Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.6 GENERAL EIGENVALUE PROBLEMS considerK=I−λJgiven by K= ∫b a [ py′ 2 −(q+λρ)y^2 ] dx. On application of the EL equation (22.5 ...

CALCULUS OF VARIATIONS
Show that ∫b a ( y′jpy′i−yjqyi ) dx=λiδij. (22.27) Letyibe an eigenfunction of (22.24), corresponding to ...

22.7 ESTIMATION OF EIGENVALUES AND EIGENFUNCTIONS λmaxis finite, orλmax=∞andλminis finite. Notice that here we have departed fro ...

CALCULUS OF VARIATIONS (a) (b) (c) (d) 0. 2 0. 2 0. 4 0. 4 0. 6 0. 6 0. 8 0. 8 1 1 x y(x) Figure 22.10 Trial solutions usedto es ...

22.8 Adjustment of parameters It is easily verified that functions (b), (c) and (d) all satisfy (22.30) but, so far as mimicking ...

CALCULUS OF VARIATIONS operatorHis called the Hamiltonian and for a particle of massmmoving in a one-dimensional harmonic oscill ...

22.9 Exercises 22.9 Exercises 22.1 A surface of revolution, whose equation in cylindrical polar coordinates isρ= ρ(z), is bounde ...

CALCULUS OF VARIATIONS 22.8 Derive the differential equations for the plane-polar coordinates,randφ,ofa particle of unit mass mo ...

22.9 EXERCISES path of a small test particle is such as to make ∫ dsstationary, find two first integrals of the equations of mot ...

CALCULUS OF VARIATIONS 22.23 For the boundary conditions given below, obtain a functional Λ(y) whose sta- tionary values give th ...

22.10 Hints and answers total energy (per unit depth) of the film consists of its surface energy and its gravitational energy, a ...

CALCULUS OF VARIATIONS 22.5 (a)∂x/∂t=0andso ̇x= ∑ i ̇qi∂x/∂qi; (b) use ∑ i ̇qi d dt ( ∂T ∂ ̇qi ) = d dt (2T)− ∑ i ̈qi ∂T ∂ ̇qi . ...

23 Integral equations It is not unusual in the analysis of a physical system to encounter an equation in which an unknown but re ...

INTEGRAL EQUATIONS We shall illustrate the principles involved by considering the differential equa- tion y′′(x)=f(x, y), (23.1) ...

23.3 Operator notation and the existence of solutions In fact, we shall be concerned with various special cases of (23.4), which ...

INTEGRAL EQUATIONS inhomogeneous Fredholm equation of the first kind may be written as 0=f+λKy, which has the unique solutiony=− ...

23.4 CLOSED-FORM SOLUTIONS 23.4.1 Separable kernels The most straightforward integral equations to solve are Fredholm equations ...

INTEGRAL EQUATIONS These two simultaneous linear equations may be straightforwardly solved forc 1 andc 2 to give c 1 = 24 +λ 72 ...

23.4 CLOSED-FORM SOLUTIONS 23.4.2 Integral transform methods If the kernel of an integral equation can be written as a function ...

INTEGRAL EQUATIONS Thus, using (23.18), we find the Fourier transform of the solution to be ̃y(k)= { ̃f(k)/(1−πλ)if|k|≤1, ̃f(k)i ...