This study refines a single-layer Feed-Forward Neural Network (FFNN) for the treatment of textile dye wastewater, concentrating on percentage decolorization (%DEC) and percentage chemical oxygen demand (%COD) reduction. The optimized neural network configuration comprises four input and one output neuron, fine-tuned based on the mean squared error (MSE). The training phase demonstrates a consistent MSE decline, reaching its lowest at epoch 209 for %DEC and epoch 34 for %COD, with corresponding MSEs of 1.799×10−5 and 1.4×10−3, respectively. The maximum absolute errors for %DEC and %COD were found to be 4.0787 and 2.4486, while the mean absolute errors were 0.4821 and 0.7256, respectively. In contrast to second-degree polynomial regression, the FFNN model exhibits enhanced predictive accuracy, as indicated by higher R2 values of 0.99363 for %DEC and 0.99716 for %COD, and reduced error metrics.
Citation: Ali S. Alkorbi, Muhammad Tanveer, Humayoun Shahid, Muhammad Bilal Qadir, Fayyaz Ahmad, Zubair Khaliq, Mohammed Jalalah, Muhammad Irfan, Hassan Algadi, Farid A. Harraz. Comparative analysis of feed-forward neural network and second-order polynomial regression in textile wastewater treatment efficiency[J]. AIMS Mathematics, 2024, 9(5): 10955-10976. doi: 10.3934/math.2024536
[1] | Tekin Toplu, Mahir Kadakal, İmdat İşcan . On n-Polynomial convexity and some related inequalities. AIMS Mathematics, 2020, 5(2): 1304-1318. doi: 10.3934/math.2020089 |
[2] | Naila Mehreen, Matloob Anwar . Some inequalities via Ψ-Riemann-Liouville fractional integrals. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403 |
[3] | Hengxiao Qi, Muhammad Yussouf, Sajid Mehmood, Yu-Ming Chu, Ghulam Farid . Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Mathematics, 2020, 5(6): 6030-6042. doi: 10.3934/math.2020386 |
[4] | Ghulam Farid, Saira Bano Akbar, Shafiq Ur Rehman, Josip Pečarić . Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity. AIMS Mathematics, 2020, 5(2): 966-978. doi: 10.3934/math.2020067 |
[5] | Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable fractional integral inequalities for GG- and GA-convex functions. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322 |
[6] | Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable integral version of Hermite-Hadamard-Fejér inequalities via η-convex functions. AIMS Mathematics, 2020, 5(5): 5106-5120. doi: 10.3934/math.2020328 |
[7] | Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334 |
[8] | Muhammad Uzair Awan, Nousheen Akhtar, Artion Kashuri, Muhammad Aslam Noor, Yu-Ming Chu . 2D approximately reciprocal ρ-convex functions and associated integral inequalities. AIMS Mathematics, 2020, 5(5): 4662-4680. doi: 10.3934/math.2020299 |
[9] | Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang . Generalized inequalities for integral operators via several kinds of convex functions. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297 |
[10] | Xiuzhi Yang, G. Farid, Waqas Nazeer, Muhammad Yussouf, Yu-Ming Chu, Chunfa Dong . Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions. AIMS Mathematics, 2020, 5(6): 6325-6340. doi: 10.3934/math.2020407 |
This study refines a single-layer Feed-Forward Neural Network (FFNN) for the treatment of textile dye wastewater, concentrating on percentage decolorization (%DEC) and percentage chemical oxygen demand (%COD) reduction. The optimized neural network configuration comprises four input and one output neuron, fine-tuned based on the mean squared error (MSE). The training phase demonstrates a consistent MSE decline, reaching its lowest at epoch 209 for %DEC and epoch 34 for %COD, with corresponding MSEs of 1.799×10−5 and 1.4×10−3, respectively. The maximum absolute errors for %DEC and %COD were found to be 4.0787 and 2.4486, while the mean absolute errors were 0.4821 and 0.7256, respectively. In contrast to second-degree polynomial regression, the FFNN model exhibits enhanced predictive accuracy, as indicated by higher R2 values of 0.99363 for %DEC and 0.99716 for %COD, and reduced error metrics.
The general theory of convex functions is the origin of powerful tools for the study of problems in analysis. Convex functions are thoroughly associated to the theory of inequalities. Inequalities involving convex functions are the most efficient tools in the development of several branches of mathematics and has been given considerable attention in the literature (see [1,2]) and references therein). We start by giving some renowned definitions:
Definition 1.1. A function f:I⊆R→R is said convex if
f(ta+(1−t)b)≤tf(a)+(1−t)f(b) |
for all a,b∈Iand t∈[0,1].
In [3] Toader defines the m−convexity:
Definition 1.2. A functionf:[0,d]→R0=[0,∞) is said m−convex on [0,d] for some m∈(0,1],if
f(ta+m(1−t)b)≤tf(a)+m(1−t)f(b), ∀a,b∈[0,d] andt∈[0,1]. |
In [4] Miheşan introduced the following class of functions:
Definition 1.3. A functionf:[0,d]→R0=[0,∞) is said (α,m)− convex on [0,d],(α,m)∈(0,1]2, if
f(ta+m(1−t)b)≤tαf(a)+m(1−tα)f(b), ∀a,b∈[0,d]andt∈[0,1]. |
Let Kmα be the set of (α,m)−convex functions on [0,d].
In [5] Park J. introduced the following class of (s,m)−convex functions in the second sense:
Definition 1.4. For some fixed s∈(0,1] and m∈[0,1] a mapping f:[0,d]→R is said (s,m)− convex in the second sense on [0,d], if
f(ta+m(1−t)b)≤tsf(a)+m(1−t)sf(b) |
holds for all a,b∈[0,d] and t∈[0,1].
In [6], Özdemir et al. gave the following lemmas for twice differentiable functions.
Lemma 1.1. Let f:I⊆R→R be a twice differentiable mapping on I∘(interior of I), a≠b∈I and f′′∈L[a,b], then the following equality holds:
f(a)+f(b)2−1b−a∫baf(x)dx=(b−a)22∫10t(1−t)f′′(ta+(1−t)b)dt. |
Lemma 1.2. Let f:I⊆R→R be a twice differentiable mapping on I∘, where a,b∈I with a<mb and m∈(0,1].If f′′∈L[a,b],then the following equality holds:
f(a)+f(mb)2−1mb−a∫mbaf(x)dx=(mb−a)22∫10t(1−t)f′′(ta+m(1−t)b)dt. | (1.1) |
In [7], M. Z. Sarıkaya and N. Aktan obtained the following Trapezoid inequality for convex functions
|f(a)+f(b)2−1b−a∫baf(t)dt|≤(b−a)224[|f′′(a)|+|f′′(b)|]. | (1.2) |
In [8], I. Işcan gave a refinement of the Hölder's integral inequality as follows:
Theorem 1.1. (Hölder- İşcan integral inequality) Let p>1 and q−1+p−1=1. If f and g are real functions defined on interval [a,b] and if |f|p and |g|q are integrable functions on [a,b], then
∫ba|f(t)g(t)|dt≤ 1b−a(∫ba(b−t)|f(t)|pdt )1p(∫ba(b−t)|g(t)|qdt )1q+1b−a(∫ba(t−a)|f(t)|pdt )1p(∫ba(t−a)|g(t)|qdt )1q. |
An refinement of power-mean integral inequality as a different version of the Hölder- İşcan integral inequality can be given as follows:
Theorem 1.2. (Improved power-mean integral inequality [9]) Let q≥1. If f and g are real functions defined on [a,b] and if |f|,|f||g|q are integrable functions on [a,b], then
∫ba|f(t)g(t)|dt≤1b−a(∫ba(b−t)|f(t)|dt )1−1q(∫ba(b−t)|f(t)||g(t)|qdt )1q+1b−a(∫ba(t−a)|f(t)|dt )1−1q(∫ba(t−a)|f(t)||g(t)|qdt )1q. |
There is a massive literature on Hermite – Hadamard type inequalities via different kind of convexities. The most notable are m–convex, (α,m)−convex, s –convex and extended s− convex (see [10,11,12,13,14,15,16,17,18,19]). Besides this, we mention recent results related to the Hermite–Hadamard type inequalities, for example, see [20,21,22] and the references therein. In the next section we consider the most recent generalized convex functions and its properties which covers all the above as particular cases.
Definition 2.1. [23] The function f:[0,d]→R is said to be (α,s,m)−convex, if we have
f(ta+m(1−t)b)≤tαsf(a)+m(1−tα)sf(b), |
where a,b∈[0,d], t∈(0,1)and for some s∈[−1,1],(α,m)∈(0,1]2.
Here Ks,mα is the set of (α,s,m) convex functions on [0,d].
Remark 2.1. (i) If s=1, then f is an (α,m) - convex function on (0,d].
(ii) If α=1, then f is an extended (s,m) - convex function on (0,d].
(iii) If α=m=1, then f is an extended s -convex function on (0,d].
(iv) If α=s=m=1, then f is a convex function on (0,d].
Proposition 2.1. If a function f is (α,m) – convex for all s∈[−1,1], then it is also (α,s,m) – convex.
Proof. Since f∈Kmα, we have
f(ta+m(1−t)b)≤tαf(a)+m(1−tα)f(b) |
∀a,b∈[0,d],t∈[0,1]and (α,m)∈(0,1]2. On the other hand, we have tα≤tαsand 1−tα≤(1−tα)s,for s∈[−1,1],thus
f(ta+m(1−t)b)≤tαf(a)+m(1−tα)f(b)≤tαsf(a)+m(1−tα)sf(b), |
i.ef∈Ks,mα.
Proposition 2.2. Let α∈(0,1]and s∈[−1,1],then
(i)when s∈(−1,1],we have
Mβ(α,s)≈∫10t(1−t)(1−tα)sdt=1α{β(2α,s+1)−β(3α,s+1)}, |
(ii)whens=−1, we have ∫10t(1−t)(1−tα)dt=1α{Ψ(3α)−Ψ(2α)},
where Γ(.),β(.,.),Ψ(.)are the classical Euler-Gamma, Beta and Psi-Gamma functions, respectively.
Γ(x)=∫∞0zx−1ezdz, β(x,y)=∫10zx−1(1−z)y−1dz(x>0,y>0),Ψ(x)=dlnΓ(x)dx=−∫∞011−e−ze−zxdz. |
Proof. (i)
Mβ(α,s)≈∫10t(1−t)(1−tα)sdt. |
If we change the variable tα=z⇒t=z1α, we get dt=1αz1α−1dz. Hence
∫10z1α(1−z)s1αz1α−1dz−∫10z2α(1−z)s1αz1α−1dz=1α∫10z2α−1(1−z)sdz−1α∫10z3α−1(1−z)sdz=1α{β(2α,s+1)−β(3α,s+1)}, |
which completes the proof of proposition (i).
(ii)Whens=−1
∫10t(1−t)(1−tα)−1dt=∫10t(1−t)1−tαdt. |
Again using the change of variable tα=z⇒t=z1α, we have
∫10t(1−t)1−tαdt=∫10z1α−z2α1−z1αz1α−1dz=1α∫10(1−z3α−1)−(1−z2α−1)1−zdz=1α∫10(1−z3α−1)1−zdz−1α∫10(1−z2α−1)1−zdz=1α{[Ψ(3α)+γ]−[Ψ(2α)+γ]}=1α{[Ψ(3α)]−[Ψ(2α)]}, |
where Ψ(x)+γ=∫101−tx−11−tdt and γ=∫∞0(11+t−e−t)1t dtis Euler–Mascheroini constant (see p.258 [24]). Which completes the proof of proposition (ii).
Theorem 3.1. If f:[0,d]→Ris (α,s,m)−convex, we have
f(a+mb2)≤1mb−a∫mbaf(t)dt≤1αs+1(f(a)+mf(b)), | (3.1) |
where a,b∈[0,d], a<mbandforsomes∈[0,1],(α,m)∈(0,1]2.
Proof. The left-hand side of (3.1) is easy to prove:
1mb−a∫mbaf(t)dt=1mb−a(∫a+mb2af(t)dt+∫mba+mb2f(t)dt)=12∫10[f(a+mb−t(mb−a)2)+f(a+mb+t(mb−a)2)]dt≥f(a+mb2). |
The proof of the right-hand side (3.1).
By the (α,s,m)−convexity of fwe observe that
f(ta+m(1−t)b)≤tαsf(a)+m(1−tα)sf(b),∀t∈[0,1]. |
Integrating the resulting inequality with respect to t, we get
∫10f(ta+m(1−t)b)dt≤∫10[tαsf(a)+m(1−tα)sf(b)]dt |
as
∫10f(ta+m(1−t)b)dt=1mb−a∫mbaf(t)dt |
and
∫10[tαsf(a)+m(1−tα)sf(b)]dt=∫10tαsf(a)dt+∫10m(1−tα)sf(b)dt≤∫10tαsf(a)dt+mf(b)∫10(1−t)αsdt=1αs+1(f(a)+mf(b)), |
which proved the theorem as we used the fact
∫10(1−tα)sdt≤∫10(1−t)αsdt=1αs+1. |
Remark 3.1. If we take α=m=s=1.The function fbecomes (1,1,1)−convex. Hence the inequality (3.1) reduces to the Hermite- Hadamard inequality :
f(a+b2)≤1b−a∫baf(x)dx≤12(f(a)+f(b)). |
Theorem 3.2. Let f:I⊂(0,d]→R be a differentiable function on I∘(I∘is interior of I) for a,b∈I∘ such that f′′∈L[a,b] with 0<a<mb. If |f′′| is an (α,s,m)−convex function on [a,b], then the following inequality holds:
|f(a)+f(mb)2−1mb−a∫mbaf(x)dx|≤(mb−a)22|f′′(a)|+m|f′′(b)|(αs+2)(αs+3), | (3.2) |
where(α,m)∈(0,1]2,s∈(0,1].
Proof. From Lemma 1.2 and using the (α,s,m)−convexity of |f′′|, we have
|f(a)+f(mb)2−1mb−a∫mbaf(x)dx|≤(mb−a)22∫10t(1−t)[tαs|f′′(a)|+m(1−tα)s|f′′(b)|]dt=(mb−a)22[|f′′(a)|∫10tαs+1(1−t)dt+m|f′′(b)|∫10t(1−t)(1−tα)sdt]≤(mb−a)22[|f′′(a)|∫10tαs+1(1−t)dt+m|f′′(b)|∫10t(1−t)(1−t)αsdt]=(mb−a)22|f′′(a)|+m|f′′(b)|(αs+2)(αs+3), |
where we used the facts that, since tα≥tfor all α∈(0,1]and t∈[0,1],we have −tα≤−tor1−tα≤1−t, 1−tα≤(1−t)αand (1−tα)s≤(1−t)αs,s∈(0,1].If we multiply the resulting inequality with t(1−t)and taking integral on [0,1],we have
∫10t(1−t)(1−tα)sdt≤∫10t(1−t)(1−t)αsdt=1(αs+2)(αs+3) |
and
∫10tαs+1(1−t)dt=1(αs+2)(αs+3). |
Corollary 3.1. From Remark 2.1 (iv), since f(.) is a convex function on [a,b] for m=α=s=1, we obtain the inequality (1.2).
Corollary 3.2. Under the same conditions in Theorem 3.2 with |f′′(x)|≤M for all x∈[a,b], we have
|f(a)+f(mb)2−1mb−a∫mbaf(x)dx|≤M(mb−a)22×(m+1)(αs+2)(αs+3). |
The following Theorem presents a upper bound for (α,s,m)-convex functions.
Theorem 3.3. Let f:I⊂(0,d]→R be a differentiable function on I∘(I∘is interior of I) for a,b∈I∘, f′′∈L[a,b],(α,m)∈(0,1]2,s∈(−1,1] with 0<a<mb. If |f′′|q is an (α,s,m)−convex function on [a,b] for q≥1, then the following inequality holds:
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22(16)1−1q× [|f′′(a)|q(αs+2)(αs+3)+mMβ(α,s)|f′′(bm)|q]1q, | (3.3) |
where Mβ(α,s) is given in Proposition 2.2(i).
Proof. First, we assume that q=1.From Lemma 1.1 and using the (α,s,m)−convexity with properties of modulus we have
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22∫10t(1−t)|f′′(ta+m(1−t)bm)|dt≤(b−a)22∫10t(1−t)(tαs|f′′(a)|+m(1−tα)s|f′′(bm)|)dt=(b−a)22[|f′′(a)|(αs+2)(αs+3)+mMβ(α,s)|f′′(bm)|], |
which completes the proof for q = 1. Here we also used the Proposition 2.2 (ⅰ).
Secondly, suppose now that q>1. From Lemma 1 and using the Hölder's integral inequality for q>1 with properties of modulus, we have
∫10(t−t2)|f′′(ta+m(1−t)bm)|dt=∫10(t−t2)1−1q(t−t2)1q|f′′(ta+m(1−t)bm)|dt≤[∫10(t−t2)dt]1−1q[∫10(t−t2)|f′′(ta+m(1−t)bm)|qdt]1q, | (3.4) |
where p−1+q−1=1.
Since |f′′|qis (α,s,m)−convex on [a,b], we know that for every t∈[0,1].
|f′′(ta+m(1−t)bm)|q≤tαs|f′′(a)|q+m(1−tα)s|f′′(bm)|q. | (3.5) |
From (3.4), (3.5) and Proposition 2.2 (ⅰ)
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22[∫10(t−t2)dt]1−1q[∫10(t−t2)|f′′(ta+m(1−t)bm)|qdt]1q≤(b−a)22[∫10(t−t2)dt]1−1q[∫10(t−t2)(tαs|f′′(a)|q +m(1−tα)s|f′′(bm)|q)dt]1q=(b−a)22(16)1−1q[|f′′(a)|q(αs+2)(αs+3)+mMβ(α,s)|f′′(bm)|q]1q, |
which completes the inequality (3.3).
Corollary 3.3. Under the same conditions in Theorem 3.3 with |f′′(x)|≤M and m=1 for all x∈[a,b], we have
|f(a)+f(b)2−1b−a∫baf(x)dx|≤M(b−a)22(16)1−1q[1(αs+2)(αs+3)+Mβ(α,s)]1q. |
Theorem 3.4. When s=−1, under the same conditions of Theorem 3.3, we obtain the following result
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22(16)1−1q | (3.6) |
×[|f′′(a)|q(2−α)(3−α)+m|f′′(bm)|q1α{Ψ(3α)−Ψ(2α)}]1q. |
Proof. Using the (α,s,m)−convexity of |f′′|q and Proposition 2.2(ⅱ) with the properties of modulus, we have
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22[∫10(t−t2)]1−1q ×[∫10(t−t2)(t−α|f′′(a)|q+m(1−tα)−1|f′′(bm)|q)dt]1q=(b−a)22(16)1−1q[∫10(t−t2)(t−α|f′′(a)|q+m(1−tα)−1|f′′(bm)|q)dt]1q=(b−a)22(16)1−1q[|f′′(a)|q∫10(t1−α−t2−α)dt+m|f′′(bm)|q∫10t−t21−tαdt]1q=(b−a)22(16)1−1q[|f′′(a)|q(2−α)(3−α)+m|f′′(bm)|q1α{Ψ(3α)−Ψ(2α)}]1q, |
which completes the inequality (3.6).
Corollary 3.4. Under the same conditions in Theorem 3.4 with |f′′(x)|≤M and m=1 for all x∈[a,b], we have
|f(a)+f(b)2−1b−a∫baf(x)dx| ≤M(b−a)22(16)1−1q×[1(2−α)(3−α)+1α{Ψ(3α)−Ψ(2α)}]1q. |
Theorem 3.5. Let f:I⊂(0,d]→R be a twice differentiable function on I∘ for a,b∈I∘ such that f′′∈L[a,b],(α,m)∈(0,1]2,s∈(−1,1]with 0<a<mb. If |f′′|q is an (α,s,m)−convex function on [a,b]for q≥1 andq≥r>0, then the following inequality holds:
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22(β(2q−r−1q−1,2q−1q−1))1−1q | (3.7) |
×[|f′′(a)|qαs+r+1+mα|f′′(bm)|q(β(r+1α,s+1))]1q. |
Proof. Using the Lemma 1.1, Hölder's inequality and (α,s,m)−convexity of |f′′|q we have
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22[∫10(tq−rq−1(1−t)qq−1)dt]1−1q×[∫10tr(tαs|f′′(a)|q+m(1−tα)s|f′′(bm)|q)dt]1q=(b−a)22[β(2q−r−1q−1,2q−1q−1)]1−1q×[|f′′(a)|q∫10tαs+rdt+m|f′′(bm)|q∫10tr(1−tα)sdt]1q. |
After simplifying we get inequality (3.7).
An immediate consequences of Theorem 3.5 by considering special cases for (r=0) and (m=1) can be given as:
Corollary 3.5. Under the assumptions of Theorem 3.5, the following inequality holds
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22(β(2q−1q−1,2q−1q−1))1−1q | (3.8) |
×[|f′′(a)|qαs+1+mα|f′′(bm)|q(β(1α,s+1))]1q. |
Corollary 3.6. Under the assumptions of Theorem 3.5, the following inequality holds
|f(a)+f(b)2−1b−a∫baf(x)dx|≤(b−a)22(β(2q−1q−1,2q−1q−1))1−1q | (3.9) |
×[|f′′(a)|qαs+1+1α |f′′(b)|q(β(1α,s+1))]1q. |
Corollary 3.7. Under the same conditions in Theorem 3.5 with |f′′(x)|≤M and m=1, for all x∈[a,b], we have
|f(a)+f(b)2−1b−a∫baf(x)dx|≤M(b−a)22[β(2q−r−1q−1,2q−1q−1)]1−1q×[1αs+r+1+1aβ(r+1α,s+1)]1q. |
Theorem 3.6. Let f:I⊂(0,d]→R be a twice differentiable function on I∘ such that f′′∈L[a,b] for a,b∈I∘,(α,m)∈(0,1]2,s∈(−1,1] with 0 < a < mb . If \left\vert f^{\prime \prime }\right\vert ^{q}\; is an \left(\alpha, s, m\right)- convex function on [a, b]\; for q\ge1, \; q\geq r > 0 , then the following inequality holds:
\begin{eqnarray} \left\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a}\int_{a}^{b}f\left( t\right) dt\right\vert &\leq &K \left[ \beta \left( \frac{2q-r-1}{q-1},\frac{2q-1}{q-1}\right) \right] ^{1-\frac{1}{q}} \\ &&\times \left[ \frac{1}{\alpha s+r+1}\left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}+\frac{m}{\alpha }\cdot \beta \left( \frac{r+1}{ \alpha },s+1\right) \left\vert f^{\prime \prime }\left( \frac{b}{m}\right) \right\vert ^{q}\right] ^{\frac{1}{q}}, \end{eqnarray} | (3.10) |
where \ K = \frac{\left(b-a\right) ^{2}}{2}.
Proof. By using Lemma 1.1 and Hölder's inequality with \left(\alpha, s, m\right)- convexity of \left\vert f^{\prime \prime }\right\vert ^{q} , we have
\begin{eqnarray*} &&\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{b-a}\int_{a}^{b}f\left( t\right) dt\right\vert \\ &\leq&K\left[ \int_{0}^{1}t^{\frac{q-r}{q-1}}\left( 1-t\right) ^{\frac{q}{ q-1}}dt\right] ^{1-\frac{1}{q}}\left[ \int_{0}^{1}t^{r}\left( \begin{array}{c} t^{\alpha s}\left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q} \\ +m\left( 1-t^{\alpha }\right) ^{s}\left\vert f^{\prime \prime }\left( \frac{b }{m}\right) \right\vert ^{q} \end{array} \right) dt\right] ^{\frac{1}{q}} \\ &\leq &K\left\{ \left[ \int_{0}^{1}t^{\frac{q-r}{q-1}}\left( 1-t\right) ^{ \frac{q}{q-1}}dt\right] ^{1-\frac{1}{q}}\left[ \begin{array}{c} \left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}\int_{0}^{1}t^{r+\alpha s}dt \\ +m\left\vert f^{\prime \prime }\left( \frac{b}{m}\right) \right\vert ^{q}\int_{0}^{1}t^{r}\left( 1-t^{\alpha }\right) ^{s}dt \end{array} \right] ^{\frac{1}{q}}\right\} \\ & = &K\left\{ \left[ B\left( \frac{2q-r-1}{q-1},\frac{2q-1}{q-1}\right) \right] ^{1-\frac{1}{q}}\left[ \begin{array}{c} \left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}\int_{0}^{1}t^{r+\alpha s}dt \\ +m\left\vert f^{\prime \prime }\left( \frac{b}{m}\right) \right\vert ^{q}\int_{0}^{1}t^{r}\left( 1-t^{\alpha }\right) ^{s}dt \end{array} \right] ^{\frac{1}{q}}\right\}. \end{eqnarray*} |
Here we used the fact that
\begin{eqnarray*} \int_{0}^{1}t^{\frac{q-r}{q-1}}\left( 1-t\right) ^{\frac{q}{q-1}}dt = B\left( \frac{2q-r-1}{q-1},\frac{2q-1}{q-1}\right) \end{eqnarray*} |
and with a formal change of variables t^{\alpha } = z
\begin{eqnarray*} \int_{0}^{1}t^{r}\left( 1-t^{\alpha }\right) ^{s}dt = \frac{1}{\alpha } \int_{0}^{1}z^{\frac{r}{\alpha }}\left( 1-z\right) ^{s}dz = \frac{1}{\alpha } B\left( \frac{r+1}{\alpha },\;s+1\right). \end{eqnarray*} |
If we write the above integrals places we obtain the inequality (3.10).
Theorem 3.7. Let \ f:I\subset (0, d]\rightarrow R be a twice differentiable function on I^{\circ} such that \; f^{\prime \prime }\in L[a, b] for a, b\in I^{\circ}, \; (\alpha, m)\in (0, 1]^{2}, s\in [0, 1]\; with 0 < a < mb . Also let p > 1 such that q = \frac{p}{p-1} , if \left\vert f^{\prime \prime }\right\vert ^{q} is an \left(\alpha, s, m\right) - convex on [a, b] , then the following inequality holds:
\begin{equation} \left\vert \frac{f(a)+f(mb)}{2}-\frac{1}{mb-a}\int_{a}^{mb}f\left( t\right) dt\right\vert \end{equation} | (3.11) |
\begin{eqnarray*} \leq G \left\{ \begin{array}{c} \left( \frac{3p}{ 2\left( p+1\right) \left( p+2\right) \left( 2p+1\right) }\right) ^{\frac{1}{p }}\left( \frac{\left\vert f^{{\prime \prime }}\left( a\right) \right\vert ^{q}}{\left( \alpha s+1\right) \left( \alpha s+2\right) }+\frac{m\left\vert f^{{\prime \prime }}\left( b\right) \right\vert ^{q}}{\left( \alpha s+2\right) }\right) ^{\frac{1}{q}} \\ +\left( \frac{p}{2\left( p+1\right) \left( p+2\right) }\right) ^{ \frac{1}{p}}\left( \frac{\left\vert f^{^{\prime \prime }}\left( a\right) \right\vert ^{q}}{\left( \alpha s+2\right) }+\frac{m\left\vert f^{^{\prime \prime }}\left( b\right) \right\vert ^{q}}{\left( \alpha s+1\right) \left( \alpha s+2\right) }\right) ^{\frac{1}{q}} \end{array} \right\}, \end{eqnarray*} |
where G = \frac{\left(mb-a\right) ^{2}}{2}.
Proof. Using Lemma 1.2 and Hölder- İşcan integral inequality along with \left(\alpha, s, m\right) - convexity of \left\vert f^{\prime \prime }\right\vert ^{q} , we have
\begin{eqnarray*} &&\left\vert \frac{f(a)+f(mb)}{2}-\frac{1}{mb-a}\int_{a}^{mb}f\left( t\right) dt\right\vert \leq G \int_{0}^{1}\left( t-t^{2}\right) \Big\vert f^{{\prime \prime }}\left( ta+m\left( 1-t\right) b\right) dt\Big\vert \ dt \\ &\leq &G \left\{ \begin{array}{c} \left( \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert ^{p}dt\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left( 1-t\right) \Big\vert f^{{\prime \prime }}\left( ta+m\left( 1-t\right) b\right) dt\Big\vert ^{q}dt\right) ^{\frac{1}{q}} \\ +\left( \int_{0}^{1}t\left\vert t-t^{2}\right\vert ^{p}dt\right) ^{\frac{1 }{p}}\left( t\Big\vert f^{{\prime \prime }}\left( ta+m\left( 1-t\right) b\right) dt\Big\vert ^{q}dt\right) ^{\frac{1}{q}}. \end{array} \right\} \\ \end{eqnarray*} |
Using the fact that \left\vert m-n\right\vert ^{c}\leq m^{c}-n^{c} for c > 1 and for m, n > \ 0; \ \ m > n . The right-hand side of above inequality becomes
\begin{eqnarray*} &\leq &G \left\{ \begin{array}{c} \left( \int_{0}^{1}\left( 1-t\right) \left( t^{p}-t^{2p}\right) dt\right) ^{\frac{1}{p}}\left( \int_{0}^{1}\left( 1-t\right) \left\{ t^{\alpha s}\left\vert f^{{\prime \prime }}\left( a\right) \right\vert ^{q}+m\left\vert f^{{\prime \prime }}\left( b\right) \right\vert ^{q}\left( 1-t\right) ^{\alpha s}\right\} dt\right) ^{\frac{1}{q} } \\ +\ \left( \int_{0}^{1}t\left( t^{p}-t^{2p}\right) dt\right) ^{\frac{1}{p} }\left( t\left\{ t^{\alpha s}\left\vert f^{{\prime \prime }}\left( a\right) \right\vert ^{q}+m\left\vert f^{{\prime \prime }}\left( b\right) \right\vert ^{q}\left( 1-t\right) ^{\alpha s}\right\} dt\right) ^{\frac{1}{q} }. \end{array} \right\} \\ \end{eqnarray*} |
Simplifying above integrals, we obtain the result of the Theorem 3.7.
Corollary 3.8. Under the same conditions in Theorem 3.7 with \left\vert f^{\prime \prime }\left(x\right) \right\vert\leq M for all x\in \lbrack a, b] , we have
\begin{equation*} \left\vert \frac{f\left( a\right) +f\left( mb\right) }{2}-\frac{1}{mb-a} \int_{a}^{mb}f\left( x\right) dx\right\vert \leq G\cdot M\cdot \left\{ \begin{array}{c} \left( \frac{3p}{ 2\left( p+1\right) \left( p+2\right) \left( 2p+1\right) }\right) ^{\frac{1}{p }}\left( \frac{1}{\left( \alpha s+1\right) \left( \alpha s+2\right) }+\frac{m}{\left( \alpha s+2\right) }\right) ^{\frac{1}{q}} \\ +\left( \frac{p}{2\left( p+1\right) \left( p+2\right) }\right) ^{ \frac{1}{p}}\left( \frac{1}{\left( \alpha s+2\right) }+\frac{m}{\left( \alpha s+1\right) \left( \alpha s+2\right) }\right) ^{\frac{1}{q}} \end{array} \right\} \end{equation*} |
Theorem 3.8. Let \ f:I\subset (0, d]\rightarrow R be a twice differentiable function on I^{\circ} such that \; f^{\prime \prime }\in L[a, b] for a, b\in I^{\circ}, \; (\alpha, m)\in (0, 1]^{2}, s\in [0, 1]\; with 0 < a < mb . If \left\vert f^{\prime \prime }\right\vert ^{q} is an \left(\alpha, s, m\right) - convex on [a, b] for q\ge1 , then the following inequality holds:
\begin{equation} \left\vert \frac{f(a)+f(mb)}{2}-\frac{1}{mb-a}\int_{a}^{mb}f\left( t\right) dt\right\vert \leq \frac{G}{12}\left( \frac{12}{\left( \alpha s+3\right) \left( \alpha s+4\right) }\right) ^{\frac{1}{q}} \end{equation} | (3.12) |
\begin{equation*} \times \left\{ \left( \frac{2\left\vert f^{^{\prime \prime }}\left( a\right) \right\vert ^{q}}{\alpha s+2}+m\ \left\vert f^{^{\prime \prime }}\left( b\right) \right\vert ^{q}\right) ^{\frac{1}{q}}+\left( \left\vert f^{^{\prime \prime }}\left( a\right) \right\vert ^{q}+\frac{2m\ \left\vert f^{^{\prime \prime }}\left( b\right) \right\vert ^{q}}{\alpha s+2}\right) ^{ \frac{1}{q}}\right\}, \end{equation*} |
where G = \frac{\left(mb-a\right) ^{2}}{2}.
Proof. Using Lemma 1.2 and improved power mean integral inequality with \left(\alpha, s, m\right)- convexity of \left\vert f^{\prime \prime }\right\vert ^{q} , we have
\begin{equation*} \ \ \ \left\vert \frac{f(a)+f(mb)}{2}-\frac{1}{mb-a}\int_{a}^{mb}f\left( t\right) dz\right\vert \end{equation*} |
\begin{equation*} \leq G\cdot\left\{ \begin{array}{c} \left( \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert \ dt\right) ^{1-\frac{1}{q}}\left( \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert \ \left\vert f^{^{\prime \prime }}\left( ta+m\left( 1-t\right) b\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \\ +\ \left( \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ dt\right) ^{1-\frac{1 }{q}}\left( \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ \left\vert f^{^{\prime \prime }}\left( ta+m\left( 1-t\right) b\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \end{array} \right\} \end{equation*} |
By employing \left(\alpha, s, m\right)- convexity and simplifying, we have
\begin{eqnarray} &\leq & G\cdot\left\{ \begin{array}{c} \left( \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert dt\right) ^{1-\frac{1}{q}} \\ \times \left[ \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert \ \left( t^{\alpha s}\left\vert f^{^{\prime \prime }}\left( a\right) \right\vert ^{q}+m\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}\left( 1-t^{\alpha }\right) ^{s}dt\right) \right] ^{\frac{1}{q}} \\ +\left( \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ dt\right) ^{1-\frac{1}{q }}\left[ \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ \left( t^{\alpha s}\left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}+m\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}\left( 1-t^{\alpha }\right) ^{s}\right) dt\right] ^{\frac{1}{q}} \end{array} \right\} \end{eqnarray} | (3.13) |
It can be noticed that
\begin{equation} \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert \ dt = \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ dt = \frac{1}{12} \end{equation} | (3.14) |
and
\begin{equation} \int_{0}^{1}t^{\alpha s}\left( 1-t\right) \left\vert t-t^{2}\right\vert \ dt\ \ = \ \frac{2}{\left( \alpha s+2\right) \left( \alpha s+3\right) \left( \alpha s+4\right) }\ \end{equation} | (3.15) |
and
\begin{eqnarray} \int_{0}^{1}\left( 1-t\right) \left\vert t-t^{2}\right\vert \ \left( 1-t^{\alpha }\right) ^{s}dt &\leq &\int_{0}^{1}\left\vert t-t^{2}\right\vert \ \left( 1-t\right) ^{\alpha s+1}dt \\ & = &\int_{0}^{1}t\left( 1-t\right) ^{\alpha s+2}dt = \frac{1}{\left( \alpha s+3\right) \left( \alpha s+4\right) } \end{eqnarray} | (3.16) |
\begin{eqnarray} \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ t^{\alpha s}\ dt & = &\int_{0}^{1}\left( 1-t\right) \ t^{\alpha s+2}\ dt\ = \frac{1}{\left( \alpha s+3\right) \left( \alpha s+4\right) } \\ \int_{0}^{1}t\left\vert t-t^{2}\right\vert \ \left( 1-t^{\alpha }\right) ^{s}dt\ &\leq &\int_{0}^{1}t\left\vert t-t^{2}\right\vert \ \left( 1-t\right) ^{\alpha s}dt \\ & = &\int_{0}^{1}t^{2}\ \left( 1-t\right) ^{\alpha s+1}dt = \frac{2}{\left( \alpha s+2\right) \left( \alpha s+3\right) \left( \alpha s+4\right) } \end{eqnarray} | (3.17) |
Replacing the values of the integrals computed in (3.14), (3.15), (3.16) and (3.17) in (3.13), we get (3.12).
Corollary 3.9. Under the same conditions in Theorem 3.8 with \left\vert f^{^{\prime \prime }}\left(x\right) \right\vert\leq M for all x\in \lbrack a, b] , we have
\begin{eqnarray*} &&\left\vert \frac{f\left( a\right) +f\left( mb\right) }{2}-\frac{1 }{mb-a}\int_{a}^{mb}f\left( x\right) dx\right\vert \ \\ &\leq &M\cdot \frac{\left( mb-a\right) ^{2}}{24}\left( \frac{12}{\left( \alpha s+3\right) \left( \alpha s+4\right) }\right) ^{\frac{1}{q}}\left[ \left( \frac{2}{\alpha s+2}+m\right) ^{\frac{1}{q}}+\left( 1+\frac{2m}{ \alpha s+2}\right) ^{\frac{1}{q}}\right]. \end{eqnarray*} |
Now we let us consider some special means for arbitrary positive real numbers c and d .
The\ arithmetic\ mean\ \ \ :
\ \ \ \ \ \ \ A\left(c, d\right) = \frac{c+d}{2}, \qquad \qquad c, d\in R\ with\ c, d > 0.
The\ geometric\ mean\ \ \ :
G\left(c, d\right) = \left(c \cdot d\right) ^{\frac{1}{2}}, \qquad c, d\in R\ with\ c, d > 0.
The\ harmonic\ mean\ \ \ :
H\left(c, d\right) = \frac{2\ c\ d}{c + d}, \qquad \qquad c, d\in R\ \backslash \{0\}.
The\ logarithmic\ mean\ \ \ :
L\left(c, d\right) = \left\{ \begin{array}{c} c, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ c = d \\ \frac {d-c}{ln d - ln c}, \ \ \ \ \ \ \ \ if\ \ c\neq d. \end{array} \ \ \ \ \right.
The\ Generalized\ logarithmric\ mean\ \ \ :
L_{n}\left(c, d\right) = \left\{ \begin{array}{c} c, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ c = d, \\ \left(\frac{d^{n+1}-c^{n+1}}{\left(n+1\right) \left(d-c\right) }\right)^{1/n}, \ \ \ \ \ \ \ \ if\ \ c\neq d, \ \ n\in Z\backslash \{-1, 0\}; c, d > 0. \end{array} \ \ \ \ \right.
The\ \ Identric\ mean\ \ \ :
I\left(c, d\right) = \left\{ \begin{array}{c} c, \qquad if\ \ c = d \\ \frac{1}{e}\left(\frac{d^{d}}{c^{c}}\right) ^{\frac{1}{d-c}}, \qquad if\ \ c\neq d \; \ c, d > 0. \end{array} \ \ \ \ \right.
Proposition 4.1. Let a, b\in R, \ \ a < b\ , \ a, b > 0\ , then we have the following inequality holds
\begin{eqnarray*} &&\left\vert A\left( e^{a},\; e^{b}\right) -L\left( e^{a},\; e^{b}\right) \right\vert \\ && \leq \frac{\left( b-a\right) ^{2}}{2}\left( \frac{1}{6}\right) ^{1-\frac{1}{q}}.\left[ \frac{\left\vert e^{a}\right\vert ^{q}}{\left( \alpha s+2\right) \left( \alpha s+3\right) }+mM_{\beta }\left( \alpha ,s\right) \left\vert e^{\frac{b}{m}}\right\vert ^{q}\right] ^{\frac{1 }{q}}. \end{eqnarray*} |
Proof. This assertion follows from Theorem 3.3 for f\left(x\right) = e^{x} and \ x\in R .
Proposition 4.2. Let a, b\in R, \ \ a < b , a, b > 0 , then we have the following inequality
\begin{eqnarray*} &&\left\vert A\left( \ln a^{-1},\; \ln b^{-1}\right) +\ln \left( I\right) \right\vert\\ && \leq \frac{\left( b-a\right) ^{2}}{2}\left( \frac{1}{6} \right) ^{1-\frac{1}{q}}\left[ \frac{\left\vert \frac{1}{a^{2}}\right\vert ^{q}}{\left( 2-\alpha \right) \left( 3-\alpha \right) }+\frac{m}{\alpha }\left\vert \frac{m}{b}\right\vert ^{2q}\left\{ \Psi \left( \frac{3}{ \alpha }\right) -\Psi \left( \frac{2}{\alpha }\right) \right\} \right] ^{ \frac{1}{q}}. \end{eqnarray*} |
Proof. This assertion follows from Theorem 3.4 for f\left(x\right) = -ln{x} and \ x > 0 .
Proposition 4.3. Let a, b\in R, \ \ a < b\ and 0\notin \left[ a, b\right] , then the following inequality holds
\begin{eqnarray*} &&\left\vert H^{-1}\left( a,b\right) \ -\ L^{-1}\left( a,b\right) \right\vert \leq \frac{\left( b-a\right) ^{2}}{2}\left( \frac{1}{6}\right) ^{1-\frac{1}{q}}\\ && \times\left[ \frac{\left\vert \frac{2}{a^{3}}\right\vert ^{q}}{ \left( 2-\alpha \right) \left( 3-\alpha \right) }+m\; \left\vert \frac{2m^{3}}{ b^{3}}\right\vert ^{q}.\frac{1}{\alpha }\left\{ \Psi \left( \frac{3}{\alpha } \right) -\Psi \left( \frac{2}{\alpha }\right) \right\} \right] ^{\frac{1}{q} }. \end{eqnarray*} |
Proof. This assertion follows from Theorem 3.4 for f\left(x\right) = \frac{1}{x} ; \ x \in[a, b] .
Proposition 4.4. Let a, b\in R, \ \ a < b and 0\notin \left[ a, b\right] , then the following inequality holds
\left\vert H^{-1}\left( a,b\right) \ \ -\ \ L^{-1}\left( a,b\right) \right\vert \leq \frac{\left( b-a\right) ^{2}}{2}\left( B\left( \frac{2q-r-1}{q-1},\frac{2q-1}{q-1}\right) \right) ^{1-\frac{1}{q}}\\\times\left[ \frac{\left\vert \frac{2}{a^{3}}\right\vert ^{q}}{\alpha s+r+1}+\frac{m}{ \alpha }\left\vert \frac{2m^{3}}{b^{3}}\right\vert ^{q}B\left( \frac{r+1}{ \alpha },s+1\right) \right] ^{\frac{1}{q}} . |
Proof. This assertion follows from Theorem 3.5 for f\left(x\right) = \frac{1}{x} ; x \in[a, b] .
In many practical studies, it is necessary to evaluate the difference between the two quantities. From the point of view of programming an algorithm for solving optimization problems, classical inequalities with the smallest upper limit play an important role.
We have proved several new generalized integral inequalities involving \left(\alpha, s, m\right)- convex functions for extended case of s where s \in[-1, 1] , we obtained Hermite–Hadamrad type inequalities. The most interesting case is extended case for s = -1 , when we get connected to Psi-Gamma functions. We analyze new upper bounds involving special functions by employing different variants of Hölder inequality such as Hölder-Işcan inequality and Improved Power mean inequality.
The author declares no conflict of interest in this paper.
[1] |
G. M. Walker, L. R. Weatherley, Adsorption of acid dyes on to granular activated carbon in fixed beds, Water Res., 31 (1997), 2093–2101. https://doi.org/10.1016/S0043-1354(97)00039-0 doi: 10.1016/S0043-1354(97)00039-0
![]() |
[2] | A. H. Hassani, S. Seye, A. H. Javid, M. Borgheei, Comparison of adsorption process by GAC with novel formulation of coagulation–flocculation for color removal of textile wastewater, Int. J. Environ. Res., 2 (2008), 239–248. |
[3] |
S. Wijannarong, S. Aroonsrimorakot, P. Thavipoke, S. Sangjan, Removal of reactive dyes from textile dyeing industrial effluent by ozonation process, APCBEE procedia, 5 (2013), 279–282. https://doi.org/10.1016/j.apcbee.2013.05.048 doi: 10.1016/j.apcbee.2013.05.048
![]() |
[4] |
V. K. Gupta, A. Nayak, S. Agarwal, I. Tyagi, Potential of activated carbon from waste rubber tire for the adsorption of phenolics: Effect of pre-treatment conditions, J. Colloid Interf. Sci., 417 (2014), 420–430. https://doi.org/10.1016/j.jcis.2013.11.067 doi: 10.1016/j.jcis.2013.11.067
![]() |
[5] |
X. R. Xu, H. B. Li, W. H. Wang, J. D. Gu, Decolorization of dyes and textile wastewater by potassium permanganate, Chemosphere, 59 (2005), 893–898. https://doi.org/10.1016/j.chemosphere.2004.11.013 doi: 10.1016/j.chemosphere.2004.11.013
![]() |
[6] | A. A. Ansari, B. D. Thakur, Bio-chemical reactor for treatment of concentrated textile effluent, Colourage, 49 (2002), 27–30. |
[7] |
F. Zhang, A. Yediler, X. Liang, A. Kettrup, Ozonation of the purified hydrolyzed azo dye reactive red 120 (CI), J. Environ. Sci. Health Part A, 37 (2002), 707–713. https://doi.org/10.1081/ESE-120003248 doi: 10.1081/ESE-120003248
![]() |
[8] |
R. Rajeshkannan, M. Rajasimman, N. Rajamohan, Sorption of acid blue 9 using Hydrilla verticillata biomass—optimization, equilibrium, and kinetics studies, Bioremediat. J., 15 (2011), 57–67. https://doi.org/10.1080/10889868.2010.548002 doi: 10.1080/10889868.2010.548002
![]() |
[9] |
R. Krull, M. Hemmi, P. Otto, D. C. Hempel, Combined biological and chemical treatment of highly concentrated residual dyehouse liquors, Water Sci. Technol., 38 (1998), 339–346. https://doi.org/10.1016/S0273-1223(98)00517-4 doi: 10.1016/S0273-1223(98)00517-4
![]() |
[10] |
P. Verma, D. Madamwar, Decolourization of synthetic dyes by a newly isolated strain of Serratia marcescens, World J. Micro. Biotechnol., 19 (2003), 615–618. https://doi.org/10.1023/A:1025115801331 doi: 10.1023/A:1025115801331
![]() |
[11] | V. Arutchelvan, D. J. Albino, V. Muralikaishnan, S. Nagarajan, Decolourization of textile mill effluent by Sporotrichum pulverulentum, Indian J. Environ. Ecoplaning, 7 (2003), 59–62. |
[12] |
J. P. Jadhav, S. P. Govindwar, Biotransformation of malachite green by Saccharomyces cerevisiae MTCC 463, Yeast, 23 (2006), 315–323. https://doi.org/10.1002/yea.1356 doi: 10.1002/yea.1356
![]() |
[13] |
J. S. Chang, C. Chou, S. Y. Chen, Decolorization of azo dyes with immobilized pseudomonas luteola, Process Biochem., 36 (2001), 757–763. https://doi.org/10.1016/S0032-9592(00)00274-0 doi: 10.1016/S0032-9592(00)00274-0
![]() |
[14] |
M. S. Khehra, H. S. Saini, D. K. Sharma, B. S. Chadha, S. S. Chimni, Decolorization of various azo dyes by bacterial consortium, Dyes Pigments, 67 (2005), 55–61. https://doi.org/10.1016/j.dyepig.2004.10.008 doi: 10.1016/j.dyepig.2004.10.008
![]() |
[15] |
S. Pointing, Feasibility of bioremediation by white-rot fungi, Appl. Microbiol. Biotechnol., 57 (2001), 20–33. https://doi.org/10.1007/s002530100745 doi: 10.1007/s002530100745
![]() |
[16] |
S. Sathian, G. Radha, V. Shanmugapriya, M. Rajasimman, C. Karthikeyan, Optimization and kinetic studies on treatment of textile dye wastewater using Pleurotus floridanus, Appl. Water Sci., 3 (2013), 41–48. https://doi.org/10.1007/s13201-012-0055-0 doi: 10.1007/s13201-012-0055-0
![]() |
[17] |
E. Pérez-Santín, L. de-la-Fuente-Valentín, M. G. García, K. A. S. Bravo, F. C. L. Hernández, J. I. L. Sánchez, Applicability domains of neural networks for toxicity prediction, AIMS Mathematics, 8 (2023), 27858–27900. https://doi.org/10.3934/math.20231426 doi: 10.3934/math.20231426
![]() |
[18] |
H. Najafi, A. Bensayah, B. Tellab, S. Etemad, S. K. Ntouyas, S. Rezapour, et al., Approximate numerical algorithms and artificial neural networks for analyzing a fractal-fractional mathematical model, AIMS Mathematics, 8 (2023), 28280–28307. https://doi.org/10.3934/math.20231447 doi: 10.3934/math.20231447
![]() |
[19] |
N. Ruttanaprommarin, Z. Sabir, R. A. S. Núñez, S. Salahshour, J. L. G. Guirao, W. Weera, et al., Artificial neural network procedures for the waterborne spread and control of diseases, AIMS Mathematics, 8 (2023), 2435–2452. https://doi.org/10.3934/math.2023126 doi: 10.3934/math.2023126
![]() |
[20] |
S. D. Mourtas, E. Drakonakis, Z. Bragoudakis, Forecasting the gross domestic product using a weight direct determination neural network, AIMS Mathematics, 8 (2023), 24254–24273. https://doi.org/10.3934/math.20231237 doi: 10.3934/math.20231237
![]() |
[21] |
N. Alruwais, H. Alamro, M. M. Eltahir, A. S. Salama, M. Assiri, N. A. Ahmed, Modified arithmetic optimization algorithm with Deep Learning based data analytics for depression detection, AIMS Mathematics, 8 (2023), 30335–30352. https://doi.org/10.3934/math.20231549 doi: 10.3934/math.20231549
![]() |
[22] |
J. E. F. Moraes, F. H. Quina, C. A. O. Nascimento, D. N. Silva, O. Chiavone-Filho, Treatment of saline wastewater contaminated with hydrocarbons by the photo-Fenton process, Environ. Sci. Technol., 38 (2004), 1183–1187. https://doi.org/10.1021/es034217f doi: 10.1021/es034217f
![]() |
[23] |
V. K. Pareek, M. P. Brungs, A. A. Adesina, R. Sharma, Artificial neural network modeling of a multiphase photodegradation system, J. Photoch. Photobio. A: Chem., 149 (2002), 139–146. https://doi.org/10.1016/S1010-6030(01)00640-2 doi: 10.1016/S1010-6030(01)00640-2
![]() |
[24] |
A. M. Ghaedi, A. Vafaei, Applications of artificial neural networks for adsorption removal of dyes from aqueous solution: A review, Adv. Colloid Interfac. Sci., 245 (2017), 20–39. https://doi.org/10.1016/j.cis.2017.04.015 doi: 10.1016/j.cis.2017.04.015
![]() |
1. | Mohsen Ayyash, Dawood Khan, Saad Ihsan Butt, Youngsoo Seol, Superquadratic stochastic processes and their fractional perspective with applications in information theory, 2025, 10, 2473-6988, 13695, 10.3934/math.2025617 | |
2. | Mohsen Ayyash, Dawood Khan, Saad Ihsan Butt, Youngsoo Seol, Fractional Inclusion Analysis of Superquadratic Stochastic Processes via Center-Radius Total Order Relation with Applications in Information Theory, 2025, 9, 2504-3110, 375, 10.3390/fractalfract9060375 |