SOLVING FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS IN HEAT CONDUCTION AND BIOLOGICAL MODELING

Аннотация

This article analyzes the method of solving fractional integro-differential equations using the Neumann series. The convergence of the solution for equations based on the Caputo fractional derivative is proven by applying the Banach fixed-point theorem. The efficiency of the method is demonstrated through numerical examples and is applied to problems of heat conduction and biological population dynamics. The results are of significant importance in modern materials science and biology, and contribute to the advancement of fractional mathematics within thescientificcommunity ofUzbekistan. Keywords: fractional integro-differential equations, Neumann series, Caputo derivative, convergence, heat conduction, biological modeling. Introduction.In recent years, fractional differential and integro-differential equations have taken on an important role in mathematics and applied sciences [1, 2]. They are used to model memory-effect processes, such as anomalous diffusion, viscoelastic materials, biological population dynamics, and signal transmission [3, 4]. In Uzbekistan’s mathematical school, research on fractional equations is evolving, particularly through the works of Kadirkulov and Khudaybergenov [5]. Compared to classical differential equations, fractional equations provide a more accurate description of the behavior of complex systems. However, obtaining analytical solutions for such equations is challenging; hence, iterative methods, especially the Neumann series, are widely employed [6]. The Neumann series is an effective tool for solving integral equations, and its convergence isjustified through theBanach fixed-point theorem [7]. This article examines the method of solving fractional integro-differential equations via the Neumann series. The convergence of the solution is analyzed, and the method is applied to heat conduction and biological population dynamics problems. The aim of this paper is to demonstrate the effectiveness of the method, verify it through numerical simulations, and contributeto thedevelopment of fractional mathematics in thescientific contextofUzbekistan. Definition1.The Caputo fractional derivative oforder α\alphaαis defined as follows: t 1 Dau(t) = (t -s)-au'(s)ds Г(1-a) 0 where Г(z) = e-ttz-1dt is thegammafunction [1]. 0 Definition2.An integral equation of theform 280 Volume 4, issue 6, 2025 t u(t) = f (t)+ K(t,s)Dau(s)ds , t [ 0,T ] 0 in which the upper limit of integration depends on a free variable, is called a Volterra integral equation of the second kind. Here, u(t) is the unknown function to be determined, f (t) is a given function, K(t,s) is the kernel [ ] [ ] [ ] function ,where f (t) C 0,T ,K(t,s) C( 0,T 0,T ). Definition3.The Neumann series is an iterative method forsolving integral equations andis defined as: t u (t) = f (t), u (t) = f (t)+ K(t,s)Dau (s)ds , n =0,1,2,....... 0 n+1 n 0 If the convergence conditions are satisfied, thesolutioncan beexpressed as: u(t)=lim u (t) [7]. n® n Theorem1.(Banach Fixed-PointTheorem). If theoperator t A:u ® f (t)+ K(t,s)Da(s)ds 0 is a contraction, that is, Au - Av q u -v , q <1 hen the equation has a unique solution, andtheNeumann series converges to thissolution[8]. Now, let us consider the application of the Neumann series to fractional integro-differential equations and analyze the convergence of the solution. Let us examinethefollowingequation: t u(t) = f (t)+ K(t,s)Dau(s)ds ,t [ 0,T ] 0 here, f (t) C [ 0,T ] ,K(t,s) C( [ 0,T ] [ 0,T ] ) and Da is theCaputo fractional derivative. Theoperatoris defined as follows: t Au(t) = f (t)+ K(t,s)Dau(s)ds 0 TheNeumann series iterations are given by:u (t) = f (t), 0 t u (t) = f (t)+ K(t,s)Dau (s)ds n+1 n 0 281 Volume 4, issue 6, 2025 To demonstratethat theoperator Aiscontractive, weverify thefollowing condition: t Au - Av = K(t,s)-Da(u(s)-v(s))ds 0 Regarding theproperties oftheCaputo derivative: T1-a Da(u-v) u-v Г(2-a) t T1-a If sup K(t,s)ds K then Au- Av K u-v =q u-v , t [0,T] 0 0 Г(2-a) 0 T1-a where q = K <1.Therefore, the operator A is a contraction, and by Banach's 0 Г(2-a) fixed-point theorem, theNeumann series converges toa uniquesolution. Example. Let us consider the equation for the parameters f (t) =t2 , K(t,s) =es-t , a=0,5,T =1. t u(t)=t2 + es-tD0,5u(s)ds 0 tamizFirst, let us examinethefirst two iterations oftheNeumann series.. t 1 s 4 s3 u (t) =t2 , u (t)=t2 + es-tD0,5(s2)ds, D0,5(s2) = (s -t)-0,52tdt= 0 1 Г(0,5) 3 p 0 0 t 4 s3 u (t) =t2 + es-t ds 1 3 p 0 282 Volume 4, issue 6, 2025 The convergence of the iterated kernels mentioned above can be visualized by generating their graphs using Python. importnumpyasnp importmatplotlib.pyplotasplt fromscipy.integrateimportquad fromscipy.specialimportgamma 283 Volume 4, issue 6, 2025 T=1;alpha=0.5;n=100 t=np.linspace(0,T,n) f=lambdat:t**2 K=lambdat,s:np.exp(s-t) Gamma=gamma(1.5) u0=f(t) u1=np.zeros(n) foriinrange(n): integrand=lambdas:K(t[i],s)*(4*np.sqrt(s**3)/(3*np.sqrt(np.pi)))#D^0.5t^2 u1[i]=f(t[i])+quad(integrand,0,t[i])[0] plt.plot(t,