This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative. The model is often adopted for describing the anomalous process of nearly sonic speed gas dynamics. The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode. The local collocation method approximates the differential operators using a weighted sum of the function values over a local collection of nodes (named stencil) through a radial basis function expansion.
Journal of Advanced Research 25 (2020) 205–216 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics O Nikan a, J.A Tenreiro Machado b, Z Avazzadeh c,d,⇑, H Jafari e a School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran Department of Electrical Engineering, ISEP-Institute of Engineering, Polytechnic of Porto, Porto, Portugal c Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam d Faculty of Natural Sciences, Duy Tan University, Da Nang 550000, Vietnam e Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa b h i g h l i g h t s g r a p h i c a l a b s t r a c t The fractional Tricomi-type model is adopted for describing the anomalous process of nearly sonic speed gas dynamics A new hybrid scheme based LRBF-FD method is formulated to solve the model The LRBF-FD method useful for irregular domains with good accuracy is proposed The stability and convergence of the proposed method are analyzed using the energy method a r t i c l e i n f o Article history: Received 15 April 2020 Revised June 2020 Accepted 21 June 2020 Available online 23 June 2020 Keywords: Caputo fractional derivative Time fractional Tricomi-type model LRBF-FD Stability analysis a b s t r a c t This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative The model is often adopted for describing the anomalous process of nearly sonic speed gas dynamics The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode The local collocation method approximates the differential operators using a weighted sum of the function values over a local collection of nodes (named stencil) through a radial basis function expansion This technique considers merely the discretization nodes of each subdomain around the collocation node This leads to sparse systems and tackles the ill-conditioning produced of global collocation The theoretical convergence and stability analyses of the proposed time semi-discrete scheme are proved by means of the discrete energy method Numerical results confirm the accuracy and efficiency of the new approach Ó 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction This paper proposes an efficient numerical formulation for solving the time fractional Tricomi-type model (TFTTM), that can be written as ⇑ Corresponding author E-mail addresses: omidnikan77@yahoo.com (O Nikan), jtm@isep.ipp.pt (J.A.T Machado), zakiehavazzadeh@duytan.edu.vn (Z Avazzadeh), jafari.usern@gmail com (H Jafari) @ a ux; t ị t 2c Dux; t ị ẳ f ðx; t Þ; @ta https://doi.org/10.1016/j.jare.2020.06.018 2090-1232/Ĩ 2020 The Authors Published by Elsevier B.V on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) x ẳ x; yị X & R2 ; < t T; ð1Þ 206 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 along with the initial and Dirichlet boundary conditions given by uðx; 0ị ẳ g xị; @ux; 0ị ẳ wxị; @t ux; tị ẳ hx; t ị; ẳ X [ @ X; x2X ¼ X [ @ X; x2X x @ X; t > 0; ð2Þ ð3Þ ð4Þ where T is the final time, D denotes the Laplace operator with respect to space variables x; c is real non-negative number, X is a bounded domain in R2 , and @ X represents the boundary of X The fractional derivative @ a uðx; t Þ=@t a of order < a < is defined in the Caputo sense as @ a ux; t ị ẳ @t a C2 À aÞ Z t @ uðx; sÞ ðt À sÞ1Àa ds; @s2 where CðÁÞ represents the Euler’s gamma function [1,2] During the 20s, Tricomi [3] started the work on the linear partial differential equations of variable type with boundary condition Later, Frankl [4] showed that the gas flows with nearly sonic speeds could be described by the Tricomi model For the numerical solution of the TFTTM, we find some works published during the last years Zhang et al [5] formulated a local discontinuous Galerkin finite element, Zhang et al [6] used the finite element scheme and Liu et al [7] applied the reduced-order finite element technique to approximate the TFTTM More recently, Dehghan and Abbaszadeh [8] adopted the element-free Galerkin technique and Ghehsareh et al [9] implemented the local Petrov– Galerkin formulation Numerical techniques are extensively applied to approximate partial differential equation (PDE) and we can mention the finite element, finite difference, finite volume, and pseudo-spectral methods However, usually these techniques are defined on data point meshes meaning that a grid generation is often required, which in turn increases the computation time Moreover, these schemes have insufficient accuracy over irregular and nonsmooth domains because they provide the problem solution only on mesh points As a result, meshless techniques have been developed to overcome these problems One important meshless technique is the radial basis function (RBF) method The RBF is a very efficient instrument for interpolating a scattered set of points and, due to these characteristics, has received attention during last years [10–13] Indeed, the RBF approximation is a powerful tool that is particularly relevant for high-dimensional problems Rolland Hardy [14] proposed the RBF technique in 1971 by introducing the multiquadric (MQ) algorithm as a meshless interpolation method using the MQ radial function Richard Franke [15] popularized this approach in 1982 with a review of the 32 most used interpolation techniques Franke performed a set of comprehensive tests and concluded that the MQ method had the best overall performance Furthermore, he advanced that the interpolation matrix related to the MQ radial function has unconditional nonsingularity Later, in 1986, Micchelli [16] proved this using research from the 30s and 40s by Schoenberg Kansa [17,18] considered the MQ method for approximating elliptic and parabolic PDE Nonetheless, the well conditioned of the RBF interpolation matrix and good accuracy are not verified simultaneously This is known as the Uncertainty Principle following the work of Schaback [19] Fornberg and Larsson [20] implemented this technique to elliptic PDE The existence, uniqueness, and convergence of the RBF approximation were discussed in detail in several works [21,16,22] Hereafter, this paper shows that the local RBF is an efficient computational technique to numerically approximate the TFTTM with high accuracy and low computational complexity Following these ideas, this paper is arranged as follows Section formulates the temporal discretization via finite difference and discusses its convergence and error analysis Section applies the local RBFfinite difference (LRBF-FD) for space discretization Section illustrates the method with three numerical examples that show its efficiency and verify the theoretical analysis Finally, Section concludes with a summary of the key conclusions Temporal discretization To apply the numerical scheme for the solution of Eq (1), let dt ¼ T=M; tk ¼ kdt; k ¼ 1; ; M, for a positive integer M Therefore, the time domain ½0; T is covered by temporal discretization points tk The following lemmas will be used in the derivation of the time difference scheme [23] Lemma ([23].) If < a < and g tị C ẵ0; T , then it follows Z tn g ðsÞðtn À sị1a ds ẳ where n R 1 dt 3Àa max jg 00 ðt ịj: ỵ 06t6tn 22 aị Lemma bk ẳ Z n X g ðt k Þ À g ðt k1 ị tk t n sị1a ds ỵ Rn ; dt tkÀ1 k¼1 dt 2Àa 2Àa ([23].) Suppose that < a < a0 ẳ dtC12aị h i 2Àa 2Àa ; k ¼ 0; 1; 2; Then it follows k ỵ 1ị kị and R tn 1Àa Cð2ÀaÞ g ðsÞðt n À sÞ ds À a0 ! nÀ1 À Á P  b0 g ðt n Þ À ðbnÀkÀ1 À bnÀk Þg t j À bnÀ1 g 0ị kẳ1 3a 1 C2 max jg 00 t ịj: aị 22aị ỵ dt 06t6t n h i 2Àa 2Àa 2Àa ; Lemma ([23].) If < a < and bk ẳ dt2a k ỵ 1ị kị k ẳ 0; 1; 2; ; then it follows b0 > b1 > b2 > > bk ! 0; as k ! 1: We introduce the following notation: À Á2c uk À ukÀ1 ; dt dt ukÀ2 ¼ lk ¼ tk ; f kÀ12 k ¼ f ỵf k1 5ị Let us consider v x; tị ẳ @ux; t ị ; @t Vx; tk Þ ¼ Cð2 À aÞ ð6Þ Z t @ v ðx; sÞ ðt À sÞ1Àa ds: @s ð7Þ From (6) it results that Taylor expansion at t ¼ t kÀ1 can be written as: v kÀ k1 ẳ dt uk2 ỵ R1 ; ð8Þ where 1 kÀ2 R1 C dt2 : ð9Þ Based on Lemma 2, we have " V ¼ a0 b0 v k n # kÀ1 X À Á j À bkÀjÀ1 À bkÀj v bk1 v ỵ Rk2 jẳ1 207 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 and also " VkÀ2 ¼ a0 b0 v kÀ2 À 1 kÀ1 X À bkÀjÀ1 À bkÀj Á where L2 ðXÞ represents the space of measurable functions whose square is Lebesgue integrable in X and a ¼ ða1 ; ; ad Þ denotes a P d-tuple of non-negative integer with jaj ¼ di¼1 Let us consider # v jÀ À bkÀ1 v ỵ R2 ; k1 jẳ1 Da v ẳ where 1 kÀ2 R2 C dt 3Àa : v kÀ ; q " ¼ a0 b0 v kÀ2 À kÀ1 X À bkÀjÀ1 À bkÀj Á À bkÀ1 q : !12 L ðXÞ jaj6m : Now, let us examine the analysis of stability and the error estimates for the difference algorithm Vk ỵ Vk1 k1 ẳ a0 P v k2 w ỵ R2 ; 11ị where v xị ẳ v x; 0ị ẳ wxị ẳ w If we substitute (8) into (11), we obtain VkÀ2 d X a 2 D v ð10Þ Using Lemma 2, the expression (10) can be written as jjv jjHm ðXÞ ¼ # v jÀ j¼1 VkÀ2 ¼ The norm kv km of the space Hm ðXÞ can be written as We define the operator P @ jaj v Á @xa @xa @xa kÀ1 kÀ1 ¼ a0 P dt ukÀ2 ; w þ a0 P R1 ; þ R2 : Corollary (Poincaré inequality [24]) Suppose that p and that X is a bounded open set Then, there exists a constant CX (depending on X and p) such that k g CX $gk : ð12Þ Lemma ([23].) For any G ¼ fG1 ; G2 ; g and q, we have Substituting the above result (12) into (1) yields 1 kÀ1 kÀ1 a0 P dt uk2 ; w ẳ Duk2 ỵ f ỵ R2 ; ð13Þ M M a X À Á t 1Àa X t 2À M P Gj ; q Gj P M dt G2j À q2 : 2ð2 aị jẳ1 jẳ1 where k1 n o kÀ1 RkÀ2 ¼ À a0 P R1 ; ỵ R2 : Based on Lemma 2, and inequalities (9) we can write ( " kÀ12 R k1 a0 b0 R1 ỵ bkj1 bkj j¼1 ( " kÀ1 PÀ a0 b0 C dt2 ỵ k1 P # vj bkj1 bkj jẳ1 k1 bk1 R1 ỵ Lemma ([25].) If xn is nonnegative sequence and the sequence yn fulfills > < y0 d0 ; ) kÀ1 R2 # v j À bkÀ1 C dt2 ) ỵ C dt 3a ẩ ẫ ẳ a0 b0 C dt ỵ b0 bk1 ịC dt ỵ C dt3a ẩ Â Ã É a0 2b0 C dt þ C dt3Àa n h i o 2Àa ¼ dtC12aị dt2a C dt2 ỵ C dt 3a ỵ C dt3a : ð2Àa2C ÞCð2ÀaÞ u by its numerical approximation U semi-discrete recursive algorithm: , leads to the following kÀ1 1 kÀ1 a0 P dt U kÀ2 ; w ¼ l DU k2 ỵ f ; a0 b0 U ỵa0 dt dt jẳ1 k l DU ¼ a0 b0 U kÀ1 PÀ xk yk ; k¼0 then yn satisfies > < y1 d0 ỵ x0 ị ỵ y0 ; n2 n2 n1 Q Q P > ỵ xk ị ỵ zk ỵ xs ị ỵ zn1 ; : yn d0 ỵ kẳ0 n P 2: sẳkỵ1 yn ! ! n1 n1 X X d0 ỵ zk exp xk : k¼0 k¼0 Making use of these lemmas, we can derive the following result of stability Theorem If U k H10 ðXÞ, then the difference formula (14) is unconditional stable with respect to the H1 -norm kÀ1 k1 ỵ l DU k k1 bjk1 bjk dt U j2 ỵ a0 bk1 dtw þ 12 dt f þ f : k k¼0 nÀ1 P 14ị or, equivalently, we get k zk ỵ Moreover, if d0 P and zn P for n P 0, then it holds kÀ12 nÀ1 P k¼0 Dropping the error term RkÀ2 and approximating the exact value k12 > : yn d0 ỵ dt kÀ1 Proof The following variational weak formulation will be obtained by multiplying both sides of Eq (14) by m and integrating over X D Theoretical analysis of the time discretization scheme We star by defining some functional spaces that will be used in the subsequent discussion Let us define the functional space endowed with the standard norms and inner products n o H1 Xị ẳ v L2 Xị; ddxv L2 Xị ; n o H10 Xị ẳ v H1 Xị; v j@X ẳ ; n o Hm Xị ẳ v L2 Xị; Da v L2 ðXÞ; for all positive integerjaj m ; E D E 1 a0 P dt nkÀ2 ; w ; m ¼ lkÀ2 DnkÀ2 ; m ; ð15Þ $ À Á 1 where nkÀ2 ¼ U kÀ2 À U kÀ2 denotes the perturbation at the k À 12 th $ 1 time level, so that U kÀ2 and U kÀ2 are the exact and approximate solutions of Eq (14), respectively Using the divergence theorem Z X rv rx ¼ where Z @X v @x À @n Z X v Dx ; 208 O Nikan et al / Journal of Advanced Research 25 (2020) 205216 @x @x @x ẳ n1 ỵ n2 @n @x @y Theorem Let uk and U k be the solutions of (13) and (14), respec- is the normal derivative, that is, representing the derivative in the outward normal direction to the boundary @ X, we get ( ) kÀ1 D E X E À ÁD 1 a0 b0 dt nkÀ2 ; m À bkÀjÀ1 À bkÀj dt nkÀ2 ; m Proof Taking the inner product of Eqs (13) and (14) with m on the both sides, we obtain their corresponding variational weak form as follows: j¼1 D E 1 ¼ ÀlkÀ2 rnkÀ2 ; rm Á Letting ( ð16Þ kÀ1 D E X E À ÁD 1 1 a0 b0 dt nkÀ2 ; dt nkÀ2 À bkÀjÀ1 À bkÀj dt njÀ2 ; dt nkÀ2 ) j¼1 E 1 ¼ ÀlkÀ2 rnkÀ2 ; rdt nkÀ2 Á Summing on k; k ¼ 1; ; M, and applying Cauchy–Schwarz inequality, we deduce that ( ) M kÀ1 X Á jÀ1 kÀ1 kÀ12 2 X À 2 a0 b0 dt n À bkÀjÀ1 À bkÀj dt n dt n E D D E D E 1 kÀ a0 P dt U kÀ2 ; w ; m ẳ lk2 DU k2 ; m ỵ f ; m ; M X l kÀ12 2dt kẳ1 17ị 18ị 1 where fk2 ẳ uk2 À U kÀ2 Subtracting Eq (17) from Eq (18) and using the divergence theorem again, we arrive at ( ) kÀ1 D E X E D E D E À ÁD 1 1 a0 b0 dt fkÀ2 ; m À bkÀjÀ1 À bkÀj dt fkÀ2 ; m ẳ lk2 rfk2 ; rm ỵ Rk2 ; m 19ị jẳ1 Setting m ẳ dt fk2 in Eq (19) yields ( ) D E kP E À1 À ÁD 1 1 a0 b0 dt fkÀ2 ; dt fkÀ2 À bkÀjÀ1 À bkÀj dt fjÀ2 ; dt fkÀ2 j¼1 k¼1 E D D E D E D E 1 kÀ a0 P dt ukÀ2 ; w ; m ¼ lk2 Duk2 ; m ỵ f ; m ỵ RkÀ2 ; m and m ¼ dt nkÀ2 in Eq (16), we obtain D tively, such that both belong to H10 ðXÞ Then, the difference formula À Á (14) has convergence order O dt 3Àa kÀ1 2 k 2 rn À rn : ¼ Àl kÀ12 j¼1 D kÀ12 rf E D E 1 ; rdt fk2 ỵ Rk2 ; dt fk2 Now, we sum from k ¼ to M to get Making use of Lemma 4, we can conclude that M M a X t 1À kÀ12 2 X lkÀ2 kÀ1 2 k 2 m 06 d n r n À r n t 2Cð2 aị kẳ1 2dt kẳ1 a0 M P ( kÀ1 P 2 b0 dt fkÀ2 À bkÀjÀ1 bkj ịdt fj2 dt fk2 ) jẳ1 k¼1 2 2 P M M P kÀ12 kÀ12 lkÀ2 À rfk À rfkÀ1 ỵ R dt f : 2dt kẳ1 20ị kẳ1 and then 2 lMÀ2 rnM M X 2 lkÀ2 rnk k¼1 M X 2 lkÀ2 rnkÀ1 k¼1 2 þ lMÀ2 dt 2aþ1 rn0 : If we change the index from M to k, then we arrive at 2 lkÀ2 rnk k X 2 ljÀ2 rnj T 2a j¼0 k X 2 j 2 rn ỵ lk2 dt 2aỵ1 rn0 : jẳ0 This expression can be rewritten as: 2a k 2 2 k 2 X T j 2 rn rn ỵ dt2aỵ1 rn0 dt2aỵ1 n0 k1 jẳ0 ỵ k X l j¼0 After applying the discrete Gronwall’s lemma to this inequality, it yields k h i 2 X 2 k 2 ỵ T 2a dt 2a ẳ dt2a n0 rn dt2aỵ1 n0 ỵT n ing # ẳ 2C2aị, we deduce that M M M a X X t 1À kÀ12 kÀ12 Cð2 À aÞ X k12 2 k12 2 M R ỵ d f R dt f : t a 4Cð2 À aÞ kẳ1 t M kẳ1 kẳ1 21ị Inserting Eq (21) into Eq (20), it follows that M M a X X t1À lkÀ12 k 2 kÀ1 2 kÀ12 2 M dt f À rf À rf 2Cð2 À aÞ kẳ1 2dt kẳ1 M C a ị X a t 1À M and using the Poincaré inequality, we obtain the desired result rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2cỵ1 n : n exp T Hence the proof is complete The convergence order of the time-discrete approach is given in the following theorem R 2 ỵ k12 k¼1 M a X t 1À kÀ12 2 M dt f : 4C2 aị kẳ1 ð22Þ Multiplying Eq (22) by 2dt, changing M to k, and simplifying results in k k P P k 2 2 jÀ1 2 rf 2dtL Cð2 À aịt ak fj2 ỵ 2dtC2 aịt ak À1 R j¼1 j¼1 2 k P 2 2dtL2 C2 aịtak fj2 ỵ 2kdtC2 aịtak À1 max fjÀ2 : j¼0 2 2 dt 2a T 2cỵ1 ỵ T 2cỵ1 n0 exp T 2cỵ1 n0 r22 , by choos- a t M ỵ 2 T 2a dt2a rnj : 2aỵ1 2 In virtue of the Young’s inequality, jr1 r2 j 2#12 r21 ỵ #2 jẳ1 16j6k Employing a similar technique to the one adopted in the previous theorem yields 2 k 2 X k 2 T 2a dtÀ2a rfj rf dt 2aỵ1 f0 ỵ jẳ0 ỵ k X j¼1 2 2kdt Cð2 À aÞt kaÀ1 max RjÀ2 : 16j6k 209 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 Noticing that f0 ¼ 0, we get 2 k k 2 X 2a À2a j 2 T dt rf þ 2kdt Cð2 À aÞtkaÀ1 max RjÀ2 ; rf 16j6k j¼0 and applying the Poincaré inequality results in 2 k X Á2 k jÀ12 2 2À f C2X dtL C2 aịt ka1 f ỵ C2X C2 aịTC dt 3a : jẳ1 23ị Using the discrete Gronwall inequality, the expression (23) can be rewritten as the following form Fig Schematic diagram of a stencil used for approximating the differential operator on a non-uniform nodes Fig The computational domains fX1 ; X2 ; X3 ; X4 g Table Numerical errors L1 and temporal accuracy C dt with h ¼ 1=10 and a ¼ 1:3 on X1 at T ¼ TPS-RBF r lnðr Þ dt 1=10 1=20 1=40 1=80 1=160 1=320 1=640 1=1280 MQ-RBF p ỵ e2 r L1 C dt L1 C dt 4:9210e À 03 1:7509e À 03 5:8335e À 04 1:9061e À 04 6:1506e À 05 1:9502e À 05 6:0741e À 06 1:8520e À 06 – 1:4909 1:5857 1:6137 1:6318 1:6571 1:6829 1:7136 4:0081e À 03 1:4613e À 03 5:1632e À 04 1:6843e À 04 5:3772e À 05 1:6807e À 05 5:1996e À 06 1:5644e À 06 – 1:4557 1:5009 1:6161 1:6472 1:6778 1:6926 1:7328 Table Numerical errors L1 and spatial accuracy C h on X1 h 1=4 1=8 1=16 1=8 1=16 dt 1=4 1=40 1=400 1=8 1=80 a ¼ 1:45 L1 Ch 3:3808e À 01 1:6495e À 02 5:6526e À 04 2:6559e À 02 1:6845e À 03 – 4:3573 4:8670 – 3:9788 h dt a ¼ 1:65 L1 Ch 1=4 1=8 1=16 1=8 1=16 1=4 1=48 1=576 1=8 1=96 3:2919e À 01 7:9865e À 03 4:9705e À 04 2:0182e À 02 2:1633e À 03 – 5:3652 4:0061 – 3:2218 210 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 Table The absolute errors of the LRBF-FD with a ¼ 1:8; N ¼ 801 and dt ¼ 1=100 at T ¼ on X4 NI L1 CPU (s) 51 71 91 101 5:4671e À 04 5:5065e À 04 5:6804e À 04 5:7043e À 04 38.61 45.32 52.17 65.83 Table The obtained condition number and the CPU time for the GRBF and LRBF-FD with N ¼ 381 and dt ¼ 1=200 at T ¼ Domain Method CPU (s) Condition number X2 GRBF LRBF-FD GRBF LRBF-FD GRBF LRBF-FD 53:42 40:21 60:04 48:51 57:21 45:34 8:6071e ỵ 06 3:4948e ỵ 02 6:7234e þ 07 5:5923e þ 02 4:6629e þ 07 5:5911e þ 02 X3 X4 2 kÀ1 À Á2 P 2 k CX L Cð2 À aÞtkaÀ1 f TC C2X Cð2 À aÞ dt 3Àa exp j¼0 À Á2 TC CX Cð2 À aÞ dt3Àa exp C2X L2 Cð2 À aÞkdtt kaÀ1 À Á2 ¼ TC C2X Cð2 À aÞ dt 3Àa exp C2X L2 Cð2 À aÞt ak À Á2 TC C2X Cð2 À aÞ dt 3Àa exp C2X L2 Cð2 À aÞT À Á2 C ðT; a; CX Þ dt3Àa : 2 As a result, we obtain: k f C ðT; a; CX Þdt 3Àa : The proof is completed Fig The absolute error with dt ¼ 1=100; N ¼ 151 and a ¼ 1:5, at T f0:25; 0:5; 075; 1g on the rectangular domain X1 ð24Þ O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 211 Fig Sparsity pattern of the coefficient matrix when N ¼ 400 Fig The approximated solutions and their corresponding absolute errors with dt ¼ 1=100, at T ¼ on X3 Spatial discretization by the local radial basis function in a finite difference mode È É Given a set of distinct nodes X C ¼ xc1 ; ; xcN # Rd and the corresponding function values uxi ị; i ẳ 1; 2; ; N, the RBF interpolant is represented in the form uxị Sxị ẳ N X aj /j x; eị; Luxi ị 25ị jẳ1 where /j x; eị ẳ / kx xcj k2 ; e ; j ¼ 1; ; N; is a RBF corresponding th center with shape parameter e [26] The expansion coeffiÈ ÉN cients aj j¼1 , can be obtained by enforcing the interpolation condiÀ Á tion S xci ¼ uci ; i ¼ 1; ; N; at a set of nodes that usually coincides with the N centers It is worth to mention that the associated matrix the j / is a non-singular and invertible for any arbitrarily set of distinct scattered point [16,27] Kansa [17,18] adopted the linear partial differential operator L on the interpolation (25) to approximate Lu at the N scatter nodes, namely N X bj L/j xi ; eị: 26ị jẳ1 The relation (26) defines a global RBF (GRBF) approximation, i.e for approximating L at reference point xi , all points in the domain are involved The GRBF meshless methods have the disadvantage of dense and ill-conditioned interpolation matrices, but, on the other hand, the sparse matrices of these techniques have better condition numbers Nonetheless, the differentiation matrices associated with local meshless methods, that are used for solving PDE, require the multiplication of the interpolation matrix by its inverse This results 212 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 Fig The approximated solutions and their corresponding absolute errors with dt ¼ 1=100 and N ¼ 451 at T ¼ on X4 Table Numerical errors L1 and temporal accuracy C dt with h ¼ 1=10 and a ¼ 1:7 on X1 at T ¼ p ỵ e2 r dt TPS-RBF r lnðrÞ L1 C dt L1 C dt 1=10 1=20 1=40 1=80 1=160 1=320 1=640 1=1280 6:1551e À 03 3:8996e À 03 1:7470e À 03 7:6426e À 04 3:2383e À 04 1:3565e À 04 5:6581e À 05 2:3258e À 05 – 1:1460 1:1301 1:1919 1:2021 1:2214 1:2652 1:3164 6:7029e À 02 3:0289e À 03 1:3839e À 03 6:0576e À 04 2:6329e À 04 1:1292e À 04 4:6981e À 05 1:8865e À 05 – 1:1460 1:1301 1:1919 1:2021 1:2214 1:2652 1:3164 MQ-RBF Table Numerical errors L1 and spatial accuracy C h on X1 h 1=4 1=8 1=16 1=8 1=16 dt 1=4 1=40 1=400 1=8 1=80 a ¼ 1:65 a ¼ 1:45 L1 Ch h dt L1 Ch 4:7480e À 01 9:4222e À 03 7:7252e À 04 1:2036e À 01 4:3565e À 03 – 5:6551 3:6084 – 4:7880 1=4 1=8 1=16 1=8 1=16 1=4 1=48 1=576 1=8 1=96 5:0888e À 01 1:0732e À 02 9:5886e À 04 1:3354e À 01 5:2140e À 03 – 5:5673 3:4845 – 4:7227 in dense matrices again, and one may use the generalized inverse to solve this limitation Nonetheless, we must note that the discussion of this subject falls outside the scope of the present work [28–31] An innovative method named the LRBF-FD has been proposed in [32] to overcome this issue The new technique was also brought up and examined more extensively in [32–37] The discretization in LRBF-FD (as a local meshless method) is obtained for a set of local differentiation matrices and adding them up forms a large, sparse system matrix In order to calculate the differentiation matrix at each point, merely the neighboring points are taken into consideration Let us now discuss the proposed method in more detail For each node N ¼ fx1 ; ; xN g # Rd in space, we consider a subset n o ðiÞ iị SI ẳ x1 ; ; xNI # N consisting of N I À surrounding nodes and ðiÞ xk itself, and we define it as a stencil Fig illustrates the influence domain of every reference point xi In the LRBF-FD, the derivatives of a function in a node requires to be only a list of its nearest stencil The approximation of an operator L at the central node xi is obtained as a weighted sum of function values of u at the N I stencil nodes Luðxi Þ ’ NI X ð iị iị wj u xj : 27ị jẳ1 ẩ ÉNI Following [32,33], by using a set of RBF /j x; eị jẳ1 centered at SI n oNI iị for obtaining the LRBF-FD weights, wj , in Eq (27) j¼1 L/k xi ; eị ẳ NI X iị wj /j xk ; eị; jẳ1 k ẳ 1; ; NI : ð28Þ O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 213 Fig The approximated solutions and their corresponding absolute errors with dt ¼ 1=100; N ¼ 200 and a ¼ 1:5 at T ¼ on the rectangular domain X1 Fig The approximated solutions and their corresponding absolute errors with dt ¼ 1=100, at T ¼ on X2 The unknown weights of LRBF-FD can be determined by solving the system of linear equations in the following form: UwI ¼ ½LUI ; ð29Þ where the coefficient matrix UNI ÂNI has entries /kj ẳ /j xk ; eị, wI n oN I ðiÞ represents the NI  vector of differential weights wj , called j¼1 algorithm can be used to determine the NI À closest neighboring points in the computation of the differentiation weights for the stencils We find the kd-tree algorithm named knnsearch in the statistical toolbox of MATLAB Additionally, the algorithm by Sarra [38] is used to find the optimal shape parameter Results and discussion I LRBF-FD weights, and ½LU is the N I  vector for the values L/k xi ; eị; k ẳ 1; ; NI Due to the nonsingularity of the matrix U [27], we calculate the weights vector wI given by wI ẳ U1 ẵLUI : 30ị The derivatives are approximated in the LRBF-FD as for the classical FD method In brief, the derivatives are discretized at any node via the RBF interpolation by means of a small collection of neighboring nodes forming a stencil similar to those obtained with the FD In the n oNI ðiÞ FD the weights wj in the node xi are obtained on its stencil j¼1 values, with the difference that in the LRBF-FD instead of polynomials, the RBF interpolation are used A fast and effective kd-tree This section investigates three problems to highlight the high efficiency of the proposed method and to illustrate the theoretical analysis established in the previous section for different values of h and dt The rate of convergence in time and space [39] are calculated by using the formulae: ịjj C dt ẳ log2 jjLjjL112dt;h ; dt;hÞjj Á À jjL1 23Àa dt;2h jj ; C h ẳ log2 jjL1 dt;hịjj Á where L1 ¼ max U xj ; T À u xj ; T All numerical results are 16j6NÀ1 obtained using MATLAB 2016a 214 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 Fig The approximated solutions and their absolute errors with dt ¼ 1=100 and N ¼ 451 at T ¼ on X4 Example Consider the following TFTTM: NI L1 CPU (s) 11 15 21 31 5:0349e À 03 5:1589e À 03 5:1635e À 03 5:1670e À 03 15.1 17.4 18.3 19.5 Àt Method CPU (s) Condition number X2 GRBF LRBF-FD GRBF LRBF-FD GRBF LRBF-FD 828:69 502:35 751:14 463:29 678:47 352:07 3:3064e ỵ 10 1:3069e ỵ 03 3:4601e ỵ 10 1:3442e þ 03 4:8933e þ 10 1:3468e þ 03 X3 X4 Fig shows the computational domains in with two kinds of distribution points that are considered in the follow-up The domain X1 ẳ ẵ0; 12 denotes a rectangular domain with uniformly distributed points The irregular domain X2 is created using the relation rhị ẳ 0:8 ỵ 0:1sin6hị ỵ sin3hịị with uniformly distributed points The relation rhị ẳ 14 cosð4hÞ; h 2p, produces the irregular domain X3 which is covered by Halton distributed points [40] The domain X4 represents a set of Halton ÀÀ points in the unit circle in ½À1; 1 including Halton non-uniform points The LRBF-FD is applied here with several values for h; dt and a, at T on X1 ; X2 ; X3 and X4 The main results are presented in Tables 1–4 and Figs 3–6 Tables and report the values achieved for the absolute error and the convergence rates for several values dt and h when T ¼ on X1 From Table 1, one can conclude that the obtained computational orders support the theoretical order Table lists the absolute errors L1 of the LRBF-FD for various values of local points N I Table exhibits the achieved condition number and CPU time for the GRBF and LRBF-FD on the irregular domains It is observed that coefficient matrix of LRBF-FD collocation procedure is more well-posed than the coefficient matrix of GRBF method Fig shows the sparsity pattern of the matrix associated with the LRBF-FD Fig includes the graphs of the absolute Table 10 Numerical errors EkU and temporal accuracy C dt with h ¼ 1=15 on X1 at T ¼ dt 1=10 1=20 1=40 1=80 a ¼ 1:2 a ¼ 1:8 EkU C dt EkU C dt 1:0866e À 02 5:6681e À 03 2:3676e À 03 9:5950e À 04 – 0:9389 1:2594 1:3031 8:3954e À 03 3:7239e À 03 1:4702e À 03 4:9881e À 04 – 1:1728 1:3408 1:5595 Table Comparison of the absolute error in the solution for several values of h; dt and a at T ¼ on X1 a h dt LRBF-FD Ref [9] Ref [7] 1:2 1=10 1=20 1=40 1=10 1=20 1=40 1=10 1=20 1=40 1=10 1=20 1=40 4:4471e À 03 9:0283e À 04 5:6298e À 04 3:6245e À 03 5:6761e À 04 1:9843e À 04 2:7161e À 03 8:1844e À 04 3:1115e À 04 2:5158e À 03 6:3084e À 04 9:8684e À 05 8:63934e À 02 2:2133e À 02 5:5200e À 03 8:7656e À 02 2:2912e À 02 5:9500e À 03 1:99 ð31Þ The initial and boundary conditions corresponding to this example can be calculated from an exact solution À ÁÀ Á ux; y; t ị ẳ t 2ỵa x4 x2 y4 À y2 Table The obtained condition number and the CPU time for the GRBF and LRBF-FD with N ¼ 1781 and dt ¼ 1=1024 at T ¼ Domain À ÁÀ Á x À x2 y À y Á À Á À Á À Á Á 12x2 À y4 À y2 À 12y2 À x4 À x2 ; x;y X; < t T: @ a uðx;y;tÞ À t2 Dux;y;t ị ẳ C6t 4aị @ta Table The absolute errors of the LRBF-FD with N ¼ 401and dt ¼ 1=80 at T ¼ on X2 215 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 Table 11 Numerical errors L1 and spatial accuracy C h with on X1 at T ¼ h a ¼ 1:3 a ¼ 1:7 L1 Ch L1 Ch 2=4; 2=8 2=8; 2=16 2=16; 2=32 2:3540e À 03 8:7995e À 04 3:5412e À 04 – 1:1460 1:1301 2:2039e À 03 6:4571e À 04 2:3653e À 04 – 1:7711 1:4489 errors by choosing h ¼ 1=10; dt ¼ 1=100 and a ¼ 1:5, when T ¼ on X1 Fig depicts the approximate solutions and their absolute errors with a ¼ 1:15; N ¼ 451 and dt ¼ 1=100 when T ¼ on X3 Finally, Fig shows the approximate solutions and their absolute errors with a ¼ 1:75; N ¼ 353 and dt ¼ 1=100, when T ¼ on X4 where the error at dt is evaluated as a difference between the solutions U dt and U 2dt for time steps dt and 2dt at T given by Edt ¼ kU dt À U 2dt k The following predictor–corrector procedure and norm are illustrated to obtain the error in spatial variable Example We consider the following TFTTM: where U h1 and U h2 are the approximate solutions with respect to h1 and h2 , respectively The spatial convergence rate can be calculated as: a @ uðx; y; tÞ À tDuðx; y; t Þ @t a ẳ 2t2 sin2pxịsin2pyị t a ỵ 4p2 t ; x; y X; < t T: Cð3 À aÞ ð32Þ The initial and boundary conditions corresponding to this example can be obtained from the exact solution ux; y; tị ẳ t sin2pxị sin2pyị The LRBF-FD is formulated here for different quantities of dt; h and a, at T on X1 ; X2 ; X3 and X4 The results are illustrated in Tables and and Figs 7–9 Tables and report the values achieved for the absolute error and the convergence rates for several values dt and h when T ¼ on X1 From Table 5, one can conclude that the computational order is in good agreement with the theoretical order Table illustrates the absolute errors L1 of the LRBF-FD for several values of local points N I Table reports the obtained condition number and the CPU time for the GRBF and LRBF-FD on the irregular domains It can be mentioned that coefficient matrix of the LRBF-FD collocation procedure is more smaller than the coefficient matrix of the GRBF Table compares the absolute errors of the RBF-FD and those from [7,9] with several values of dt and h Fig represents the resulting absolute errors by choosing h ¼ 1=10; dt ¼ 1=100 and a ¼ 1:5, when T ¼ on X1 Fig includes the absolute errors achieved with N ¼ 451; dt ¼ 1=100 for a ¼ 1:65, when T ¼ on X2 Finally, Fig displays the approximate solutions and their absolute errors with a ¼ 1:45; N ¼ 353 and dt ¼ 1=100, when T ¼ on X4 Eh1 ;h2 ¼ kU h1 À U h2 k; C h ¼ log2 Eh;h=2 ; Eh=2;h=4 ð34Þ ð35Þ where Eh;h=2 and Eh=2;h=4 are the absolute error between the solutions with mesh sizes fh; h=2g and fh=2; h=4g, respectively We apply the LRBF-FD to obtain the numerical results for several quantities of dt; h and a Table 10 reports the achieved errors and convergence orders with respect to the temporal domain Table 11 lists the achieved errors and convergence orders with respect to the spatial domain Concluding remarks This paper presented a novel method for finding an approximate solution of the TFTTM One of the key results that emerges from this work is that the method is robust and has a reliable accuracy even for a complex domain using irregular nodal distributions It ought to be said here that the irregularly nodal distribution and complex domain lead to considerable difficulties for standard techniques The proposed algorithm includes two parts, namely, a first one where the problem is discretized based on finite difference scheme in the temporal direction, and a second one where the LRBF-FD is used for the spatial approximation The stability and convergence of the semi-discrete scheme are rigorously investigated Numerical results highlight the efficiency of the method Declaration of Competing Interest Example Lastly, we consider the following TFTTM: @ a uðx; y; t ị t Dux; y; tị ẳ 0; @ta x; y X; The authors declare that there is no conflict of interests regarding the publication of this manuscript < t T; with the initial conditions uðx; y; 0ị ẳ 0; @ux; y; 0ị ẳ 0; @t Compliance with Ethics Requirements x; y X This manuscript does not contain any studies with human participants or animals performed by any of the authors and boundary condition uðx; y; t ị ẳ t 2ỵa cos2pxị cos2pyị; Acknowledgement x; y @ X: Since the analytical solution of the above problem is unknown, we apply the relation presented by Kamranian et al [41] for the criterion convergence of the solution: EkU ẳ kU kỵ1 U k k kU kỵ1 k ; 33ị The authors are very grateful to the reviewers for their valuable comments on the manuscript that led to many improvements References [1] Samko SG, Kilbas AA, Marichev OI, et al Fractional integrals and derivatives Gordon and Breach Science Publishers, Vol Switzerland: Yverdon Yverdonles-Bains; 1993 216 O Nikan et al / Journal of Advanced Research 25 (2020) 205–216 [2] Kilbas AA, Srivastava HM, Trujillo JJ Theory and applications of fractional differential equations, Vol 204 Elsevier Science Limited; 2006 [3] Tricomi F Sulle equazioni lineari alle derivate Parziali di secondo ordine, di tipo misto Rend Reale Accad Lincei Cl Sci Fis Mat Natur 1923;5(14):134–247 [4] Frankl F On the problems of Chaplygin for mixed sub-and supersonic flows Bull Acad Sci USSR Ser Math 1945;8:195–224 [5] Zhang X, Liu J, Wen J, Tang B, He Y Analysis for one-dimensional timefractional Tricomi-type equations by LDG methods Numer Algor 2013;63 (1):143–64 [6] Zhang X, Huang P, Feng X, Wei L Finite element method for two-dimensional time-fractional Tricomi-type equations Numer Methods Partial Diff Eqs 2013;29(4):1081–96 [7] Liu J, Li H, Liu Y, Fang Z Reduced-order finite element method based on POD for fractional Tricomi-type equation Appl Math Mech 2016;37(5):647–58 [8] Dehghan M, Abbaszadeh M Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition Comput Math Appl 2017;73 (6):1270–85 [9] Ghehsareh HR, Raei M, Zaghian A Application of meshless local PetrovGalerkin technique to simulate two-dimensional time-fractional Tricomi-type problem J Brazil Soc Mech Sci Eng 2019;41(6):252 [10] Hosseininia M, Heydari M Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag-Leffler non-singular kernel Chaos, Solitons & Fractals 2019;127:389–99 [11] Shekari Y, Tayebi A, Heydari MH A meshfree approach for solving 2D variableorder fractional nonlinear diffusion-wave equation Comput Methods Appl Mech Eng 2019;350:154–68 [12] Hosseininia M, Heydari M, Avazzadeh Z Numerical study of the variable-order fractional version of the nonlinear fourth-order 2D diffusion-wave equation via 2D Chebyshev wavelets Eng Comput 2020:1–10 [13] Hosseininia M, Heydari M, Rouzegar J, Cattani C A meshless method to solve nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag-Leffler kernel Eng Comput 2019:1–13 [14] Hardy RL Multiquadric equations of topography and other irregular surfaces J Geophys Res 1971;76(8):1905–15 [15] Franke R Scattered data interpolation: tests of some methods Math Comput 1982;38(157):181–200 [16] Micchelli CA Interpolation of scattered data: distance matrices and conditionally positive definite functions In: Approximation theory and spline functions Springer; 1984 p 143–5 [17] Kansa EJ Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates Comput Math Appl 1990;19(8–9):127–45 [18] Kansa EJ Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations Comput Math Appl 1990;19(8–9):147–61 [19] Schaback R Error estimates and condition numbers for radial basis function interpolation Adv Comput Math 1995;3(3):251–64 [20] Larsson E, Fornberg B A numerical study of some radial basis function based solution methods for elliptic PDEs Comput Math Appl 2003;46(5–6):891–902 [21] Franke C, Schaback R Convergence order estimates of meshless collocation methods using radial basis functions Adv Comput Math 1998;8(4):381–99 [22] Madych W, Nelson S Multivariate interpolation and conditionally positive definite functions ii Math Comput 1990;54(189):211–30 [23] Sun Z-Z, Wu X A fully discrete difference scheme for a diffusion-wave system Appl Numer Math 2006;56(2):193–209 [24] Brezis H Functional analysis, Sobolev spaces and partial differential equations Springer Science & Business Media; 2010 [25] Der van Houwen P, Quarteroni A, Valli A Numerical approximation of partial differential equations Berlin etc., springer-verlag 1994 xvi, 543 pp., dm 128, 00 isbn 3-540-57111-6 (springer series in computational mathematics 23), Zeitschrift Angewandte Mathematik und Mechanik 1995;75: 550–550 [26] Rasoulizadeh MN, Rashidinia J Numerical solution for the Kawahara equation using local RBF-FD meshless method J King Saud Univ-Sci 2020;32 (4):2277–83 [27] Wendland H Scattered data approximation, Vol 17 Cambridge University Press; 2004 [28] Oanh DT, Davydov O, Phu HX Adaptive RBF-FD method for elliptic problems with point singularities in 2D Appl Math Comput 2017;313:474–97 [29] Bayona V, Flyer N, Fornberg B On the role of polynomials in RBF-FD approximations: III Behavior near domain boundaries J Comput Phys 2019;380:378–99 [30] Dehghan M, Abbaszadeh M The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations J Comput Phys 2017;351:478–510 [31] Dehghan M, Mohammadi V A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: RungeKutta method Comput Phys Commun 2017;217:23–34 [32] Tolstykh A, Shirobokov D On using radial basis functions in a ”finite difference mode” with applications to elasticity problems Comput Mech 2003;33 (1):68–79 [33] Wright GB, Fornberg B Scattered node compact finite difference-type formulas generated from radial basis functions J Comput Phys 2006;212(1):99–123 [34] Nikan O, Machado JT, Golbabai A, Nikazad T Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media Int Commun Heat Mass Transf 2020;111:104443 [35] Nikan O, Golbabai A, Machado JT, Nikazad T Numerical approximation of the time fractional cable equation arising in neuronal dynamics Eng Comput 2020:1–19 doi: https://doi.org/10.1007/s00366-020-01033-8 [36] Nikan O, Golbabai A, Machado JT, Nikazad T Numerical solution of the fractional Rayleigh-Stokes model arising in a heated generalized second-grade fluid Eng Comput 2020:1–14 doi: https://doi.org/10.1007/s00366-01900913-y [37] Nikan O, Machado JT, Golbabai A, Nikazad T Numerical investigation of the nonlinear modified anomalous diffusion process Nonlinear Dynam 2019;97 (4):2757–75 [38] Sarra SA A local radial basis function method for advection–diffusion–reaction equations on complexly shaped domains Appl Math Comput 2012;218 (19):9853–65 [39] Cui M Compact finite difference schemes for the time fractional diffusion equation with nonlocal boundary conditions Comput Appl Math 2018;37 (3):3906–26 [40] Fasshauer GE Meshfree Approximation Methods with Matlab: (With CDROM), Vol World Scientific Publishing Company; 2007 [41] Kamranian M, Dehghan M The finite point method for reaction-diffusion systems in developmental biology CMES Comput Model Eng Sci 2011;82 (1):1–27 ... equations of variable type with boundary condition Later, Frankl [4] showed that the gas flows with nearly sonic speeds could be described by the Tricomi model For the numerical solution of the... matrix of the LRBF-FD collocation procedure is more smaller than the coefficient matrix of the GRBF Table compares the absolute errors of the RBF-FD and those from [7,9] with several values of dt... fractional nonlinear diffusion-wave equation Comput Methods Appl Mech Eng 2019;350:154–68 [12] Hosseininia M, Heydari M, Avazzadeh Z Numerical study of the variable-order fractional version of