Mntz Spectral Methods with Applications to Some Singular Problems - PowerPoint PPT Presentation

Müntz Spectral Methods with Applications to Some Singular Problems Chuanju Xu 许传炬 School of Mathematical Sciences, Xiamen University Collaborators: Dianming Hou (Xiamen U) Brown U June 20, 2018 Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 1

Summary 1 Motivation Integro-differential equations Fractional differential equations Related works 2 Generalized fractional Jacobi polynomials Preliminary Fundamental properties of GFJPs Projection, interpolation, and related error estimates 3 Müntz spectral method for some singular problems Fractional integro-differential equations Fractional elliptic equations Time fractional diffusion equations Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 2

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works Motivation We aim at constructing efficient numerical methods for a class of equations having singular solutions: { u t = a 1 u ( t ) + a 2 I µ t u ( t ) + f ( t ) , t ∈ I , µ ≥ 0 , u (0) = 0 . { bu ( x ) − D ρ x u ( x ) = f ( x ) , x ∈ I , 1 < ρ < 2 , u (0) = 0 , u x (0) = u 1 .   t u ( x , t ) − ∂ 2 D α x u ( x , t ) = f ( x , t ) , I × Λ , 0 < α < 1 ,   u ( x , 0) = u 0 ,    u ( x , t ) | ∂ Λ = 0 . where I = [0 , 1] , a 1 and a 2 are real coefficients, and the operators I µ t , D ρ x denote the fractional integral and derivative. Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 3

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works Volterra integral equation ∫ x ( x − s ) − µ K ( x , s ) u ( s ) = g ( x ) , x ∈ Λ := (0 , 1) , 0 < µ < 1 , u ( x ) + 0 where K ( x , s ) is a kernel function. It has been well known [Brunner 2004] that: if g ∈ C m (¯ Λ) and K ∈ C m (¯ Λ × ¯ Λ) with K ( s , s ) ̸ = 0 in ¯ Λ , then the solution can be expressed as ∑ γ j , k x j + k (1 − µ ) + u r ( x ) , u ( x ) = ( j , k ) ∈ G where G := { ( j , k ) : j , k are non-negative integers s.t. j + k (1 − µ ) < m } , γ j , k are constants, and u r ( · ) ∈ C m (¯ Λ) . Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 4

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works TFDE { R ∂ α t u − ∂ 2 x u = f t ∈ I , x ∈ Λ , α ∈ (0 , 1) , u ( − 1 , t ) = u (1 , t ) = 0 t ∈ I . or  C ∂ α t u − ∂ 2 x u = f t ∈ I , x ∈ Λ , α ∈ (0 , 1) ,    u ( − 1 , t ) = u (1 , t ) = 0 t ∈ I ,    u ( x , 0) = u 0 ( x ) x ∈ Λ . Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 5

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works Solution singularity Solution representation in term of Mittag-Leffler function: [ ∫ t ] ∞ ∑ ( f ( · , τ ) , ψ i )( t − τ ) α − 1 E α,α ( − λ i ( t − τ ) α ) d τ u ( x , t )= ψ i ( x ) 0 i =1 [ ∫ 1 ] ∞ ∑ ( f ( · , τ t ) , ψ i )(1 − τ ) α − 1 E α,α ( − λ i t α (1 − τ ) α ) d τ = t α ψ i ( x ) , 0 i =1 where − ∂ 2 x ψ i ( x ) = λ i ψ i ( x ) , ψ i ( ± 1) = 0 . Even the forcing function f is smooth, the solution u may exhibit singularity with the leading order t α at the starting point t = 0 like the one for the Volterra integral equations. Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 6

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works The main difficulties: - the operators I µ t and D ρ x are non-local; - the solutions are usually singular near the boundary or at the starting time. Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 7

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works Spectral methods Weak form of the TFDE { R ∂ α t u ( x , t ) − △ u ( x , t ) = f ( x , t ) , t ∈ I := (0 , T ) , x ∈ Λ := ( − 1 , 1) , u ( − 1 , t ) = u (1 , t ) = 0 , t ∈ I . α 2 ( Q ) := H s (Λ , L 2 ( I )) ∩ L 2 (Λ , H 1 Weak form: find u ∈ B 0 ( I )) , such that α A ( u , v ) + B ( u , v ) = ( f , v ) , ∀ v ∈ B 2 ( Q ) , (1) where Q = Λ × I , α α A ( u , v ) := ( 0 ∂ t u , t ∂ T v ) Q , B ( u , v ) := ( ∂ x u , ∂ x v ) Q . 2 2 Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 8

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works Spectral approximation Let L := ( M , N ) , the space-time Galerkin spectral method reads: Find u L ∈ P 0 M (Λ) ⊗ P N ( I ) , such that A ( u L , v L ) + B ( u L , v L ) = F ( v L ) , ∀ v L ∈ P 0 M (Λ) ⊗ P N ( I ) . Theorem ( Li & Xu , 2009) If u ∈ L 2 ( I , H σ (Λ)) ∩ H γ ( I , H 1 0 (Λ)) , γ > 1 , σ ⩾ 1 , then √ cos πα α t ( u − u L ) ∥ 0 , Q + ∥ ∂ x ( u − u L ) ∥ 0 , Q 2 ∥ ∂ 2 α α 2 − γ ∥ u ∥ 0 ,γ + N 2 − γ M − σ ∥ u ∥ σ,γ ≲ N + M − σ ∥ u ∥ σ, α 2 + M 1 − σ ∥ u ∥ σ, 0 + N − γ ∥ u ∥ 1 ,γ . Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 9

Motivation Integro-differential equations Generalized fractional Jacobi polynomials Fractional differential equations Müntz spectral method for some singular problems Related works Other related works ▶ Polynomial spline collocation method for IDEs: [Brunner 1986], [Tang 1993], [Brunner et al. 2001], [Rawashdeh et al. 2004], [Tarang 2004]. ▶ Spectral method for Volterra integral equations(VIEs) with nonsmooth solution: [Chen and Tang 2010], [Li, Tang, and Xu 2015], [Stynes and Huang 2016]. ▶ Non-polynomial basis for FDEs: [Zayernouri and Karniadakis 2013, 2014, …], [Chen, Shen, and Wang 2016]. ▶ Mapped Jacobi and Müntz-Legendre functions for Elliptic equations: [Shen and Wang 2016]. Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 10

Motivation Preliminary Generalized fractional Jacobi polynomials Fundamental properties of GFJPs Müntz spectral method for some singular problems Projection, interpolation, and related error estimates Müntz polynomials The well-known Weierstrass theorem states : every continuous function on a compact interval can be uniformly approximated by algebraic polynomials . This result was generalized by Bernstein 1912, and proved by Müntz (theorem) 1914 : n ∑ a k x λ k with real coefficients, the Müntz polynomials of the form k =0 i.e., span { x λ k , k = 0 , 1 , . . . } , are dense in C 0 [0 , 1] if and only if ∞ ∑ λ − 1 = + ∞ , where { λ 0 , λ 1 , λ 2 , . . . } is a sequence of distinct k k =1 positive numbers such that 0 = λ 0 < λ 1 < ... → ∞ . Extension to L 2 (0 , 1) by Szász 1916 . Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 11

Motivation Preliminary Generalized fractional Jacobi polynomials Fundamental properties of GFJPs Müntz spectral method for some singular problems Projection, interpolation, and related error estimates Generalized fractional Jacobi polynomials (GFJPs) We will make new use of Müntz polynomial spaces defined by N ( I ) = span { 1 , x λ , x 2 λ , · · · , x N λ } , P λ 0 < λ ≤ 1 . Generalized fractional Jacobi polynomials  (2 x λ − 1) , J α,β  α, β > − 1 ,  n    n + α +1 n (2 x λ − 1) , α > − 1 , β = − 1 , n +1 x λ J α, 1 J α,β,λ n + ℓ ( x ) = n + β +1 n (2 x λ − 1) , α = − 1 , β > − 1 , n +1 (1 − x λ ) J 1 ,β      n (2 x λ − 1) , α = β = − 1 , − (1 − x λ ) x λ J 1 , 1 where J α,β ( x ) denote the classical Jacobi polynomials, and n   0 , α, β > − 1 ,  1 , α = − 1 , β > − 1 or α > − 1 , β = − 1 , ℓ =   2 , α = β = − 1 . Chuanju Xu 许传炬 (Xiamen University) June 20, 2018 ICERM, Brown University Müntz Spectral Methods 12