In this article, we propose a numerical method that is completely based on the operational matrices of fractional integral and derivative operators of fractional Legendre function vectors (FLFVs). The proposed method is independent of the choice of the suitable collocation points and expansion of the residual function as a series of orthogonal polynomials as required for Spectral collocation and Spectral tau methods. Consequently, the high efficient numerical results are obtained as compared to the other methods in the literature. The other novel aspect of our article is the development of the new integral and derivative operational matrices in Riemann-Liouville and Caputo senses respectively. The proposed method is computer-oriented and has the ability to reduce the fractional differential equations (FDEs) into a system of Sylvester types matrix equations that can be solved using MATLAB builtin function lyap(.). As an application of the proposed method, we solve multi-order FDEs with initial conditions. The numerical results obtained otherwise in the literature are also improved in our work.
Citation: Imran Talib, Md. Nur Alam, Dumitru Baleanu, Danish Zaidi, Ammarah Marriyam. A new integral operational matrix with applications to multi-order fractional differential equations[J]. AIMS Mathematics, 2021, 6(8): 8742-8771. doi: 10.3934/math.2021508
[1] | Maryam AlKandari . Nonlinear differential equations with neutral term: Asymptotic behavior of solutions. AIMS Mathematics, 2024, 9(12): 33649-33661. doi: 10.3934/math.20241606 |
[2] | Taher S. Hassan, Emad R. Attia, Bassant M. El-Matary . Iterative oscillation criteria of third-order nonlinear damped neutral differential equations. AIMS Mathematics, 2024, 9(8): 23128-23141. doi: 10.3934/math.20241124 |
[3] | Zuhur Alqahtani, Insaf F. Ben Saud, Areej Almuneef, Belgees Qaraad, Higinio Ramos . New criteria for the oscillation of a class of third-order quasilinear delay differential equations. AIMS Mathematics, 2025, 10(2): 4205-4225. doi: 10.3934/math.2025195 |
[4] | M. Sathish Kumar, V. Ganesan . Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay. AIMS Mathematics, 2020, 5(4): 3851-3874. doi: 10.3934/math.2020250 |
[5] | A. A. El-Gaber, M. M. A. El-Sheikh, M. Zakarya, Amirah Ayidh I Al-Thaqfan, H. M. Rezk . On the oscillation of solutions of third-order differential equations with non-positive neutral coefficients. AIMS Mathematics, 2024, 9(11): 32257-32271. doi: 10.3934/math.20241548 |
[6] | Lin Fan, Shunchu Li, Dongfeng Shao, Xueqian Fu, Pan Liu, Qinmin Gui . Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations. AIMS Mathematics, 2022, 7(7): 11972-11991. doi: 10.3934/math.2022667 |
[7] | Yibing Sun, Yige Zhao . Oscillatory and asymptotic behavior of third-order neutral delay differential equations with distributed deviating arguments. AIMS Mathematics, 2020, 5(5): 5076-5093. doi: 10.3934/math.2020326 |
[8] | Ahmed M. Hassan, Clemente Cesarano, Sameh S. Askar, Ahmad M. Alshamrani . Oscillatory behavior of solutions of third order semi-canonical dynamic equations on time scale. AIMS Mathematics, 2024, 9(9): 24213-24228. doi: 10.3934/math.20241178 |
[9] | Pengshe Zheng, Jing Luo, Shunchu Li, Xiaoxu Dong . Elastic transformation method for solving ordinary differential equations with variable coefficients. AIMS Mathematics, 2022, 7(1): 1307-1320. doi: 10.3934/math.2022077 |
[10] | Ali Muhib, Hammad Alotaibi, Omar Bazighifan, Kamsing Nonlaopon . Oscillation theorems of solution of second-order neutral differential equations. AIMS Mathematics, 2021, 6(11): 12771-12779. doi: 10.3934/math.2021737 |
In this article, we propose a numerical method that is completely based on the operational matrices of fractional integral and derivative operators of fractional Legendre function vectors (FLFVs). The proposed method is independent of the choice of the suitable collocation points and expansion of the residual function as a series of orthogonal polynomials as required for Spectral collocation and Spectral tau methods. Consequently, the high efficient numerical results are obtained as compared to the other methods in the literature. The other novel aspect of our article is the development of the new integral and derivative operational matrices in Riemann-Liouville and Caputo senses respectively. The proposed method is computer-oriented and has the ability to reduce the fractional differential equations (FDEs) into a system of Sylvester types matrix equations that can be solved using MATLAB builtin function lyap(.). As an application of the proposed method, we solve multi-order FDEs with initial conditions. The numerical results obtained otherwise in the literature are also improved in our work.
A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and 0≤(−1)k−1f(k−1)(x)<∞ for x∈I and k∈N, where f(0)(x) means f(x) and N is the set of all positive integers. See [1,2,3]. Theorem 12b in [3] states that a necessary and sufficient condition for a function f to be completely monotonic on the infinite interval (0,∞) is that the integral f(t)=∫∞0e−tsdτ(s) converges for s∈(0,∞), where τ(s) is nondecreasing on (0,∞). In other words, a function is completely monotonic on (0,∞) if and only if it is a Laplace transform of a nonnegative measure. This is one of many reasons why many mathematicians have been investigating completely monotonic functions for many decades.
Definition 1.1 ([4,5,6,7,8,9]). Let f(x) be a completely monotonic function on (0,∞) and denote f(∞)=limx→∞f(x). If for some r∈R the function xr[f(x)−f(∞)] is completely monotonic on (0,∞) but xr+ε[f(x)−f(∞)] is not for any positive number ε>0, then we say that the number r is completely monotonic degree of f(x) with respect to x∈(0,∞); if for all r∈R each and every xr[f(x)−f(∞)] is completely monotonic on (0,∞), then we say that completely monotonic degree of f(x) with respect to x∈(0,∞) is ∞.
The notation degdegxcm[f(x)] has been designed in [4] to denote completely monotonic degree r of f(x) with respect to x∈(0,∞). It is clear that completely monotonic degree degdegxcm[f(x)] of any completely monotonic function f(x) with respect to x∈(0,∞) is at leat 0. It was proved in [6] that completely monotonic degree degdegxcm[f(x)] equals ∞ if and only if f(x) is nonnegative and identically constant. This definition slightly modifies the corresponding one stated in [4] and related references therein. For simplicity, in what follows, we sometimes just say that degdegxcm[f(x)] is completely monotonic degree of f(x).
Why do we compute completely monotonic degrees? One can find simple but significant reasons in the second paragraph of [7] or in the papers [10,11,12,13] and closely related references therein. Completely monotonic degree is a new notion introduced in very recent years. See [4,6,9,11,12,14,15,16,17,18,19,20,21,22] and closely related references. This new notion can be used to more accurately measure and differentiate complete monotonicity. For example, the functions 1xα and 1xβ for α,β>0 and α≠β are both completely monotonic on (0,∞), but they are different completely monotonic functions. How to quantitatively measure their differences? How to quantitatively differentiate them from each other? The notion of completely monotonic degrees can be put to good use: The completely monotonic degrees of 1xα and 1xβ with respect to x∈(0,∞) for α,β>0 and α≠β are α and β respectively.
The classical Euler's gamma function Γ(x) can be defined for x>0 by Γ(x)=∫∞0tx−1e−tdt. The logarithmic derivative of Γ(x), denoted by ψ(x)=Γ′(x)Γ(x), is called the psi or digamma function, the derivatives ψ′(x) and ψ″(x) are respectively called the tri- and tetragamma functions. As a whole, the derivatives ψ(k)(x) for k≥0 are called polygamma functions. For new results on Γ(z) and ψ(k)(x) in recent years, please refer to [7,11,23,24,25,26,27,28,29] and closely related references therein.
Why do we still study the gamma and polygamma functions Γ(z) and ψ(k)(z) for k≥0 nowadays? Because this kind of functions are not elementary and are the most applicable functions in almost all aspects of mathematics and mathematical sciences.
Let
Ψ(x)=[ψ′(x)]2+ψ″(x),x∈(0,∞). | (2.1) |
In [30], it was established that the inequality
Ψ(x)>p(x)900x4(x+1)10 | (2.2) |
holds for x>0, where
p(x)=75x10+900x9+4840x8+15370x7+31865x6+45050x5+44101x4+29700x3+13290x2+3600x+450. |
It is clear that the inequality
Ψ(x)>0 | (2.3) |
for x>0 is a weakened version of the inequality (2.2). This inequality was deduced and recovered in [31,32]. The inequality (2.3) was also employed in [31,32,33,34]. This inequality has been generalized in [33,35,36,37]. For more information about the history and background of this topic, please refer to the expository and survey articles [11,38,39,40,41] and plenty of references therein.
In the paper [42], it was proved that, among all functions [ψ(m)(x)]2+ψ(n)(x) for m,n∈N, only the function Ψ(x) is nontrivially completely monotonic on (0,∞).
x+1212x4(x+1)−Ψ(x),Ψ(x)−x2+1212x4(x+1)2,Ψ(x)−p(x)900x4(x+1)10 |
were proved to be completely monotonic on (0,∞). From this, we obtain
max{x2+1212x4(x+1)2,p(x)900x4(x+1)10}<Ψ(x)<x+1212x4(x+1) | (2.4) |
for x>0. In [45], the function
hλ(x)=Ψ(x)−x2+λx+1212x4(x+1)2 | (2.5) |
was proved to be completely monotonic on (0,∞) if and only if λ≤0, and so is −hλ(x) if and only if λ≥4; Consequently, the double inequality
x2+μx+1212x4(x+1)2<Ψ(x)<x2+νx+1212x4(x+1)2 | (2.6) |
holds on (0,∞) if and only if μ≤0 and ν≥4. The inequality (2.6) refines and sharpens the right-hand side inequality in (2.4).
It was remarked in [40] that a divided difference version of the inequality (2.3) has been implicitly obtained in [46]. The divided difference form of the function Ψ(x) and related functions have been investigated in the papers [47,48,49,50,51] and closely related references therein. There is a much complete list of references in [52].
In [14,16], among other things, it was deduced that the functions x2Ψ(x) and x3Ψ(x) are completely monotonic on (0,∞). Equivalently,
degdegxcm[Ψ(x)]≥2anddegdegxcm[Ψ(x)]≥3. | (2.7) |
Motivated by these results, we naturally pose the following two questions:
1. is the function x4Ψ(x) completely monotonic on (0,∞)?
2. is α≤4 the necessary and sufficient condition for the function xαΨ(x) to be completely monotonic on (0,∞)?
In other words, is the constant 4 completely monotonic degree of Ψ(x) with respect to x∈(0,∞)?
In order to affirmatively and smoothly answer the above questions, we need five lemmas below.
Lemma 3.1 ([29]). For n∈N and x>0,
ψ(n)(x)=(−1)n+1∫∞0tn1−e−te−xtdt. | (3.1) |
Lemma 3.2 ([3,29]). Let fi(t) for i=1,2 be piecewise continuous in arbitrary finite intervals included in (0,∞) and suppose that there exist some constants Mi>0 and ci≥0 such that |fi(t)|≤Miecit for i=1,2. Then
∫∞0[∫t0f1(u)f2(t−u)du]e−stdt=∫∞0f1(u)e−sudu∫∞0f2(v)e−svdv. | (3.2) |
Lemma 3.3 ([53]). Let f(x,t) is differentiable in t and continuous for (x,t)∈R2. Then
ddt∫tx0f(x,t)dx=f(t,t)+∫tx0∂f(x,t)∂tdx. |
Lemma 3.4 ([54,55,56]). If fi for 1≤i≤n are nonnegative Lebesgue square integrable functions on [0,a) for all a>0, then
f1∗⋯∗fn(x)≥xn−1(n−1)!exp[n−1xn−1∫x0(x−u)n−2n∑j=1lnfj(u)du] | (3.3) |
for all n≥2 and x≥0, where fi∗fj(x) denotes the convolution ∫x0fi(t)fj(x−t)dt.
Lemma 3.5 ([29]). As z→∞ in |argz|<π,
ψ′(z)∼1z+12z2+16z3−130z5+142z7−130z9+⋯,ψ″(z)∼−1z2−1z3−12z4+16z6−16z8+310z10−56z12+⋯,ψ(3)(z)∼2z3+3z4+2z5−1z7+43z9−3z11+10z13−⋯. |
The formulas listed in Lemma 3.5 are special cases of [29].
Now we are in a position to compute completely monotonic degree of the function Ψ(x).
Theorem 4.1. Completely monotonic degree of Ψ(x) defined by (2.1) with respect to x∈(0,∞) is 4. In other words,
degdegxcm[Ψ(x)]=4. | (4.1) |
Proof. Using the integral representation (3.1) and the formula (3.2) gives
Ψ(x)=[∫∞0t1−e−te−xtdt]2−∫∞0t21−e−te−xtdt=∫∞0[∫t0s(t−s)(1−e−s)[1−e−(t−s)]ds−t21−e−t]e−xtdt=∫∞0q(t)e−xtdt, |
where
q(t)=∫t0σ(s)σ(t−s)ds−tσ(t)andσ(s)={s1−e−s,s≠01,s=0. | (4.2) |
Direct calculations reveal
σ′(s)=1+1−ses−1−s(es−1)2,σ″(s)=s−2es−1+3s−2(es−1)2+2s(es−1)3,σ(3)(s)=3−ses−1+9−7s(es−1)2−6(2s−1)(es−1)3−6s(es−1)4,σ(4)(s)=s−4es−1+15s−28(es−1)2+2(25s−24)(es−1)3+12(5s−2)(es−1)4+24s(es−1)5,σ(5)(s)=5−ses−1+75−31s(es−1)2−10(18s−25)(es−1)3−30(13s−10)(es−1)4−120(3s−1)(es−1)5−120s(es−1)6,σ(6)(s)=s−6es−1+3(21s−62)(es−1)2+2(301s−540)(es−1)3+60(35s−39)(es−1)4+240(14s−9)(es−1)5+360(7s−2)(es−1)6+720s(es−1)7, |
and
σ(0)=1,σ′(0)=12,σ″(0)=16,σ(3)(0)=0,σ(4)(0)=−130,σ(5)(0)=0,σ(6)(0)=142. |
Further differentiating consecutively brings out
[lnσ″(s)]′=−(s−3)e2s+4ses+s+3[(s−2)es+s+2](es−1),[lnσ″(s)]″=−e4s−4(s2−3s+4)e3s−(4s2−30)e2s−4(s2+3s+4)es+1(es−1)2[(s−2)es+s+2]2≜−h1(s)(es−1)2[(s−2)es+s+2]2,h′1(s)=4[e3s−(3s2−7s+9)e2s−(2s2+2s−15)es−s2−5s−7]es≜4h2(s)es,h′2(s)=3e3s−(6s2−8s+11)e2s−(2s2+6s−13)es−2s−5,h″2(s)=9e3s−2(6s2−2s+7)e2s−(2s2+10s−7)es−2,h(3)2(s)=[27e2s−8es(3s2+2s+3)−2s2−14s−3]es≜h3(s)es,h′3(s)=54e2s−8(3s2+8s+5)es−2(2s+7),h″3(s)=4[27e2s−2(3s2+14s+13)es−1],h(3)3(s)=8(27es−3s2−20s−27)es>0 |
for s∈(0,∞), and
h″3(0)=h′3(0)=h3(0)=h(3)2(0)=h″2(0)=h′2(0)=h2(0)=h′1(0)=h1(0)=0. |
This means that
h″3(s)>0,h′3(s)>0,h3(s)>0,h(3)2(s)>0,h″2(s)>0,h′2(s)>0,h2(s)>0,h′1(s)>0,h1(s)>0 |
for s∈(0,∞). Therefore, the derivative [lnσ″(s)]″ is negative, that is, the function σ″(s) is logarithmically concave, on (0,∞). Hence, for any given number t>0,
1. the function σ″(s)σ″(t−s) is also logarithmically concave with respect to s∈(0,t);
2. the function σ″(s) is decreasing and σ(s) is not concave on (0,∞).
By Lemma 3.3 and integration-by-part, straightforward computations yield
q′(t)=∫t0σ(s)σ′(t−s)ds+σ(0)σ(t)−[tσ′(t)+σ(t)]=∫t0σ(s)σ′(t−s)ds−tσ′(t),q″(t)=∫t0σ(s)σ″(t−s)ds+σ(t)σ′(0)−[σ′(t)+tσ″(t)]=−∫t0σ(s)dσ′(t−s)dsds+σ(t)σ′(0)−[σ′(t)+tσ″(t)]=∫t0σ′(s)σ′(t−s)ds−tσ″(t),q(3)(t)=∫t0σ′(s)σ″(t−s)ds+12σ′(t)−σ″(t)−tσ(3)(t),q(4)(t)=∫t0σ′(s)σ(3)(t−s)ds+16σ′(t)+12σ″(t)−2σ(3)(t)−tσ(4)(t)=−∫t0σ′(s)dσ″(t−s)dsds+16σ′(t)+12σ″(t)−2σ(3)(t)−tσ(4)(t)=∫t0σ″(s)σ″(t−s)ds+σ″(t)−2σ(3)(t)−tσ(4)(t)=2∫t/20σ″(s)σ″(t−s)ds+σ″(t)−2σ(3)(t)−tσ(4)(t), |
and
q(0)=q′(0)=q″(0)=0,q(3)(0)=112,q(4)(0)=16. |
Applying Lemma 3.4 to f1=f2=σ″ and n=2 leads to
∫t0σ″(s)σ″(t−s)ds≥texp[2t∫t0lnσ″(u)du]. |
Hence, the validity of the inequality
texp[2t∫t0lnσ″(u)du]+σ″(t)−2σ(3)(t)−tσ(4)(t)>0 | (4.3) |
implies the positivity of q(4)(t) on (0,∞).
When tσ(4)(t)+2σ(3)(t)−σ″(t)≤0, the inequality (4.3) is clearly valid.
When tσ(4)(t)+2σ(3)(t)−σ″(t)>0, the inequality (4.3) can be rearranged as
∫t0lnσ″(u)du>t2lntσ(4)(t)+2σ(3)(t)−σ″(t)t. |
Let
F(t)=∫t0lnσ″(u)du−t2lntσ(4)(t)+2σ(3)(t)−σ″(t)t. |
Differentiating twice produces
F′(t)=lnσ″(t)−12lntσ(4)(t)+2σ(3)(t)−σ″(t)t−t2σ(5)(t)+2tσ(4)(t)−(t+2)σ(3)(t)+σ″(t)2[tσ(4)(t)+2σ(3)(t)−σ″(t)] |
and
F″(t)=σ(3)(t)σ″(t)−t2σ(5)(t)+2tσ(4)(t)−(t+2)σ(3)(t)+σ″(t)2t[tσ(4)(t)+2σ(3)(t)−σ″(t)]−12[tσ(4)(t)+2σ(3)(t)−σ″(t)]2([t2σ(6)(t)+4tσ(5)(t)−tσ(4)(t)][tσ(4)(t)+2σ(3)(t)−σ″(t)]−[t2σ(5)(t)+2tσ(4)(t)−(t+2)σ(3)(t)+σ″(t)]×[tσ(5)(t)+3σ(4)(t)−σ(3)(t)])≜Q(t)2tσ″(t)[tσ(4)(t)+2σ(3)(t)−σ″(t)]2, |
where
Q(t)=2tσ(3)(t)[tσ(4)(t)+2σ(3)(t)−σ″(t)]2−σ″(t)[tσ(4)(t)+2σ(3)(t)−σ″(t)][t2σ(5)(t)+2tσ(4)(t)−(t+2)σ(3)(t)+σ″(t)]−tσ″(t){[t2σ(6)(t)+4tσ(5)(t)−tσ(4)(t)][tσ(4)(t)+2σ(3)(t)−σ″(t)]−[t2σ(5)(t)+2tσ(4)(t)−(t+2)σ(3)(t)+σ″(t)][tσ(5)(t)+3σ(4)(t)−σ(3)(t)]}≜e3tR(t)(et−1)15 |
and
R(t)=e9t(t5−12t4+70t3−160t2+192t−128)−e8t(16t7−220t6+1219t5−3220t4+4490t3−3248t2+1152t−768)−4e7t(37t7−423t6+1397t5−1409t4−1020t3+2632t2−732t+456)−4e6t(225t7−1281t6+1213t5+3127t4−4372t3−2648t2+1020t−504)−2e5t(908t7−1514t6−6493t5+8710t4+12754t3−1216t2−1656t+336)−2e4t(908t7+1710t6−5489t5−12370t4+594t3+4880t2+696t+336)−4e3t(225t7+1263t6+1771t5−887t4−3208t3−728t2+12t−168)−4e2t(37t7+353t6+1099t5+1337t4+272t3−632t2−108t+24)−et(16t7+180t6+827t5+1864t4+2226t3+1312t2+240t+96)+t5+8t4+30t3+48t2+48t+32. |
Differentiating and taking the limit t→0 about 76 times respectively by the same approach as either the proof of the positivity of θ(t) in [43], or proofs of the absolute monotonicity of the functions f1,f2,f3 and h1,h2,h3,h4 in [57], or the proof of the positivity of h1(s) on page 3396 in this paper, we can verify the positivity of R(t) on (0,∞). In [58], a stronger conclusion than the positivity of R(t) on (0,∞) was proved in details. This means that Q(t)>0 on (0,∞) and F″(t)>0. Accordingly, the derivative F′(t) is strictly increasing. Because
F′(8)=4+3(6e32+729e24+2825e16+1483e8+77)8e32+270e24+150e16−374e8−54+12ln8(5+3e8)(e8−1)(27+214e8+139e16+4e24)=−0.24428… |
and
F′(10)=5+72e40+4715e30+16563e20+8241e10+40919e40+440e30+186e20−568e10−77+12ln80(3+2e10)2(e10−1)(77+645e10+459e20+19e30)=0.20823…, |
which are numerically calculated with the help of the software MATHEMATICA, the unique zero of F′(t) locates on the open interval (8,10). Consequently, the unique minimum of the function F(t) attains on the interval (8,10). Since
F(t)=F(t0)+(t−t0)F′(t0)+(t−t0)22F″(ξ)>F(t0)+(t−t0)F′(t0) |
for t,t0∈[8,10], where ξ locates between t0 and t, numerically calculating with the help of the software MATHEMATICA gains
2F(t)>[F(8)+(t−8)F′(8)]+[F(10)+(t−10)F′(10)]=F(8)+F(10)−[8F′(8)+10F′(10)]+[F′(8)+F′(10)]t>∫80lnσ″(u)du−4lne8(27+214e8+139e16+4e24)2(e8−1)5+∫100lnσ″(u)du−5lne10(77+645e10+459e20+19e30)5(e10−1)5−0.1281−0.0361t>∫80lnσ″(u)du+∫100lnσ″(u)du+72.492−0.1281−0.361>∫80lnσ″(u)du+∫100lnσ″(u)du+72>13[24∑k=1lnσ″(k3)+30∑k=1lnσ″(k3)]+72>−29−43+72=0 |
on the interval [8,10]. In conclusion, the inequality (4.3) is valid and the fourth derivative q(4)(t) is positive on (0,∞).
Integrating by parts successively results in
x4Ψ(x)=x4∫∞0q(t)e−xtdt=−x3∫∞0q(t)de−xtdtdt=−x3[q(t)e−xt|t=∞t=0−∫∞0q′(t)e−xtdt]=x3∫∞0q′(t)e−xtdt=x2∫∞0q″(t)e−xtdt=x∫∞0q(3)(t)e−xtdt=−∫∞0q(3)(t)de−xtdtdt=−[q(3)(t)e−xt|t=∞t=0−∫∞0q(4)(t)de−xtdtdt]=112+∫∞0q(4)(t)e−xtdt. |
From the positivity of q(4)(t) on (0,∞), it follows that the function x4Ψ(x) is completely monotonic on (0,∞). In other words,
degdegxcm[Ψ(x)]≥4. | (4.4) |
Suppose that the function
fα(x)=xαΨ(x) |
is completely monotonic on (0,∞). Then
f′α(x)=xα−1{αΨ(x)+x[2ψ′(x)ψ″(x)+ψ(3)(x)]}≤0 |
on (0,∞), that is,
α≤−x[2ψ′(x)ψ″(x)+ψ(3)(x)]Ψ(x)≜ϕ(x),x>0. |
From Lemma 3.5, it follows
limx→∞ϕ(x)=−limx→∞{x[1x+12x2+O(1x2)]2+[−1x2−1x3+O(1x3)]×[2(1x+12x2+O(1x2))(−1x2−1x3+O(1x3))+(2x3+3x4+O(1x4))]}=4. |
As a result, we have
degdegxcm[Ψ(x)]≤4. | (4.5) |
Combining (4.4) with (4.5) yields (4.1). The proof of Theorem 4.1 is complete.
Recall from [59] that a function f is said to be strongly completely monotonic on (0,∞) if it has derivatives of all orders and (−1)nxn+1f(n)(x) is nonnegative and decreasing on (0,∞) for all n≥0.
Theorem 5.1 ([18]). A function f(x) is strongly completely monotonic on (0,∞) if and only if the function xf(x) is completely monotonic on (0,∞).
This theorem implies that the set of completely monotonic functions whose completely monotonic degrees are not less than 1 with respect to x∈(0,∞) coincides with the set of strongly completely monotonic functions on (0,∞).
Because not finding a proof for [18] anywhere, we now provide a proof for Theorem 5.1 as follows.
Proof of Theorem 5.1. If xf(x) is completely monotonic on (0,∞), then
(−1)k[xf(x)](k)=(−1)k[xf(k)(x)+kf(k−1)(x)]=(−1)kxk+1f(k)(x)−k[(−1)k−1xkf(k−1)(x)]xk≥0 |
on (0,∞) for all integers k≥0. From this and by induction, we obtain
(−1)kxk+1f(k)(x)≥k[(−1)k−1xkf(k−1)(x)]≥k(k−1)[(−1)k−2xk−1f(k−2)(x)]≥⋯≥[k(k−1)⋯4⋅3]x3f″(x)≥[k(k−1)⋯4⋅3⋅2]x2f′(x)≥k!xf(x)≥0 |
on (0,∞) for all integers k≥0. So, the function f(x) is strongly completely monotonic on (0,∞).
Conversely, if f(x) is a strongly completely monotonic function on (0,∞), then
(−1)kxk+1f(k)(x)≥0 |
and
[(−1)kxk+1f(k)(x)]′=(k+1)[(−1)kxk+1f(k)(x)]−(−1)k+1xk+2f(k+1)(x)x≤0 |
hold on (0,∞) for all integers k≥0. Hence, it follows that xf(x)≥0 and (−1)k+1[xf(x)](k+1) on (0,∞) for all integers k≥0. As a result, the function xf(x) is completely monotonic on (0,∞). The proof of Theorem 5.1 is complete.
Now we prove a property of logarithmically concave functions.
Theorem 6.1. If f(x) is differentiable and logarithmically concave (or logarithmically convex, respectively ) on (−∞,∞), then the product f(x)f(λ−x) for any fixed number λ∈R is increasing (or decreasing, respectively ) with respect to x∈(−∞,λ2) and decreasing (or increasing, respectively ) with respect to x∈(λ2,∞).
Proof. Taking the logarithm of f(x)f(λ−x) and differentiating give
{ln[f(x)f(λ−x)]}′=f′(x)f(x)−f′(λ−x)f(λ−x). |
In virtue of the logarithmic concavity of f(x), it follows that the function f′(x)f(x) is decreasing and f′(λ−x)f(λ−x) is increasing on (−∞,∞). From the obvious fact that {ln[f(x)f(λ−x)]}′|x=λ/2=0, it is deduced that {ln[f(x)f(λ−x)]}′<0 for x>λ2 and {ln[f(x)f(λ−x)]}′>0 for x<λ2. Hence, the function f(x)f(λ−x) is decreasing for x>λ2 and increasing for x<λ2.
For the case of f(x) being logarithmically convex, it can be proved similarly.
In this section, we list several remarks on our main results and pose two open prblems.
Remark 7.1. The function σ(s) defined in (4.2) is a special case of the function
ga,b(s)={sbs−as,s≠0,1lnb−lna,s=0, |
where a,b are positive numbers and a≠b. Some special cases of the function ga,b(s) and their reciprocals have been investigated and applied in many papers such as [6,8,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75]. This subject was also surveyed in [76]. Recently, it was discovered that the derivatives of the function σ(s)s=11−e−s have something to do with the Stirling numbers of the first and second kinds in combinatorics and number theory. For detailed and more information, please refer to [77,78,79,80,81,82,83,84,85,86,87,88,89].
By Theorem 6.1, it can be deduced that the function σ″(s)σ″(t−s) is increasing with respect to s∈(0,t2) and decreasing with respect to s∈(t2,t), where σ is defined in (4.2).
The techniques used in the proof of Theorem 6.1 was ever utilized in the papers [70,90,91,92] and closely related references therein.
Remark 7.2. The result obtained in Theorem 4.1 in this paper affirmatively answers those questions asked on page 3393 at the end of Section 2. Therefore, the result in Theorem 4.1 strengthens, improves, and sharpens those results in (2.7). This implies that other results established in [14,16] can also be further improved, developed, or amended.
Remark 7.3 (First open problem). Motivated by Lemma 3.4, the proof of Theorem 4.1, and Theorem 6.1, we pose the following open problem: when fi for 1≤i≤n are all logarithmically concave on [0,a) for all a>0, can one find a stronger lower bound than the one in (3.3) for the convolution f1∗f2∗⋯∗fn(x)?
Remark 7.4 (Second open problem). We conjecture that the completely monotonic degrees with respect to x∈(0,∞) of the functions hλ(x) and −hμ(x) defined by (2.5) are 4 if and only if λ≤0 and μ≥4. In other words,
degdegxcm[hλ(x)]=degdegxcm[−hμ(x)]=4 |
if and only if λ≤0 and μ≥4.
Remark 7.5. This paper is a revised and shortened version of the preprint [93].
In ths paper, the author proved that the completely monotonic degree of the function [ψ′(x)]2+ψ″(x) with respect to x∈(0,∞) is 4, verified that the set of all strongly completely monotonic functions on (0,∞) coincides with the set of functions whose completely monotonic degrees are greater than or equal to 1 on (0,∞), presented a property of logarithmically concave functions, and posed two open problems on a stronger lower bound of the convolution of finite many functions and on completely monotonic degree of a kind of completely monotonic functions on (0,∞).
The author thanks anonymous referees for their careful corrections to, helpful suggestions to, and valuable comments on the original version of this manuscript.
The author declares that he have no conflict of interest.
[1] | E. Ahmed, A. Elgazzar, On fractional order differential equations model for non-local epidemics, Phys. A., 15 (2007), 607-614. |
[2] |
W. Chen, A speculative study of 23-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures, Chaos, 16 (2006), doi: 10.1063/1.2208452. doi: 10.1063/1.2208452
![]() |
[3] |
W. Chen, H. Sun, X. Zhang, D. Korosak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754-1758. doi: 10.1016/j.camwa.2009.08.020
![]() |
[4] | M. A. Z. Raja, J. A. Khan, I. M. Qureshi, Solution of fractional order system of Bagley-Torvik equation using evolutionary computational Iintelligence, Math. Probl. Eng., (2011), doi: 10.1155/2011/675075. |
[5] | M. Gülsu, Y. O¨ztürk, A. Anapali, Numerical solution of the fractional Bagley-Torvik equation arising in fluid mechanics, Int. J. Comput. Math., (2015), doi: 10.1080/00207160.2015.1099633. |
[6] | S. Yüzbaşi, Numerical solution of the Bagley-Torvik equation by the Bessel collocation method, Math. Meth. Appl. Sci., (2012), doi: 10.1002/mma.2588. |
[7] |
H. Sun, D. Chen, Y. Zhang, L. Chen, Understanding partial bed-load transport: Experiments and stochastic model analysis, J. Hydrol., 521 (2015), 196-204. doi: 10.1016/j.jhydrol.2014.11.064
![]() |
[8] |
H. Sun, W. Chen, Y. Chen, Variable-order fractional differential operators in anomalous diffusion modeling, Phys. A., 388 (2009), 4586-4592. doi: 10.1016/j.physa.2009.07.024
![]() |
[9] |
Y. Rossikhin, M. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mech., 120 (1997), 109-125. doi: 10.1007/BF01174319
![]() |
[10] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations, New York, NY, USA, Elsevier Science B.V., 2006. |
[11] | R. El Attar, Special Functions and Orthogonal Polynomials, New York, Lulu Press, 2006. |
[12] | I. Podlubny, Fractional Differential equations, New York, Academic Press, 1998. |
[13] |
A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326-1336. doi: 10.1016/j.camwa.2009.07.006
![]() |
[14] |
E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl., 62 (2011), 2364-2373. doi: 10.1016/j.camwa.2011.07.024
![]() |
[15] |
M. H. Atabakzadeh, M. H. Akrami, G. H. Erjaee, Chebyshev Operational Matrix Method for Solving Multi-order Fractional Ordinary Differential Equations, Appl. Math. Model., 37 (2013), 8903-8911. doi: 10.1016/j.apm.2013.04.019
![]() |
[16] |
E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35 (2011), 5662-5672. doi: 10.1016/j.apm.2011.05.011
![]() |
[17] |
H. Zhang, X. Jiang, F. Zeng, G. Em Karniadakis, A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations, J. Comput. Phys., 405 (2020), 109141. doi: 10.1016/j.jcp.2019.109141
![]() |
[18] | H. Zhang, X. Jiang, X. Yang, A time-space spectral method for the time-space fractional Fokker-Planck equation and its inverse problem, Appl. Math. Comput., 320 (2018), 302-318. |
[19] |
S. Kazem, S. Abbasbandy, S. Kumar, Fractional-order Legendre functions for solving fractional-order differential equations, Appl. Math. Model., 37 (2013), 5498-5510. doi: 10.1016/j.apm.2012.10.026
![]() |
[20] |
A. H. Bhrawya, A. S. Alofi, The operational matrix of fractional integration for shifted Chebyshev polynomials, Appl. Math. Lett., 26 (2013), 25-31. doi: 10.1016/j.aml.2012.01.027
![]() |
[21] | W. Han, Y. M. Chen, D. Y. Liu, X. L. Li, D. Boutat, Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis, Adv. Difference Equ., 2018 (2018). Available from: https://doi.org/10.1186/s13662-018-1702-z. |
[22] |
S. Kazem, An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations, Appl. Math. Model., 37 (2013), 1126-1136. doi: 10.1016/j.apm.2012.03.033
![]() |
[23] | S. Z. Rida, A. M. Yousef, On the fractional order Rodrigues formula for the Legendre polynomials, Adv. Appl. Math. Sci., 10 (2011), 509-518. |
[24] |
R. A. Khan, H. Khalil, New method based on legendre polynomials for solution of system of fractional order partial differential equations, Int. J. Comput. Math., 91 (2014), 2554-2567. doi: 10.1080/00207160.2014.880781
![]() |
[25] |
I. Talib, C. Tunc, Z. A. Noor, operational matrices of orthogonal Legendre polynomials and their operational, J. Taibah Univ. Sci., 13 (2019), 377-389. doi: 10.1080/16583655.2019.1580662
![]() |
[26] | I. Talib, F. B. Belgacem, N. A. Asif, H. Khalil, On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions, AIP Conference Proceedings, 1798 (2017). Available from: https://doi.org/10.1063/1.4972616. |
[27] | R. Garrappa, E. Kaslik, M. Popolizio, Evaluation of fractional integrals and derivatives of elementary functions: Overview and tutorial, Mathematics, 407 (2019), 1-21. |
[28] | K. Diethelm, N. J. Ford, Numerical solution of the Bagley-Torvik equation, BIT., 42 (2002), 490-507. |
[29] |
I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 674-684. doi: 10.1016/j.cnsns.2007.09.014
![]() |
[30] | Z. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order, J. Appl. Math. Inform., 26 (2008), 15-27. |
[31] |
Z. Odibat, S. Momani, Analytical comparison between the homotopy perturbation method and variational iteration method for differentialequations of fractional order, Int. J. Mod. Phys., 22 (2008), 4041-4058. doi: 10.1142/S0217979208048851
![]() |
[32] |
A. Bolandtalat, E. Babolian, H. Jafari, Numerical solutions of multi-order fractional differential equations by Boubaker polynomials, Open Phys., 14 (2016), 226-230. doi: 10.1515/phys-2016-0028
![]() |
[33] |
D. Zeidan, S. Govekar, M. Pandey, Discontinuity wave interactions in generalized magnetogasdynamics, Acta Astronautica, 180 (2021), 110-114. doi: 10.1016/j.actaastro.2020.12.025
![]() |
[34] |
F. Sultana, D. Singh, R. K. Pandey, D. Zeidan, Numerical schemes for a class of tempered fractional integro-differential equations, Appl. Numer. Math., 157 (2020), 110-134. doi: 10.1016/j.apnum.2020.05.026
![]() |
[35] |
D. Zeidan, B. Bira, Weak shock waves and its interaction with characteristic shocks in polyatomic gas, Math. Meth. Appl. Sci., 42 (2019), 4679-4687. doi: 10.1002/mma.5675
![]() |
[36] | D. Zeidan, C. K. Chau, T. Tzer-Lu, On the characteristic Adomian decomposition method for the Riemann problem, Math. Meth. Appl. Sci., (2019). Available from: https://doi.org/10.1002/mma.5798. |
[37] | E. Goncalves, D. Zeidan, Simulation of compressible two-phase flows using a void ratio transport equation, Commun. Comput. Phys., 24 (2018), 167-203. |
[38] | H. Mandal, B. Bira, D. Zeidan, Power series solution of time-fractional Majda-Biello system using lie group analysis, Proceedings of International Conference on Fractional Differentiation and its Applications (ICFDA), 2018. Available from: https://doi.org/10.2139/ssrn.3284751. |
[39] | E. Goncalves, D. Zeidan, Numerical study of turbulent cavitating flows in thermal regime, Int. J. Numer. Meth. Fl., 27 (2017), 1487-1503. |
[40] |
S. Kuila, T. Raja Sekhar, D. Zeidan, On the Riemann problem simulation for the Drift-Flux equations of two-Phase flows, Int. J. Comput. Methods, 13 (2016), 1650009. doi: 10.1142/S0219876216500092
![]() |
[41] |
X. Zheng, H. Wang, An error estimate of a numerical approximation to a Hidden-Memory variable-order space-time fractional Diffusion equation, SIAM J. Numer. Anal., 58 (2020), 2492-2514. doi: 10.1137/20M132420X
![]() |
[42] |
H. Wang, X. Zheng, Wellposedness and regularity of the variable-order time-fractional diffusion equations, J. Math. Anal. Appl., 475 (2019), 1778-1802. doi: 10.1016/j.jmaa.2019.03.052
![]() |
[43] |
X. Zheng, H. Wang, Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions, IMA J. Numer. Anal., 41 (2021), 1522-1545. doi: 10.1093/imanum/draa013
![]() |
[44] |
X. Zheng, H. Wang, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes, SIAM J. Numer. Anal., 58 (2020), 330-352. doi: 10.1137/19M1245621
![]() |
[45] |
K. Kumar, R. K. Pandey, S. Sharma, Comparative study of three numerical schemes for fractional integro-differential equations, J. Comput. Appl. Math., 315 (2017), 287-302. doi: 10.1016/j.cam.2016.11.013
![]() |
1. | Felix Sadyrbaev, On Solutions of the Third-Order Ordinary Differential Equations of Emden-Fowler Type, 2023, 3, 2673-8716, 550, 10.3390/dynamics3030028 | |
2. | Asma Al-Jaser, Clemente Cesarano, Belgees Qaraad, Loredana Florentina Iambor, Second-Order Damped Differential Equations with Superlinear Neutral Term: New Criteria for Oscillation, 2024, 13, 2075-1680, 234, 10.3390/axioms13040234 | |
3. | Asma Al-Jaser, Insaf F. Ben Saoud, Higinio Ramos, Belgees Qaraad, Investigation of the Oscillatory Behavior of the Solutions of a Class of Third-Order Delay Differential Equations with Several Terms, 2024, 13, 2075-1680, 703, 10.3390/axioms13100703 | |
4. | Najiyah Omar, Stefano Serra-Capizzano, Belgees Qaraad, Faizah Alharbi, Osama Moaaz, Elmetwally M. Elabbasy, More Effective Criteria for Testing the Oscillation of Solutions of Third-Order Differential Equations, 2024, 13, 2075-1680, 139, 10.3390/axioms13030139 | |
5. | Mohamed Mazen, Mohamed M. A. El-Sheikh, Samah Euat Tallah, Gamal A. F. Ismail, On the Oscillation of Fourth-Order Delay Differential Equations via Riccati Transformation, 2025, 13, 2227-7390, 494, 10.3390/math13030494 |