We describe how Gröbner bases can be used to solve the reduction problem for Feynman integrals, i.e. to construct an algorithm that provides the possibility to express a Feynman integral of a given family as a linear combination of some master integrals. Our approach is based on a generalized Buchberger algorithm for constructing Gröbner-type bases associated with polynomials of shift operators...

We consider saddle point integrals in d variables whose phase functions are neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The proofs are via contour shifting in comp...

Note that the system also admits the independent first integral (c12u1− c23u3)/2, and the dynamical system can be reconstructed using Nambu mechanics. If c 6= 0, then I1, I2 are not first integrals anymore. Now many dissipative dynamical systems such as the Lorenz model and the Rikitake-two disc dynamo admit, depending on the bifurcation parameters, explicitly time-dependent first integrals. Th...

In view of the extensive use of stochastic integrals and stochastic differential equations in modeling of systems in engineering, and economic systems especially in mathematical finance and other applied problems, it is necessary to find whether there are good approximants to the stochastic integrals and the stochastic differential equations which can be used for simulation purposes. Some work ...

In this article, we provide constraints for the sum by employing a generalized modified form of fractional integrals Riemann-type via convex functions. The mean inequalities functions with absolute value derivatives are discovered. Hermite–Hadamard-type symmetric function explored. These results achieved using fresh and innovative methodology integrals. Some applications explored in paper brief...

In this article, under certain parametric conditions, we study the first integrals of the Duffing-van der Pol-type oscillator equations which include the van der Pol and the Duffing oscillator systems, as particular cases. After making a series of variable transformations and applying the PrellerSinger method, we find the first integrals of the simplified equations without complicated calculati...

In this paper, quadratic rules for obtaining approximate solution of definite integrals as well as single and double integrals using spline quasi-interpolants will be illustrated. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.