberhettinger, F and Badii, L., Tables of Laplace Transforms, Springer-Verlag, New York, 1973.
Pinkus, A and Zafrany, S., Fourier Series and Integral Transforms, Cambridge University Press, Cambridge,
1997.
Polyanin, A D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman &
Hall/CRC Press, Boca Raton, 2002.
Polyanin, A D and Manzhirov, A V., Handbook of Integral Equations, CRC Press, Boca Raton, 1998 Poularikas, A D., The Transforms and Applications Handbook, 2nd Edition, CRC Press, Boca Raton, 2000 Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 4, Direct Laplace
Transform, Gordon & Breach, New York, 1992.
Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 5, Inverse Laplace
Transform, Gordon & Breach, New York, 1992.
Samko, S G., Kilbas, A A., and Marichev, O I., Fractional Integrals and Derivatives Theory and
Applica-tions, Gordon & Breach, New York, 1993.
Sneddon, I., Fourier Transforms, Dover Publications, New York, 1995.
Sneddon, I., The Use of Integral Transforms, McGraw-Hill, New York, 1972.
Sveshnikov, A G and Tikhonov, A N., Theory of Functions of a Complex Variable [in Russian], Nauka
Publishers, Moscow, 1970.
Titchmarsh, E C., Introduction to the Theory of Fourier Integrals, 3rd Edition, Chelsea Publishing, New York,
1986.
Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002 Zwillinger, D., Handbook of Differential Equations, 3rd Edition, Academic Press, Boston, 1997.
Trang 3Chapter 12
Ordinary Differential Equations
12.1 First-Order Differential Equations
12.1.1 General Concepts The Cauchy Problem Uniqueness and
Existence Theorems
12.1.1-1 Equations solved for the derivative General solution
A first-order ordinary differential equation* solved for the derivative has the form
y
Sometimes it is represented in terms of differentials as dy = f (x, y) dx.
A solution of a differential equation is a function y(x) that, when substituted into the equation, turns it into an identity The general solution of a differential equation is
the set of all its solutions In some cases, the general solution can be represented as a
function y = ϕ(x, C) that depends on one arbitrary constant C; specific values of C define specific solutions of the equation (particular solutions) In practice, the general solution
more frequently appears in implicit form,Φ(x, y, C) =0, or parametric form, x = x(t, C),
y = y(t, C).
Geometrically, the general solution (also called the general integral) of an equation is
a family of curves in the xy-plane depending on a single parameter C; these curves are called integral curves of the equation To each particular solution (particular integral) there
corresponds a single curve that passes through a given point in the plane
For each point (x, y), the equation y x = f (x, y) defines a value of y x , i.e., the slope
of the integral curve that passes through this point In other words, the equation generates
a field of directions in the xy-plane From the geometrical point of view, the problem of
solving a first-order differential equation involves finding the curves, the slopes of which at each point coincide with the direction of the field at this point
Figure 12.1 depicts the tangent to an integral curve at a point (x0, y0); the slope of the integral curve at this point is determined by the right-hand side of equation (12.1.1.1):
tan α = f (x0, y0) The little lines show the field of tangents to the integral curves of the differential equation (12.1.1.1) at other points
12.1.1-2 Equations integrable by quadrature
To integrate a differential equation in closed form is to represent its solution in the form
of formulas written using a predefined bounded set of allowed functions and mathematical operations A solution is expressed as a quadrature if the set of allowed functions consists
of the elementary functions and the functions appearing in the equation and the allowed
* In what follows, we often call an ordinary differential equation a “differential equation” or, even shorter,
an “equation.”
453
Trang 4454 ORDINARYDIFFERENTIALEQUATIONS
D
α y
x x
y
0
0
O
Figure 12.1 The direction field of a differential equation and the integral curve passing through a point (x0, y0).
mathematical operations are the arithmetic operations, a finite number of function compo-sitions, and the indefinite integral An equation is said to be integrable by quadrature if its general solution can be expressed in terms of quadratures
12.1.1-3 Cauchy problem The uniqueness and existence theorems
The Cauchy problem: find a solution of equation (12.1.1.1) that satisfies the initial condition
where y0and x0are some numbers
Geometrical meaning of the Cauchy problem: find an integral curve of equation
(12.1.1.1) that passes through the point (x0, y0); see Fig 12.1
Condition (12.1.1.2) is alternatively written y(x0) = y0 or y|x=x0 = y0
THEOREM(EXISTENCE, PEANO) Let the function f (x, y) be continuous in an open domain D of the xy-plane Then there is at least one integral curve of equation (12.1.1.1) that passes through each point (x0, y0)D; each of these curves can be extended at both
ends up to the boundary of any closed domain D0 ⊂ D such that (x0, y0) belongs to the
interior of D0
THEOREM(UNIQUENESS) Let the function f (x, y) be continuous in an open domain D and have in D a bounded partial derivative with respect to y (or the Lipschitz condition
holds: |f (x, y) – f (x, z)| ≤M|y – z|, where M is some positive number) Then there is a
unique solution of equation (12.1.1.1) satisfying condition (12.1.1.2)
12.1.1-4 Equations not solved for the derivative The existence theorem
A first-order differential equation not solved for the derivative can generally be written as
F (x, y, y x) =0 (12.1.1.3)
THEOREM(EXISTENCE AND UNIQUENESS) There exists a unique solution y = y(x) of equation (12.1.1.3) satisfying the conditions y|x=x0 = y0 and y x |x=x0 = t0, where t0 is
one of the real roots of the equation F (x0, y0, t0) =0if the following conditions hold in a
neighborhood of the point (x0, y0, t0):
1 The function F (x, y, t) is continuous in each of the three arguments.
2 The partial derivative F t exists and is nonzero
3 There is a bounded partial derivative with respect to y, |F y| ≤M
The solution exists for |x – x0| ≤a, where a is a (sufficiently small) positive number.
Trang 512.1 F IRST -O RDER D IFFERENTIAL E QUATIONS 455
12.1.1-5 Singular solutions
1◦ A point (x, y) at which the uniqueness of the solution to equation (12.1.1.3) is violated
is called a singular point If conditions 1 and 3 of the existence and uniqueness theorem
hold, then
F (x, y, t) =0, F t (x, y, t) =0 (12.1.1.4)
simultaneously at each singular point Relations (12.1.1.4) define a t-discriminant curve in parametric form In some cases, the parameter t can be eliminated from (12.1.1.4) to give
an equation of this curve in implicit form, Ψ(x, y) =0 If a branch y = ψ(x) of the curve
Ψ(x, y) =0 consists of singular points and, at the same time, is an integral curve, then this
branch is called a singular integral curve and the function y = ψ(x) is a singular solution
of equation (12.1.1.3)
2◦ The singular solutions can be found by identifying the envelope of the family of integral
curves, Φ(x, y, C) =0, of equation (12.1.1.3) The envelope is part of the C-discriminant
curve, which is defined by the equations
Φ(x, y, C) =0, ΦC (x, y, C) =0
The branch of the C-discriminant curve at which
(a) there exist bounded partial derivatives, |Φx|< M1 and |Φy|< M2, and
(b) |Φx|+|Φy| ≠ 0
is the envelope
12.1.1-6 Point transformations
In the general case, a point transformation is defined by
x = F (X, Y ), y = G(X, Y ), (12.1.1.5)
where X is the new independent variable, Y = Y (X) is the new dependent variable, and
F and G are some (prescribed or unknown) functions.
The derivative y x under the point transformation (12.1.1.5) is calculated by
y
x = G X + G Y Y
X
F X + F Y Y X , where the subscripts X and Y denote the corresponding partial derivatives.
Transformation (12.1.1.5) is invertible if F X G Y – F Y G X ≠ 0
Point transformations are used to simplify equations and reduce them to known equa-tions Sometimes a point transformation allows the reduction of a nonlinear equation to a linear one
Example The hodograph transformation is an important example of a point transformation It is defined
by x = Y , y = X, which means that y is taken to be the independent variable and x the dependent one In this
case, the derivative is expressed as
y x = 1
X Y .
Other examples of point transformations can be found in Subsections 12.1.2 and 12.1.4– 12.1.6
Trang 6456 ORDINARYDIFFERENTIALEQUATIONS
12.1.2 Equations Solved for the Derivative Simplest Techniques of
Integration
12.1.2-1 Equations with separated or separable variables
1◦ An equation with separated variables (a separated equation) has the form
f (y)y x = g(x).
Equivalently, the equation can be rewritten as f (y) dy = g(x) dx (the right-hand side depends
on x alone and the left-hand side on y alone) The general solution can be obtained by
termwise integration:
f (y) dy = g (x) dx + C, where C is an arbitrary constant.
2◦ An equation with separable variables (a separable equation) is generally represented
by
f1(y)g1(x)y x = f2(y)g2(x).
Dividing the equation by f2(y)g1(x), one obtains a separated equation Integrating yields
f1(y)
f2(y) dy =
g2(x)
g1(x) dx + C.
lost.
12.1.2-2 Equation of the form y x = f (ax + by).
The substitution z = ax + by brings the equation to a separable equation, z x = bf (z) + a;
see Paragraph 12.1.2-1
12.1.2-3 Homogeneous equations y x = f (y/x).
1◦ A homogeneous equation remains the same under simultaneous scaling (dilatation) of the independent and dependent variables in accordance with the rule x → αx, y → αy,
where α is an arbitrary constant (α≠ 0) Such equations can be represented in the form
y
x = f
y
x
The substitution u = y/x brings a homogeneous equation to a separable one, xu x = f (u)–u;
see Paragraph 12.1.2-1
2◦ The equations of the form
y
x = f
a1x + b1y + c1
a2x + b2y + c2
can be reduced to a homogeneous equation To this end, for a1x + b1y≠k (a2x + b2y), one
should use the change of variables ξ = x – x0, η = y – y0, where the constants x0 and y0
are determined by solving the linear algebraic system
a1x0+ b1y0+ c1=0,
a2x0+ b2y0+ c2=0
Trang 712.1 F IRST -O RDER D IFFERENTIAL E QUATIONS 457
As a result, one arrives at the following equation for η = η(ξ):
η
ξ = f
a1ξ + b1η
a2ξ + b2η
On dividing the numerator and denominator of the argument of f by ξ, one obtains a homogeneous equation whose right-hand side is dependent on the ratio η/ξ only:
η
ξ = f
a1+ b1η/ξ
a2+ b2η/ξ
For a1x + b1y = k(a2x + b2y), see the equation of Paragraph 12.1.2-2
12.1.2-4 Generalized homogeneous equations and equations reducible to them
1◦ A generalized homogeneous equation (a homogeneous equation in the generalized
sense) remains the same under simultaneous scaling of the independent and dependent
variables in accordance with the rule x → αx, y → α k y , where α ≠ 0 is an arbitrary
constant and k is some number Such equations can be represented in the form
y
x = x k–1f (yx–k).
The substitution u = yx–k brings a generalized homogeneous equation to a separable
equation, xu x = f (u) – ku; see Paragraph 12.1.2-1.
Example Consider the equation
y x = ax2y4+ by2 (12 1 2 1 )
Let us perform the transformation x = α ¯x, y = α k ¯y and then multiply the resulting equation by α1 –kto obtain
¯y
¯
x = aα3(k+1)¯x2¯y4+ bα k+1 ¯y2 (12 1 2 2 )
It is apparent that if k = –1 , the transformed equation (12.1.2.2) is the same as the original one, up to notation.
This means that equation (12.1.2.1) is generalized homogeneous of degree k = –1 Therefore the substitution
u = xy brings it to a separable equation: xu x = au4+ bu2+ u.
2◦ The equations of the form
y
x = yf (e λx y)
can be reduced to a generalized homogeneous equation To this end, one should use the
change of variable z = e x and set λ = –k.
12.1.2-5 Linear equation y x + f (x)y = g(x).
A first-order linear equation is written as
y
x + f (x)y = g(x).
The solution is sought in the product form y = uv, where v = v(x) is any function that satisfies the “truncated” equation v x + f (x)v =0 [as v(x) one takes the particular solution
v = e–F , where F =7
f (x) dx] As a result, one obtains the following separable equation for u = u(x): v(x)u x = g(x) Integrating it yields the general solution:
y (x) = e–F
e F g (x) dx + C
, F = f (x) dx, where C is an arbitrary constant.