The aim of this paper is to investigate the numerical solutions of delayed fractional differential equations (FDEs). A constant time delay is taken in the Caputo derivative of the state variable and also in the state variable. To obtain solutions, we extend the block boundary value method (BBVM) and show the convergence of the resulting scheme. Further, it is found that the order of convergence is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\min \{m,q-\delta +1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>q</mi> <mo>-</mo> <mi>δ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, where m and q are the order and block-size of the BBVM, respectively, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> lies in between 0 and 1. In our methodology, we approximate the fractional order derivative (FoD) by a combination of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m^{\text {th}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>m</mi> <mtext>th</mtext> </msup> </math></EquationSource> </InlineEquation>-order BBVM and a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q^{\text {th}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mtext>th</mtext> </msup> </math></EquationSource> </InlineEquation>-order Lagrange interpolating polynomial. Lastly, we analyze the global stability of the numerical scheme and with the aid of some numerical examples, its computational effectiveness is proved. In examples, we analyze some fractional order delay differential systems and present a comparison of solutions for different FoDs. Furthermore, it is demonstrated that the global errors diminish as the FoD decreases, and this fact is supported by the theoretical order of convergence.

Numerical solutions, convergence and global stability of delayed differential system with delay in the state variable of the Caputo derivative

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2026年7月20日

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Abhishek Sharma,
Harendra Pal Singh,
Surendra Kumar

摘要

The aim of this paper is to investigate the numerical solutions of delayed fractional differential equations (FDEs). A constant time delay is taken in the Caputo derivative of the state variable and also in the state variable. To obtain solutions, we extend the block boundary value method (BBVM) and show the convergence of the resulting scheme. Further, it is found that the order of convergence is $\min \{m,q-\delta +1\}$ $min {m, q - δ + 1}$ , where m and q are the order and block-size of the BBVM, respectively, and $\delta $ $δ$ lies in between 0 and 1. In our methodology, we approximate the fractional order derivative (FoD) by a combination of the $m^{\text {th}}$ $m^{th}$ -order BBVM and a $q^{\text {th}}$ $q^{th}$ -order Lagrange interpolating polynomial. Lastly, we analyze the global stability of the numerical scheme and with the aid of some numerical examples, its computational effectiveness is proved. In examples, we analyze some fractional order delay differential systems and present a comparison of solutions for different FoDs. Furthermore, it is demonstrated that the global errors diminish as the FoD decreases, and this fact is supported by the theoretical order of convergence.