\(C^1\)-approximate solutions of second-order singular ordinary differential equations (Q2786015)

A \(C^1\)-approximation is constructed to the solution of the differential equation NEWLINENEWLINE\[NEWLINEx''(t)+a(t;p)x'(t)+b(t;p)x(t)+f(t,x(t);p)=0\tag{1}NEWLINE\]NEWLINENEWLINEsatisfying either the initial conditions NEWLINENEWLINE\[NEWLINEx(0;p)=x_0(p),\;x'(0;p)=\widehat x_0(p)\tag{2}NEWLINE\]NEWLINENEWLINEor the boundary conditionsNEWLINENEWLINE\[NEWLINEx(0;p)=x_0(p),\;x(1;p)=x_1(p).\tag{3}NEWLINE\]NEWLINENEWLINEHere, for all large values of the parameter \(p\), the function \(f(t,x,p)\) is continuous, \(a(t;p)\in C^1(I),\;b(t;p)\in C^2(I)\), where \(I=[0,T_0)\), and there is \(\theta>0\) such that \(|b(t;p)|\geq\theta\) for \(t\in[0,\infty).\)NEWLINENEWLINEIn the case \(b(t;p)>0\) for \(t\in[0,\infty)\) and large \(p\), it is supposed for the IVP (1), (2) thatNEWLINENEWLINE\[NEWLINEx_0,\widehat x_0\in A_E=\{h:[0,\infty)\to\mathbb R:\exists b\in\mathbb R \text{ such that }\lim_{p\to+\infty}\sup(E(p))^b|h(p)|<\infty\},NEWLINE\]NEWLINE NEWLINEwhere \(E(p)>0\), \(p\geq 0,\) is a suitable function, as well as that for large \(p\)NEWLINENEWLINE\[NEWLINEa(t;p)\geq0\;\;\text{for}\;\;t\in[0,\infty)NEWLINE\]NEWLINENEWLINEandNEWLINENEWLINE\[NEWLINEa(t;p),b(t;p)\in A_E \text{ for }t \text{ in compact subsets of } (0,\infty).NEWLINE\]NEWLINENEWLINEThe approximate solution is defined by using a solution of an IVP for a differential equation with constant coefficients. This new IVP is obtained by a suitable transformation of (1), (2). The error function and its first derivative are estimated under the assumptions that there exist functions \(\Phi_j\in A_E,\;j=1,2,\dots,5,\) such thatNEWLINENEWLINE\[NEWLINE|b'(t;p)|^2\leq \Phi_1(p)|b(t;p)|^3,\quad |b''(t;p)|^2\leq \Phi_2(p)|b(t;p)|^2,\quad |a(t;p)|^2\leq \Phi_3(p)|b(t;p)|,NEWLINE\]NEWLINENEWLINENEWLINE\[NEWLINE|a'(t;p)|\leq \Phi_4(p)|b(t;p)|,\quad |f(t,z,p)|\leq \Phi_5(p)|zb(t;p)|NEWLINE\]NEWLINENEWLINEfor all \(t\in[0,T),\;z\in\mathbb R\) and large \(p\).NEWLINENEWLINEThe \(C^1\)-approximate solution of IVP (1),(2) in the case \(b(t;p)<0\) for \(t\in[0,\infty)\) and large \(p\) as well as the \(C^1\)-approximate solution of BVP (1),(3) are constructed similarly.