The abstract nonlocal boundary value problem <Equation ID="Equ131"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{l} {\varepsilon }^{2}u^{^{\prime \prime }}\left( t\right) +Au(t)=f(t), 0<t<T,\\ u(0)=\alpha u(T)+\varphi , u^{^{\prime }}(0)=\beta u^{^{\prime }}(T)+\psi \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msup> <mrow> <mi>ε</mi> </mrow> <mn>2</mn> </msup> <mmultiscripts> <mi>u</mi> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>″</mo> </mmultiscripts> </mmultiscripts> <mfenced close=")" open="("> <mi>t</mi> </mfenced> <mo>+</mo> <mi>A</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mn>0</mn> <mo><</mo> <mi>t</mi> <mo><</mo> <mi>T</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>φ</mi> <mo>,</mo> <mmultiscripts> <mi>u</mi> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>β</mi> <mmultiscripts> <mi>u</mi> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>ψ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for hyperbolic equations in a Hilbert space H, where A is a self-adjoint positive definite operator and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon \in \left( 0,\infty \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mi>∞</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is a parameter multiplying the highest-order derivative term, is considered. An asymptotic formula for the solution of this problem with a small <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> parameter is established. The high order of accuracy two-step uniform difference scheme for the solution of this problem is presented. The convergence estimates for the solution of the difference scheme are obtained.

ON UNIFORM DIFFERENCE SCHEMES AND ASYMPTOTIC FORMULAS FOR THE SOLUTION OF HYPERBOLIC NONLOCAL BOUNDARY VALUE PERTURBATION PROBLEMS

期刊论文
发表日期：

2026年3月5日

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Allaberen Ashyralyev,
Ozgur Yildirim

摘要

The abstract nonlocal boundary value problem $\begin{aligned} \left\{ \begin{array}{l} {\varepsilon }^{2}u^{^{\prime \prime }}\left( t\right) +Au(t)=f(t), 0<t<T,\\ u(0)=\alpha u(T)+\varphi , u^{^{\prime }}(0)=\beta u^{^{\prime }}(T)+\psi \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} ε^{2} u^{^{″}} (t) + A u (t) = f (t), 0 < t < T, \\ u (0) = α u (T) + φ, u^{^{'}} (0) = β u^{^{'}} (T) + ψ \end{matrix}) \end{matrix}$ for hyperbolic equations in a Hilbert space H, where A is a self-adjoint positive definite operator and $\varepsilon \in \left( 0,\infty \right) $ $ε \in (0, \infty)$ is a parameter multiplying the highest-order derivative term, is considered. An asymptotic formula for the solution of this problem with a small $\varepsilon $ $ε$ parameter is established. The high order of accuracy two-step uniform difference scheme for the solution of this problem is presented. The convergence estimates for the solution of the difference scheme are obtained.