Abstract
In this paper, we have extended the operational matrix method for approximating the solution of the fractional-order two-dimensional elliptic partial differential equations (FPDEs) under nonlocal boundary conditions. We use a general Legendre polynomials basis and construct some new operational matrices of fractional order operations. These matrices are used to convert a sample nonlocal heat conduction phenomenon of fractional order to a structure of easily solvable algebraic equations. The solution of the algebraic structure is then used to approximate a solution of the heat conduction phenomena. The proposed method is applied to some test problems. The obtained results are compared with the available data in the literature and are found in good agreement.Dedicated to my father Mr. Sher Mumtaz, (1955-2021), who gave me the basic knowledege of mathematics.
Highlights
Nonlocal partial differential equations (PDEs) arise in the mathematical modeling of various problems in physics, engineering, ecology, and biological sciences [5,6,7]. e term nonlocal problems means that the solution of PDEs on the boundary is connected with the solution on some interior points of the domain. e case arises when the solution at the boundary is not known
Work e main advantage of the proposed method is its applicability to the fractional order Poisson equations. e method can handle fractional order problems with two-point boundary conditions. e method converts the heat flow phenomena to an algebraic structure, whose condition number is independent of the order of derivative
E proposed method yields a very accurate approximation when applied to fractional order Poisson equations. e comparison of results of the proposed method with some recent methods, such as, HWCM, MCTMQ, and MCTSMQ, shows that the proposed method is more appropriate for integer order problems
Summary
We present some useful results and notations which are of primary importance in our further investigation. For a given function φ(x) ∈ Cn[a, b], the Caputo fractional order derivative is defined as follows: Dαφ(x). E shifted Legendre polynomials [35] defined on [0, 1] are given by the following relation: i. Where GσM,y2×M2 is the operational matrix of the fractional integration of order σ and is defined as follows: GσM,y2×M2 Δ′q,r,. Let Ψ(x, y) be the function vector as defined in (16), the fractional integral of order σ of Ψ(x, y) w.r.t y is given by the following: Proof. Where GσM,x2×M2 is the operational matrix of derivative of order σ and is defined as follows: GσM,x2×M2 Θ′q,r,. Erefore to replace such terms with their equivalent matrix form, we need to derive two more operational matrices. E operational matrices used to replace such term by their equivalent matrix form are derived in the following lemmas. Proof. is lemma can be proved by following similar steps as in the previous lemma. is is left as an exercise for interested readers
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