Differential-equations sentence example
The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work.
During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential Equations much more complete than the first edition; and part of his last vacation was spent in the libraries of the Royal Society and the British Museum.
Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
It has been shown above that a covariant, in general, satisfies four partial differential equations.Advertisement
By considering only the particles of air found in a right line, he reduced the problem of the propagation of sound to the solution of the same partial differential equations that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject.
Laplace owned that he had despaired of effecting the integration of the differential equations relative to secular inequalities until Lagrange showed him the way.
To Lagrange, perhaps more than to any other, the theory of differential equations is indebted for its position as a science, rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the beautiful formula of interpolation which bears his name; although substantially the same result seems to have been previously obtained by Euler.
Instead of following the motion of each individual part of a material system, he showed that, if we determine its configuration by a sufficient number of variables, whose number is that of the degrees of freedom to move (there being as many equations as the system has degrees of freedom), the kinetic and potential energies of the system can be expressed in terms of these, and the differential equations of motion thence deduced by simple differentiation.
He also showed that every equation of an even degree must have at least one real quadratic factor, reduced the solution of linear differential equations to definite integrals, and furnished an elegant method by which the linear partial differential equation of the second order might be solved.Advertisement
He published a memoir on the integration of partial differential equations and a few others, which have not been noticed above, but they relate to subjects with which his name is not especially associated.
Pfaff's researches bore chiefly on the theory of series, to which he applied the methods of the so-called combinatorial school of German mathematicians, and on the solution of differential equations.
Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit.
He investigated the problem by means of the general differential equations of static equilibrium for dams of triangular and rectangular form considered as isotropic elastic solids.
Jacobi and other mathematicians have developed to a great extent, and as a question of pure mathematics only, Hamilton's processes, and have thus made extensive additions to our knowledge of differential equations.Advertisement
Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of differential equations which constantly occur in the treatment of physical questions, only a few items have been published, at intervals, in the Philosophical Magazine.
In the actual problems of celestial mechanics three co-ordinates necessarily enter, leading to three differential equations and six equations of solution.
In a case like the present one, where there are two differential equations of the second order, there will be four such constants.
After the establishment of universal gravitation as the primary law of the celestial motions, the problem was reduced to that of integrating the differential equations of the moon's motion, and testing the completeness of the results by comparison with observation.
The analytic method sought to express the moon's motion by integrating the differential equations of the dynamical theory.Advertisement
By the second general method the moon's co-ordinates are obtained in terms of the time by the direct integration of the differential equations of motion, retaining as algebraic symbols the values of the various elements.
His treatises and contributions to scientific journals (to the number of 789) contain investigations on the theory of series (where he developed with perspicuous skill the notion of convergency), on the theory of numbers and complex quantities, the theory of groups and substitutions, the theory of functions, differential equations and determinants.
These powerful and effective tools are used to solve many problems involving differential equations.
These systems can be described by a set of ordinary differential equations (ODEs) or map.
Stochastic differential equations driven by a Wiener process are studied.Advertisement
Thus, in total there are six partial differential equations to be solved.
For most of the nonlinear partial differential equations which we have tested, the running time is no more than 10 seconds.
Hamiltonian theory is an important element of integrable systems, whether discrete, ordinary differential or partial differential equations.
In the 16th and 17th chapters of the Differential Equations we find, for instance, a lucid account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844.
An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties.Advertisement
It is amusing to find him speaking jubilantly of the unexpectedly large audience of eight which assembled to hear his first lecture (in 1854) on partial differential equations and their application to physical problems.
The results are applied to a class of retarded delay differential equations.
The concept of instability of a basic state is first introduced using models which yield simple ordinary differential equations.
Detailed analysis may be undertaken in a manner similar to that for the first order ordinary differential equations.