Operational calculus approach to nonlocal Cauchy problems
\textit{I. Dimovski} published in 1990 his Convolutional calculus [2nd ed. Mathematics and Its Applications: East European Series, 43. Dordrecht etc.: Kluwer Academic Publishers (1990; Zbl 0685.44006)] and introduced the right inverse operator \(L\), defined by the solution \(y= Lf(x)\) of the elementary boundary value problem \(y'= f(x)\) and \(\Phi(y)= 0\) with a given non-local functional \(\Phi\) in the space \(C(\Delta)\) of continuous functions over the interval \(\Delta\) in \({\mathfrak R}\). With the convolution operator \(L(f)=\{1\}*f\), the convolution algebra \((C^*)\), and the ring \({\mathfrak M}\) of the convolution fractions \(f/g\) with \(f\in C\) and \(g\in D\), the non-empty subset of \(C\) of all nonzero non-divisors of \(0\) in \(C^*\), the most important element in \({\mathfrak M}\) is \(S= 1/\{1\}\), the algebraic inversion of \(L= \{1\}^*\). After the relationship between the derivative \(f'\) and the product \(Sf\) is established, the operational method for solving the elementary non-local Cauchy problem is extended to the general case: \(P(d/dt)y= F\) with \(P(\lambda)= a_0\lambda^n+ a_1\lambda^{n-1}+\cdots+ a_{n-1} \lambda+ a_n\) and \(\Phi\{y^{(k)}\}= \alpha_k\) with \(k= 0,1,2,\dots, n-1\), or to \(P(S)= F+ Q(S)\), equivalent to the single algebraic equation in the ring \({\mathfrak M}\) with \[ Q(S)= \sum^n_{J=1} \Biggl(\sum^n_{k=i} \alpha_{n-k} \alpha_{k-j}\Biggr)S^{j-1}. \] The solution can be written as \(y= F/P(S)+ Q(S)/P(S)\). The non-resonance case, where \(P(S)\) is a non-divisor of \(0\) in \({\mathfrak M}\), is further developed here, including the extension of the classical Duhamel principle. The resonance case, where \(P(S)\) is a divisor of \(0\) in \({\mathfrak M}\), will be the subiect of a later publication, where a new table of interpretation formulae will be included. Two examples, the \(\pi\)-periodic solution of \[ y^{(6)}+ 2y^{(4)}+ y''= F(t)\text{ with }F(t+\pi)= F(t)\quad\text{ and }\quad\int^\pi_0 F(\lambda)\,d\lambda= 0, \] and the \(\pi\)-antiperiodic solution of \(y^{(4)}- 4y= F(t)\) with \(F(t+\pi)= - F(t)\), illustrate the implementation of the algorithm in the program environment of the computer algebra system Mathematica.
scientific article; zbMATH DE number 5905255