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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
One of the most intriguing things about differential equations is that for an arbitrary function f, there is no general method for finding an exact formula for the solution. For many differential equations that are encountered in real-world applications, it is impossible to express their solutions via known functions. Generally speaking, every differential equation defines its solution (if it exists) as a special function not necessarily expressible by elementary functions (such as polynomial, exponential, or trigonometric functions). Only exceptional differential equations can be explicitly or implicitly integrated. For instance, such “simple” differential equations as y′=y2−x or y′=exy cannot be solved by available methods.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
One of the most intriguing things about differential equations is that for an arbitrary function f, there is no general method for finding an exact formula for the solution. For many differential equations that are encountered in real-world applications, it is impossible to express their solutions via known functions. Generally speaking, every differential equation defines its solution (if it exists) as a special function not necessarily expressible by elementary functions (such as polynomial, exponential, or trigonometric functions). Only exceptional differential equations can be explicitly or implicitly integrated. For instance, such "simple" differential equations as y'=y2-x or y'=exy cannot be solved by available methods.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
One of the most intriguing things about differential equations is that for an arbitrary function, there is no general method for finding an exact formula for the solution. For many differential equations that are encountered in real-world applications, it is impossible to express their solutions via known functions. Generally speaking, every differential equation defines its solution (if it exists) as a special function not necessarily expressible by elementary functions (such as polynomial, exponential, or trigonometric functions). Only exceptional differential equations can be explicitly or implicitly integrated. For instance, such “simple” differential equations as y′ = y2 - x or y’ = exy cannot be solved by available methods.
A Multigroup Homogeneous Flux Reconstruction Method Based on the ANOVA-HDMR Decomposition
Published in Nuclear Science and Engineering, 2023
Pavel M. Bokov, Danniell Botes, Rian H. Prinsloo, Djordje I. Tomašević
where the special function is the hyperbolic analog of the sine cardinal function and . Coefficients , , , and are then determined by requiring that the univariate function , corresponding to the transverse-integrated flux, interpolates the side-averaged fluxes, , and side-averaged currents in direction .
Optimal stocking policies for inventory systems with uncertain returns
Published in International Journal of Production Research, 2022
By differentiating both sides of (2) with respect to , we get a first-order differential equation to solve for . It can then easily be verified that a general solution to (2) can be given by: where is the exponential integral, a special function defined as (see Abramowitz and Stegun 1964) and is a constant to be determined. In general, the function does not admit a closed-form expression; nonetheless, it can be evaluated readily in many computational software packages such as Maple.
Sum of alternating-like series as definite integrals
Published in International Journal of Mathematical Education in Science and Technology, 2022
The attentive reader may wonder if it is possible to evaluate all previous integrals in a closed form. In fact, the examples considered so far suggest this idea with formulas involving logarithms and π. However, it is evidenced that, when k increases, the calculations to find a primitive for these rational functions are considerably more difficult. Also, the success of simplifying the answer depends on the known values of the trigonometric functions. Despite these obstacles, there is a way to express the integrals in terms of the famous digamma functionψ that goes back to Gauss (Andrews et al., 1999). This special function is defined by the series where is the Euler–Mascheroni constant Note that formula (12) converges absolutely for any such x: fixing an integer N>0, if , then and which is the general term of a convergent series independent of x. For the present work, we require the following theorem.