共查询到20条相似文献，搜索用时 15 毫秒
1.
Alfred Landé 《Journal of The Franklin Institute》1940,229(6):767-774
In order to quantize Dirac's classical point electron1 we supplement Einstein's classical equation (E/c)2 ? p2 = b2 with a reciprocal classical equation (CΔt)2 ? (Δr)2 = a2 where b = mc and a is Dirac's signal radius. Δt is the time saved by a light signal in various states of motion of the electron, and a/c is the rest time saved. Our former efforts2 of obtaining an integral equation for the probability amplitude have been rectified by Born.3 There is no solution of the integral equation, however, unless advanced and retarded phases are introduced simultaneously, along with Dirac's advanced and retarded potentials. We have obtained a transcendental equation for the eigen-value μ = αγ where α is the Sommerfeld fine-structure constant, and γ is the numerical factor in Dirac's signal radius a = γe2/mc2. The smallest eigen-value is μ = 0.0299.That is, ab = hγ = h/210. 相似文献
2.
Walter T. White 《Journal of The Franklin Institute》1948,245(1):25-36
In this paper integral equations are applied for the calculation of the normal modes of vibrating beams. Both exact and approximate methods of solving the integral equation are considered. The Green's function, or kernel, of the integral equation is constructed for both uniform and nonuniform beams. Solutions for the normal modes of a uniform cantilever are given. A nonuniform, naturally-twisted turbine blade is studied in detail and the first and second normal modes are calculated by the integral-equation method. 相似文献
3.
Changqing Yang 《Journal of The Franklin Institute》2012,349(3):947-956
In the current work, the Chebyshev collocation method is adopted to find an approximate solution for nonlinear integral equations. Properties of the Chebyshev polynomials and operational matrix are used in the integral equation of a system consisting of nonlinear algebraic equations with the unknown Chebyshev coefficients. Numerical examples are presented to illustrate the method and results are discussed. 相似文献
4.
《Journal of The Franklin Institute》2022,359(14):7540-7561
In this paper, we first introduce the necessary and sufficient conditions for the existence of the solution of discrete algebraic Riccati equation. Then we propose the Newton method without inversion to find the solution of the discrete algebraic Riccati equation. We show that the proposed method converges to a positive definite solution of the discrete algebraic Riccati equation. Finally, the accuracy and effectiveness of the proposed method in compare to some existing algorithms are demonstrated by various numerical examples. 相似文献
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A new method is employed to identify the unknown parameters of a bilinear system. This method expands the system input and output by block pulse functions and reduces the original identification problem to an algebraic form. Furthermore, the dyad formed by block pulse functions and its integral are in diagonal forms, whereas the integration of the “triple-product” matrix can be reduced to the upper triangular form. Consequently, only very few calculations are required to find the solution for the algebraic equation. Two examples are given to show that the use of this method is considerably more economical in computation time than the use of Walsh function expansion. 相似文献
7.
There are few techniques available to numerically solve linear Fredholm integrodifferential-difference equation of high-order. In this paper we show that the Taylor matrix method is a very effective tool in numerically solving such problems. This method transforms the equation and the given conditions into the matrix equations. By merging these results, a new matrix equation which corresponds to a system of linear algebraic equation is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the differential, difference, differential-difference and Fredholm integral equations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method. 相似文献
8.
By use of the multiple Laplace transform a partial differential equation and its associated boundary conditions characterizing a boundary value problem in n independent real variables can be transferred directly into an algebraic equation in n independent complex variables. This algebraic equation can be solved for the multiple transform of the solution of the boundary value problem. Multiple inversion of this transform then gives the desired solution. The general theory underlying such solution of boundary value problems in two and three independent variables is advanced in detail. Use of this theory is illustrated by solution of two specific problems. 相似文献
9.
Bellman's dynamic programming equation for the optimal index and control law for stochastic control problems is a parabolic or elliptic partial differential equation frequently defined in an unbounded domain. Existing methods of solution require bounded domain approximations, the application of singular perturbation techniques or Monte Carlo simulation procedures.In this paper, using the fact that Poisson impulse noise tends to a Gaussian process under certain limiting conditions, a method which achieves an arbitrarily good approximate solution to the stochastic control problem is given. The method uses the two iterative techniques of successive approximation and quasi-linearization and is inherently more efficient than existing methods of solution. 相似文献
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12.
Raimond A. Struble 《Journal of The Franklin Institute》1978,306(1):63-76
It has been tacitly understood that Fourier's representation theorem applies only to a very restricted class of functions. On the contrary, by adopting an appropriate algebraic viewpoint, it is explained in this article how the theorem can be readily extended to include all (non-pathological) functions. 相似文献
13.
James William Dunion Arjan S. Mirchandaney Domina Eberle Spencer 《Journal of The Franklin Institute》1984,318(6):393-406
A solution to the interflection problem in rooms lit by peaked ceilings is investigated. The integral equation for interflections in the peaked ceiling is formulated. A method of solving the integral equation by employing linear approximations is developed. This shows that uniform incident light on the peaked ceiling will result in nonuniformity. An equation is derived which gives the degree of nonuniformity in the incident light required for a uniform ceiling. The actual incident distribution of light from fluorescent lamps is expressed simply in prolate spheroidal coordinates. 相似文献
14.
Edmond Ghandour 《Journal of The Franklin Institute》1977,303(1):67-73
An algorithm is suggested for the determination of an effective speed of propagation for a random medium whose mean speed of propagation is perturbed by a small random term of the white-noise type. We study the propagation of disturbances in the limiting geometrical optics case, and pursue the analysis within the framework of stochastic optimal control. The statement of an appropriate Fermat's principle for the random medium and invoking the principle of dynamic programming lead to a non-linear elliptic equation—the classical Eikonal equation—perturbed by terms associated with the stochastics. This equation and estimates for the departure of the stochastic trajectories from the free-space geodesic path of propagation enable us to calculate the approximate speed. 相似文献
15.
The uniqueness of the minimal solution of the matrix equation A(λ)X(λ)+Y(λ)B(λ)=C(λ) is discussed in this paper. It is proved that Barnett's condition for uniqueness can be obtained when only A(λ) or B(λ) (not necessarily both) is regular. Furthermore, it is shown that if such a minimal solution exists, it is not unique if and only if both matrices are nonregular. 相似文献
16.
This paper deals with the computation of the values of two functionals which are defined over the sample paths of a randomly rotating rigid body. It is assumed that the body is subjected to two different kinds of perturbation. The first kind of perturbation is represented by the standard Wiener process and the second kind by a homogeneous process with independent increments, finite second-order moments, mean zero and no continuous sample functions. In order to measure quantitatively the stochastic stability of the body's motion, two functionals are defined over its sample paths. It is shown that each of these functionals is a solution to a corresponding partial integro-differential equation. A numerical procedure for the solution of these equations is suggested, and its efficiency and applicability are demonstrated with examples. 相似文献
17.
The difficulties in solving Fredholm integral equations of the first kind are well known. A classical method has been to convert the equation into a set of m linear algebraic equations in n unknowns (m?n). For computational convenience, it is customary to force m = n and solve the resulting ill-conditioned system using one technique or other. In the general case, a feasible solution, if it exists, can be found by determining the generalized inverse of the coefficient matrix. One method of finding the generalized inverse is to reformulate the problem and observe the steady state response of a system of ordinary differential equations with prescribed initial conditions. Results obtained from this reformulation are found to be comparable in quality to those obtained earlier by other methods. Analog and digital computer implementations are discussed. 相似文献
18.
F.L. N-Nagy 《Journal of The Franklin Institute》1983,315(1):49-60
This paper describes the derivation of Heaviside's decomposition in matrix frame presentation. For simplicity, the response transform having one multiple pole and any number of distinct poles is considered. There is no restriction on the number and the order of zeros. The aim is to avoid the repeated differentiations in the algorithm for the computation of both residue and constituent coefficients about the multiple pole. This will be referred to as the RRM (Residue and Resolvent Matrices) method and adopted to time response derivation of dynamic systems. The RRM method is also applicable as a mathematical catalyst to the algebraic solution of differentiation and integration of complex functions in the s domain used for system optimisation. 相似文献
19.
M.H. Heydari 《Journal of The Franklin Institute》2019,356(15):8216-8236
The main goal of this study is to develop an efficient matrix approach for a new class of nonlinear 2D optimal control problems (OCPs) affected by variable-order fractional dynamical systems. The offered approach is established upon the shifted Chebyshev polynomials (SCPs) and their operational matrices. Through the way, a new operational matrix (OM) of variable-order fractional derivative is derived for the mentioned polynomials.The necessary optimality conditions are reduced to algebraic systems of equations by using the SCPs expansions of the state and control variables, and applying the method of constrained extrema. More precisely, the state and control variables are expanded in components of the SCPs with undetermined coefficients. Then these expansions are substituted in the cost functional and the 2D Gauss-Legendre quadrature rule is utilized to compute the double integral and consequently achieve a nonlinear algebraic equation.After that, the generated OM is employed to extract some algebraic equations from the approximated fractional dynamical system. Finally, the procedure of the constrained extremum is used by coupling the algebraic constraints yielded from the dynamical system and the initial and boundary conditions with the algebraic equation extracted from the cost functional by a set of unknown Lagrange multipliers. The method is established for three various types of boundary conditions.The precision of the proposed approach is examined through various types of test examples.Numerical simulations confirm the suggested approach is very accurate to provide satisfactory results. 相似文献
20.
It is shown that Youla's theory of broadband matching between a passive load and a resistive generator with a preassigned transducer power gain characteristic is completely equivalent to Wohlers' solution to the problem of compatible impedences, which transforms a given passive impedance by a lossless coupling network into another specified one. 相似文献