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- Open Access
Computing new solutions of algebro-geometric equation using the discrete inverse Sumudu transform
- Adem Kılıçman^{1}Email authorView ORCID ID profile and
- Rathinavel Silambarasan^{2}
https://doi.org/10.1186/s13662-018-1785-6
© The Author(s) 2018
- Received: 10 May 2018
- Accepted: 30 August 2018
- Published: 14 September 2018
Abstract
The discrete inverse Sumudu transform method is designed to solve ordinary differential equations and tested for an algebro-geometric equation. The two new sets of exact analytical and complex solutions are gotten through a discrete inverse Sumudu transform, and Maple complex graphs are drawn to show the new solution simulations in the complex plane which are compared to the existing solutions. The list of inverse Sumudu transforms is added in the sequel to strengthen the study.
Keywords
- Inverse Sumudu transform
- Complex solutions
- Lommel S1 function
- Struve function
MSC
- 33E30
- 44A10
1 Introduction
Algebro-geometric lattices were constructed using Rings and Fields to obtain the solutions of KdV equation and Toda equation in [1]. For solving sine-Gordon equation, Landau–Lifshitz equation, and reducing Abelian and hyperelliptic integrals, theta functions algebro-geometric principles were employed in [2]. Nonlinear integrable equations of mathematical physics, electrical systems were studied using the algebro-geometric method in [3]. KdV equation, Toda equation AKNS, and Hill’s hierarchy were solved in [4, 5]. Soliton and quasi solutions of Dym type and water flow equations were solved for algebro-geometric solutions in [6]. Theta function notation of algebro-geometric solutions for Camassa–Holm equation and soliton solutions was given in [7–9]. Algebro-geometric Sturm–Liouville coefficients were calculated in [10]. Solutions without reflection for hierarchies of evolution equations were given in [11]. Endpoint classification of three forms of algebro-geometric equation (AGE) and eigenvalues of Sturm–Liouville differential equations were given in [12].
A Sumudu transform was applied for nonzero modulus Dixon elliptic functions to calculate their Hankel determinants in [13]. A discrete inverse Sumudu transform (DIST) method was proposed to solve ordinary differential equations to obtain their new exact solutions, and Whittaker and Zettl equations with a table of the inverse Sumudu transform of functions were solved in [14] as in [15], which gives special functions [16] as inverse Sumudu. Lane–Emden type differential equations were solved by the decomposition method using Sumudu in [17]. Fractional reaction-diffusion equation and delay differential equation were studied using the Sumudu transform in [18, 19]. Human relationships were studied numerically in [20]. Fuzzy differential equations were solved using a fuzzy Sumudu transform in [21]. Fractional differential equations, telegraph equations, and fuzzy differential equations were solved using the Sumudu transform respectively in [22–24]. Cattaneo–Vernotte with space fractional, time fractional, and space-time fractional equations were solved for integer values and rational values of φ in [25]. In [26] a reaction-diffusion equation with variable order fractionals was solved numerically by using a combined Adams and finite difference method where they took Liouville–Caputo and ABC (Atangana–Baleanu–Caputo) fractional derivatives. Liouville–Caputo, Caputo–Fabrizio–Caputo, and Mittag-Leffler kernel fractional derivatives were applied for the Bateman–Feshbach–Tikochinsky oscillator and Caldirola–Kanai oscillator and their individual behavior was studied in [27]. In [28] Atangana–Baleanu fractional derivatives were used for a nonlinear Bloch system, and the Adams–Moulton method was applied to solve it numerically. A homotopy perturbation transform method was applied to solve some nonlinear fractional differential equations in [29]. Apart from this, some of the very recent advancements in fractional calculus theories were given in [30, 31].
2 DIST method description
Inverse Sumudu transform of elementary functions
S. No | f(x) | \(\mathbb{S}^{-1}[f(x)]=F_{-1}(w)\) |
---|---|---|
1 | \(\frac{1}{x+a}\) | \(\frac{1}{a}e^{-\frac{w}{a}}\) |
2 | \(\frac{1}{x-a}\) | \(-\frac{1}{a}e^{\frac{w}{a}}\) |
3 | \(\frac{1}{(x+a)^{n}}\) | \(\frac{e^{-\frac{w}{2a}}}{a^{n}w} [ a(n+1) M_{n+1,\frac{1}{2}}( \frac{w}{a}) -(an-w)M_{n,\frac{1}{2}}( \frac{w}{a}) ]\) |
4 | \(\frac{x^{n}}{x+a}\) | \(\frac{(-1)^{-n}a^{n-1}e^{-\frac{w}{a}}}{n!} [ \Gamma ( n+1) -n\Gamma ( n,-\frac{w}{a}) ] \) |
5 | \(\frac{Ax+Ba}{x^{2}-a^{2}}\) | \(\frac{1}{2a} [ e^{-\frac{w}{a}}(A-B)-e^{\frac{w}{a}}(A+B) ] \) |
6 | \(\frac{Ax+Ba}{x^{2}+a^{2}}\) | \(\frac{1}{2a} [ e^{-\frac{iw}{a}}(iA+B)-e^{\frac{iw}{a}}(iA-B) ] \) |
7 | \(\frac{1}{\sqrt{x+a}}\) | \(\frac{1}{\sqrt{a}}e^{-\frac{w}{2a}}\mbox{I}_{0}( \frac{w}{2a}) \) |
8 | \(\frac{1}{(x+a)^{\frac{3}{2}}}\) | \(\frac{e^{-\frac{w}{2a}}}{a^{\frac{5}{2}}} [ (a-w)\mbox{I}_{0}( \frac{w}{2a}) +w\mbox{I}_{1}( \frac{w}{2a}) ]\) |
9 | \(\frac{\sqrt{x}}{x+a}\) | \(-\frac{e^{-\frac{w}{a}}}{\sqrt{a}}\mbox{erf}( \frac{i\sqrt{w}}{\sqrt{a}}) \) |
10 | \(\frac{\sqrt{x-b}}{x}\) | \(-\frac{ie^{\frac{w}{2b}}}{2\sqrt{b}} [ \mbox{I}_{0}( \frac{w}{2b}) - \mbox{I}_{1}( \frac{w}{2b}) ]\) |
11 | \(\frac{1}{\sqrt{x}(x+a)}\) | \(\frac{e^{-\frac{w}{a}}}{a^{2}\sqrt{w\pi }} [ ae^{\frac{w}{a}}+i\sqrt{aw\pi }\mbox{erf}( \frac{i\sqrt{w}}{\sqrt{a}}) ]\) |
12 | \(\frac{1}{\sqrt{x(x+a)}}\) | \(\frac{e^{\frac{w}{a}}}{\sqrt{aw\pi }}\) |
13 | \(\frac{1}{x\sqrt{x-b}}\) | \(-\frac{ie^{\frac{w}{2b}}}{2b^{\frac{3}{2}}} [ \mbox{I}_{0}( \frac{w}{2b}) +\mbox{I}_{1}( \frac{w}{2b}) ]\) |
14 | \(\frac{x}{\sqrt{x^{2}+a^{2}}}\) | \(-\frac{w\operatorname{csgn}(a)}{2a} [ \mbox{J}_{0}( \frac{w}{a})( \pi \textbf{H}_{1}( \frac{w}{a}) -2) - \pi \mbox{J}_{1}( \frac{w}{a}) \textbf{H}_{0}( \frac{w}{a}) ] \) |
15 | \(\frac{x}{\sqrt{b^{2}-x^{2}}}\) | \(\frac{w\operatorname{csgn}(b)}{2b} [ \mbox{I}_{0}( \frac{w}{b}) ( \pi \textbf{L}_{1}( \frac{w}{b}) +2) -\pi \mbox{I}_{1}( \frac{w}{b}) \textbf{L}_{0}( \frac{w}{b}) ] \) |
16 | \(\frac{x}{\sqrt{x^{2}-b^{2}}}\) | \(\frac{iw}{2b} [ \mbox{I}_{0}( \frac{w}{b}) ( \pi \textbf{L}_{1}( \frac{w}{b}) +2) -\pi \mbox{I} _{1}( \frac{w}{b}) \textbf{L}_{0}( \frac{w}{b}) ] \) |
17 | \(\frac{1}{x+\sqrt{x^{2}+a^{2}}}\) | \(\frac{1}{2a^{3}}(2\operatorname{csgn}(a)\mbox{J}_{0}( \frac{w}{a}) ( a^{2}+w^{2}-\frac{\pi w^{2}}{2}\textbf{H}_{1}( \frac{w}{a}) )-2w( \operatorname{csgn}(a)\mbox{J}_{1}( \frac{w}{a}) +a ( a-\frac{\pi w}{2} \textbf{H}_{0}( \frac{w}{a}) ) )) \) |
18 | \(( x+\sqrt{1+x^{2}}) ^{n}+( x-\sqrt{1+x^{2}})^{n} \) | \((e^{in\pi }+1){}_{2}F_{3}( \frac{n}{2},\frac{n}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w^{2}}{4}) +nw(e^{in\pi }-1){}_{2}F_{3} ( \frac{1+n}{2},\frac{1-n}{2};1,\frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4})\) |
19 | \(\frac{( x+\sqrt{1+x^{2}}) ^{n}}{\sqrt{1+x^{2}}}\) | \({}_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w^{2}}{4}) +nw{}_{2}F_{3}( 1-\frac{n}{2},1+\frac{n}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \) |
20 | \(\frac{( x-\sqrt{1+x^{2}}) ^{n}}{\sqrt{1+x^{2}}}\) | \((e^{in\pi })_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2}, \frac{1}{2},1;-\frac{w^{2}}{4}) -nw{}_{2}F_{3}( 1-\frac{n}{2},1+\frac{n}{2};1,\frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \) |
21 | \(\frac{(x-b)^{v}}{x}\) | \((-1)^{v}b^{v-1} [ \mbox{L}_{v}( \frac{w}{b}) -\mbox{L} _{v}^{1}( \frac{w}{b}) ] \) |
22 | \(\frac{x^{v-1}}{1+x^{2}}\) | \(\frac{\sqrt{w}\mbox{S1}_{\frac{3}{2},\frac{1}{2}}(w)+v(v+1)w^{v-1}-w ^{v+1}}{(v+1)!} \) |
23 | \((1+x^{2})^{v-\frac{1}{2}}\) | \({}_{1}F_{2}( \frac{1}{2}-v;\frac{1}{2},1;-\frac{w^{2}}{4}) \) |
24 | \((x^{2}-b^{2})^{v-\frac{1}{2}}\) | \((-b)^{v-\frac{1}{2}}{}_{1}F_{2}( \frac{1}{2}-v;\frac{1}{2},1;\frac{w ^{2}}{4b^{2}})\) |
25 | \((b^{2}-x^{2})^{v-\frac{1}{2}}\) | \(\frac{2(b^{2})^{v-\frac{1}{2}}}{b^{2}(2v+1)} [ w^{2}\mbox{L}_{v+ \frac{1}{2}}^{1}( \frac{w^{2}}{b^{2}}) +( ( v+ \frac{1}{2}) b^{2}-w^{2}) \mbox{L}_{v+\frac{1}{2}}( \frac{w ^{2}}{b^{2}}) ]\) |
26 | \(x^{v-1}(x+a)^{\frac{1}{2}-v}\) | \(\frac{a^{\frac{1}{2}-v}w^{v-1}e^{-\frac{w}{2a}} ( \frac{w}{a}) ^{-\frac{v}{2}}}{av(v-1)!(v+1)w} [ a(av+w)( v+\frac{3}{2})\mbox{M}_{\frac{v+3}{2},\frac{v+1}{2}}( \frac{w}{a}) +( \frac{2w^{2}-2wa-va^{2}}{2}) \mbox{M}_{\frac{v+1}{2}, \frac{v+1}{2}}( \frac{w}{a}) ] \) |
27 | \(x^{v-1}(x+a)^{-\frac{1}{2}-v}\) | \(\frac{e^{-\frac{w}{2a}}( \frac{w}{a}) ^{-\frac{v}{2}} a^{- ( v+\frac{1}{2}) }w^{v-1}}{vw(v+1)(v-1)!} [ -( \frac{w+av}{2}) \mbox{M}_{\frac{v+1}{2},\frac{v+1}{2}}( \frac{w}{a}) + ( v+\frac{3}{2}) av\mbox{M}_{\frac{v+3}{2},\frac{v+1}{2}} ( \frac{w}{a}) ] \) |
28 | \(( \sqrt{x^{2}+1}+x) ^{v}\) | \({}_{2}F_{3}( \frac{v}{2},-\frac{v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w^{2}}{4}) +vw{}_{2}F_{3}( \frac{1+v}{2},\frac{1-v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \) |
29 | \(( \sqrt{x^{2}+1}-x) ^{v}\) | \({}_{2}F_{3}( \frac{v}{2},-\frac{v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w ^{2}}{4}) -vw{}_{2}F_{3}( \frac{1+v}{2},\frac{1-v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \) |
30 | \(\frac{( \sqrt{x^{2}+1}+x) ^{v}}{\sqrt{x^{2}+1}}\) | \({}_{2}F_{3}( \frac{1-v}{2},\frac{1+v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w ^{2}}{4}) +vw{}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \) |
31 | \(\frac{( \sqrt{x^{2}+1}-x) ^{v}}{\sqrt{x^{2}+1}}\) | \({}_{2}F_{3}( \frac{1-v}{2},\frac{1+v}{2};\frac{1}{2},\frac{1}{2},1;-\frac{w ^{2}}{4}) -vw{}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1, \frac{3}{2},\frac{3}{2};-\frac{w^{2}}{4}) \) |
32 | \(\frac{( \sqrt{x^{2}-1}+x) ^{v}+( \sqrt{x^{2}-1}+x)^{-v}}{\sqrt{x^{2}-1}}\) | \(-\frac{(e^{i\pi v}+1)}{i^{v-1}}{}_{2}F_{3}( \frac{1-v}{2},\frac{1+v}{2};\frac{1}{2},\frac{1}{2},1;\frac{w^{2}}{4}) -\frac{(e^{i\pi v}-1)vw}{i^{v}}{}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1,\frac{3}{2},\frac{3}{2};\frac{w^{2}}{4})\) |
33 | \(( \sqrt{x+2a}+\sqrt{x}) ^{2v} -( \sqrt{x+2a}-\sqrt{x}) ^{2v}\) | \(\frac{4v\sqrt{w}2^{v+\frac{1}{2}}a^{v-\frac{1}{2}}}{\sqrt{\pi }}{}_{2}F_{2}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{3}{2},\frac{3}{2};-\frac{w}{2a})\) |
34 | \(( \sqrt{x+b}+\sqrt{x-b}) ^{2v} -( \sqrt{x+b}-\sqrt{x-b}) ^{2v}\) | \(-ivwb^{v-1}( (1+i)^{2v}+(1-i)^{2v}) {}_{2}F_{3}( \frac{1+v}{2}, \frac{1-v}{2};1,\frac{3}{2},\frac{3}{2};\frac{w^{2}}{4b^{2}}) -b^{v}( -(1+i)^{2v}+(1-i)^{2v}) {}_{2}F_{3}( \frac{v}{2},- \frac{v}{2};\frac{1}{2},\frac{1}{2},1;,\frac{w^{2}}{4b^{2}} )\) |
35 | \(\frac{(2a)^{2v}( x+\sqrt{x^{2}+4a^{2}}) ^{2v}}{\sqrt{x^{3}+4a^{2}x}}\) | \(\frac{1}{24}( \operatorname{csgn}^{2v+1}(a)w^{\frac{3}{2}}a^{4v-3}16^{v}( v^{2}-\frac{1}{4}) ) \times {}_{2}F_{5}( \frac{3}{2}-v, \frac{3}{2}+v;\frac{5}{4},\frac{3}{2},\frac{3}{2},\frac{7}{4},2;-\frac{w^{2}}{1024a^{2}}) +\frac{1}{2}( \operatorname{csgn}^{2v}(a)va^{4v+2}16^{v}\sqrt{w}) \times {}_{2}F_{5}( 1+v,1-v; \frac{1}{2},\frac{3}{4},1,\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{1024a^{2}})\) |
36 | \(\frac{( x+\sqrt{x^{2}-1}) ^{2v}+( x-\sqrt{x^{2}-1})^{2v}}{\sqrt{x}\sqrt{x^{2}-1}}\) | \(\frac{8iv\sqrt{w}}{\sqrt{\pi }\sin (\pi v)}( \cos^{2}(\pi v)-1){}_{2}F_{3}( 1+v,1-v;\frac{3}{4},\frac{5}{4},\frac{3}{2}; \frac{w^{2}}{4}) -\frac{2i\cos (\pi v)}{\sqrt{\pi w}}{}_{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{1},\frac{3}{4};\frac{w ^{2}}{4})\) |
37 | \(e^{-ax}\) | \(\mbox{J}_{0}(2\sqrt{aw})\) |
38 | \(xe^{-ax}\) | \(\frac{w\mbox{J}_{1}(2\sqrt{aw})}{\sqrt{aw}}\) |
39 | \(x^{v-1}e^{-ax}\) | \(\frac{w^{v-1}\Gamma (v)\mbox{J}_{v-1}(2\sqrt{aw})}{(v-1)!(aw)^{\frac{v-1}{2}}}\) |
40 | \(\frac{e^{-ax}-e^{-bx}}{x}\) | \(\frac{b\sqrt{a}\mbox{J}_{1}(2\sqrt{bw})-a\sqrt{b}\mbox{J}_{1}(2\sqrt{aw})}{\sqrt{abw}} \) |
41 | \(\frac{(1-e^{-ax})^{2}}{x^{2}}\) | \(\frac{-2a(\mbox{J}_{2}(2\sqrt{aw})-\mbox{J}_{2}(2\sqrt{2aw}))}{w}\) |
42 | \(\frac{1}{x}-\frac{(x+2)(1-e^{-x})}{2x^{2}}\) | \(\frac{w\mbox{I}_{3}(2\sqrt{-w})}{2(-w)^{\frac{3}{2}}}\) |
43 | \(e^{-\frac{x^{2}}{4a}}\) | \({}_{0}F_{2}( ;\frac{1}{2},1;-\frac{w^{2}}{16a})\) |
44 | \(xe^{-\frac{x^{2}}{4a}}\) | \(w{}_{0}F_{2}( ;1,\frac{3}{2};-\frac{w^{2}}{16a}) \) |
45 | \(\frac{e^{-\frac{x^{2}}{4a}}}{\sqrt{x}}\) | \(\frac{1}{\sqrt{\pi w}}{}_{0}F_{2}( ;\frac{1}{4},\frac{3}{4};-\frac{w ^{2}}{16a}) \) |
46 | \(x^{v-1}e^{-\frac{x^{2}}{8a}}\) | \(\frac{w^{v-1}}{(v-1)!}{}_{0}F_{2}( ;\frac{v}{2},\frac{v+1}{2};-\frac{w ^{2}}{32a}) \) |
47 | \(e^{-\frac{x}{4a}}\) | \(\mbox{I}_{0}( \sqrt{-\frac{w}{a}}) \) |
48 | \(\sqrt{x}e^{-\frac{x}{4a}}\) | \(2\sqrt{\frac{a}{\pi }}\sin ( \sqrt{\frac{w}{a}}) \) |
49 | \(\frac{e^{-\frac{x}{4a}}}{\sqrt{x}}\) | \(\frac{\cos ( \sqrt{\frac{w}{a}}) }{\sqrt{\pi w}} \) |
50 | \(\frac{e^{-\frac{x}{4a}}}{x^{\frac{3}{2}}}\) | \(-\frac{( \sqrt{aw}\sin ( \sqrt{\frac{w}{a}}) +a \cos ( \sqrt{\frac{w}{a}}) ) }{2a\sqrt{\pi }w^{ \frac{3}{2}}} \) |
51 | \(x^{v-1}e^{-\frac{x}{4a}}\) | \(\frac{(2w)^{v-1}\Gamma (v)( -\frac{w}{a}) ^{\frac{1-v}{2}} \mbox{I}_{v-1}( \sqrt{-\frac{w}{a}}) }{(v-1)!} \) |
52 | \(\frac{(e^{-\frac{x}{4a}}-1)}{\sqrt{x}}\) | \(\frac{1}{\sqrt{w\pi }} [ \cos ( \sqrt{\frac{w}{a}}) -1 ] \) |
53 | \(e^{-2\sqrt{a}\sqrt{x}}\) | \(-\frac{4\sqrt{aw}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{3}{2},\frac{3}{2};,aw) + {}_{0}F_{2}( ;\frac{1}{2},1;,aw) \) |
54 | \(\sqrt{x}e^{-2\sqrt{a}\sqrt{x}}\) | \(\frac{2\sqrt{w}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{1}{2},\frac{3}{2};,aw) - 2w\sqrt{a}_{0}F_{2}( ;\frac{3}{2},2;,aw) \) |
55 | \(\frac{e^{-2\sqrt{a}\sqrt{x}}}{\sqrt{x}}\) | \(\frac{1}{\sqrt{w\pi }} {}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};,aw) - 2\sqrt{a}_{0}F_{2}( ;1,\frac{3}{2};,aw) \) |
56 | \(\frac{e^{-2\sqrt{a}\sqrt{x}}}{\sqrt{2x}}\) | \(\frac{1}{\sqrt{2w\pi }} {}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};,aw) -\sqrt{2a}_{0}F_{2}( ;1,\frac{3}{2};,aw) \) |
57 | log(1 + ax) | Ei_{1}(aw)+ln(aw)+γ |
58 | log(x + a) | \(\mbox{Ei}_{1}( \frac{w}{a}) +\ln (w)+\gamma \) |
59 | \(\log (x^{2}-a^{2})\) | \(2 [ \ln (w)+\gamma -\mbox{Chi}( \frac{w}{a}) ] +i \pi \) |
60 | \(\log (x^{2}+a^{2})\) | \(2 [ \ln (w)+\gamma -\mbox{Ci}( \frac{w}{a}) ] \) |
61 | \(\frac{\log (x^{2}+a^{2})-\log (a^{2})}{x}\) | \(\frac{2}{w} [ 1-\cos ( \frac{w}{a}) ] \) |
62 | \(\log ( \frac{\sqrt{x+b}+\sqrt{x-b}}{\sqrt{2}\sqrt{b}})\) | \(\frac{1}{2b} [ -iw {}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1, \frac{3}{2},\frac{3}{2};\frac{w^{2}}{4b^{2}}) - b( \ln (2)+2 \ln (i+1)) ] \) |
63 | \(\log ( \frac{\sqrt{x}+\sqrt{x+2a}}{\sqrt{2}\sqrt{a}})\) | \(\sqrt{\frac{2w}{a\pi }}{}_{2}F_{2}( \frac{1}{2},\frac{1}{2}; \frac{3}{2},\frac{3}{2};-\frac{w}{2a}) \) |
64 | \(\log ( \frac{\sqrt{x+ib}+\sqrt{x-ib}}{\sqrt{2}\sqrt{b}})\) | \(\frac{1}{2b} [ w{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1, \frac{3}{2},\frac{3}{2}; -\frac{w^{2}}{4b^{2}}+2i\pi b) ] \) |
65 | sin(ax) | \(\operatorname{bei}_{0}(2\sqrt{aw}) \) |
66 | xsin(ax) | \(-\frac{\sqrt{aw}}{2} [ \mbox{ber}_{1}(2\sqrt{w}) +\operatorname{bei} _{1}(2\sqrt{w}) ] \) |
67 | \(x^{n}\sin (ax)\) | \(\frac{aw^{n+1}}{(n+1)!}{}_{0}F_{3}( ;\frac{3}{2},\frac{n}{2}+1, \frac{n+3}{2};-\frac{(aw)^{2}}{16}) \) |
68 | \(x^{v-1}\sin (ax)\) | \(\frac{aw^{v}}{v!}{}_{0}F_{3}( ;\frac{3}{2},\frac{v}{2}+1, \frac{v+1}{2};-\frac{(aw)^{2}}{16}) \) |
69 | \(\frac{\sin (ax)}{x}\) | \(-\sqrt{\frac{a}{2w}} [ \mbox{ber}_{1}(2\sqrt{aw})-\operatorname{bei} _{1}(2\sqrt{aw}) ] \) |
70 | \(\frac{\sin^{2}(ax)}{x}\) | \(-\sqrt{\frac{a}{4w}} [ \mbox{ber}_{1}(2\sqrt{2aw})+\operatorname{bei} _{1}(2\sqrt{2aw}) ] \) |
71 | \(\frac{\sin^{3}(ax)}{x}\) | \(-\frac{3\sqrt{a}}{8\sqrt{3iw}} [ -\sqrt{3}( \mbox{I} _{1}( 2(-1)^{\frac{1}{4}}\sqrt{aw}) +\mbox{J}_{1}( 2(-1)^{ \frac{1}{4}}\sqrt{aw}) ) +\mbox{I}_{1}(2\sqrt{3iaw})+ \mbox{J}_{1}(2\sqrt{3iaw}) ] \) |
72 | \(\frac{\sin^{2}(ax)}{x^{2}}\) | \(\frac{\sqrt{a}}{2w^{\frac{3}{2}}} [ \mbox{ber}_{1}(2 \sqrt{2aw})+\operatorname{bei}_{1}(2\sqrt{2aw})+2\sqrt{aw}\operatorname{bei}_{0}(2 \sqrt{2aw}) ] \) |
73 | \(\sin (x^{2})\) | \(\frac{w^{2}}{2}{}_{0}F_{5}( ;\frac{3}{4},1,\frac{5}{4}, \frac{3}{2}, \frac{3}{2};-\frac{w^{2}}{1024}) \) |
74 | \(\frac{\sin (x^{2})}{x}\) | \(w{}_{0}F_{5}( ;\frac{1}{2},\frac{3}{4},1,\frac{5}{4}, \frac{3}{2};-\frac{w ^{2}}{1024})\) |
75 | \(\frac{\sin (x^{2})}{x^{2}}\) | \(w{}_{0}F_{5}( ;\frac{1}{4},\frac{1}{2},\frac{3}{2},1,\frac{3}{2} ;-\frac{w ^{2}}{1024}) \) |
76 | \(\sin (2\sqrt{a}\sqrt{x})\) | \(\frac{4\sqrt{aw}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{3}{2}, \frac{3}{2};-aw) \) |
77 | \(x^{n}\sin (2\sqrt{a}\sqrt{x})\) | \(\frac{2\sqrt{a}w^{n+\frac{1}{2}}}{( n+\frac{1}{2}) !} {}_{0}F_{2}( ;\frac{3}{2},\frac{3}{2}+n;-aw) \) |
78 | \(\frac{\sin (2\sqrt{a}\sqrt{x})}{x}\) | \(\frac{2\sqrt{a}}{\sqrt{w\pi }}{}_{0}F_{2}( ;\frac{1}{2}, \frac{3}{2};-aw) \) |
79 | \(\sqrt{x}\sin (2\sqrt{a}\sqrt{x})\) | \(2w\sqrt{a}{}_{0}F_{2}( ;\frac{3}{2},2;-aw) \) |
80 | \(\frac{\sin (2\sqrt{a}\sqrt{x})}{\sqrt{x}}\) | \(2\sqrt{a}{}_{0}F_{2}( ;1,\frac{3}{2};-aw) \) |
81 | \(x^{v-1}\sin (2\sqrt{a}\sqrt{x})\) | \(\frac{2\sqrt{a}w^{v-\frac{1}{2}}}{( v-\frac{1}{2}) !}{}_{0}F_{2}( ;\frac{3}{2},v+\frac{1}{2};-aw) \) |
82 | cos(ax) | \(\mbox{ber}_{0}(2\sqrt{aw}) \) |
83 | cos^{2}(ax) | \(1-\frac{(aw)^{2}}{2}{}_{1}F_{4}( 1;\frac{3}{2},\frac{3}{2},2,2;- \frac{(aw)^{2}}{4}) \) |
84 | cos^{3}(ax) | \(\frac{3}{8} [ \mbox{I}_{0}( 2(-1)^{\frac{1}{4}}\sqrt{aw}) + \mbox{J}_{0}( 2(-1)^{\frac{1}{4}}\sqrt{aw}) ] + \frac{1}{8} [ \mbox{I}_{0}( 2\sqrt{3iaw}) + \mbox{J} _{0}( 2\sqrt{3iaw}) ] \) |
85 | xcos(x) | \(-\sqrt{\frac{w}{2}} [ \mbox{ber}_{1}(2\sqrt{w})-\operatorname{bei}_{1}(2 \sqrt{w}) ] \) |
86 | \(x^{n}\cos (ax)\) | \(\frac{w^{n}}{n!}{}_{0}F_{3}( ;\frac{1}{2},\frac{n}{2}+1, \frac{n+1}{2};-\frac{(aw)^{2}}{16}) \) |
87 | \(x^{v-1}\cos (ax)\) | \(\frac{w^{v-1}}{(v-1)!}{}_{0}F_{3}( ;\frac{1}{2},\frac{v}{2}, \frac{v+1}{2};-\frac{(aw)^{2}}{16}) \) |
88 | \(\frac{1-\cos (ax)}{x}\) | \(-\sqrt{\frac{a}{2w}} [ \mbox{ber}_{1}(2\sqrt{aw})+\operatorname{bei} _{1}(2\sqrt{aw}) ] \) |
89 | \(\cos (x^{2}) \) | \({}_{0}F_{5}( ;\frac{1}{4},\frac{1}{2},\frac{1}{2},\frac{3}{4},1;-\frac{w ^{2}}{1024}) \) |
90 | \(\cos (2\sqrt{a}\sqrt{x})\) | \({}_{0}F_{2}( ;\frac{1}{2},1;-aw) \) |
91 | \(x\sqrt{x}\cos (2\sqrt{a}\sqrt{x})\) | \(\frac{2\sqrt{w}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{1}{2}, \frac{3}{2};-aw) \) |
92 | \(\frac{\cos (2\sqrt{a}\sqrt{x})}{\sqrt{x}}\) | \(\frac{1}{\sqrt{w\pi }}{}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};-aw) \) |
93 | \(x^{n-\frac{1}{2}}\cos (2\sqrt{a}\sqrt{x})\) | \(\frac{w^{n-\frac{1}{2}}}{( n-\frac{1}{2}) !} {}_{0}F_{2} ( ;\frac{1}{2},n+\frac{1}{2};-aw) \) |
94 | \(x^{v-1}\cos (2\sqrt{a}\sqrt{x})\) | \(\frac{w^{v-1}}{(v-1)!} {}_{0}F_{2}( ;\frac{1}{2},v;-aw) \) |
95 | sin(ax)sin(bx) | \(\frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a-b)}) + \mbox{I}_{0}( 2\sqrt{-iw(a-b)}) ] - \frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a+b)}) +\mbox{I}_{0}( 2 \sqrt{-iw(a+b)}) ] \) |
96 | cos(ax)sin(bx) | \(\frac{1}{4} [ i\mbox{I}_{0}( 2\sqrt{iw(a-b)}) -i \mbox{I}_{0}( 2\sqrt{-iw(a-b)}) ] + \frac{1}{4} [ i\mbox{I}_{0}( 2\sqrt{-iw(a+b)}) -\mbox{I}_{0}( 2 \sqrt{iw(a+b)}) ] \) |
97 | cos(ax)cos(bx) | \(\frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a+b)}) + \mbox{I}_{0}( 2\sqrt{-iw(a+b)}) ] + \frac{1}{4} [ \mbox{I}_{0}( 2\sqrt{iw(a-b)}) +\mbox{I}_{0}( 2 \sqrt{-iw(a-b)}) ] \) |
98 | \(\frac{2ax\sin (ax)\cos (ax)-\sin^{2}(ax)}{x^{2}}\) | \(\frac{1}{2\sqrt{a}w^{\frac{5}{2}}} [aw(2aw-1)\operatorname{bei}_{1}( 2\sqrt{2aw}) - aw(2aw+1)\mbox{ber}_{1}( 2 \sqrt{2aw}) - 2(aw)^{\frac{3}{2}}\operatorname{bei}_{0}( 2 \sqrt{2aw}) ] \) |
99 | \(\frac{ax\cos (ax)-\sin (ax)}{x^{2}}\) | \(\frac{1}{2a^{\frac{3}{2}}w^{\frac{7}{2}}} [ \sqrt{2}(aw)^{2}(aw+1) \operatorname{bei}_{1}( 2\sqrt{aw}) + \sqrt{2}(aw)^{2}(aw-1) \mbox{ber}_{1}( 2\sqrt{aw}) - 2(aw)^{\frac{5}{2}} \mbox{ber}_{0}( 2\sqrt{aw}) ] \) |
100 | arcsin(x) | \(w{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1,\frac{3}{2},\frac{3}{2};\frac{w ^{2}}{4}) \) |
101 | xarcsin(x) | \(\frac{w^{2}}{2} {}_{2}F_{3}( \frac{1}{2},\frac{1}{2};\frac{3}{2}, \frac{3}{2},2;\frac{w^{2}}{4}) \) |
102 | \(\arctan ( \frac{x}{a})\) | \(\mbox{Si}( \frac{W}{a}) \) |
103 | \(\cot^{-1}( \frac{x}{a})\) | \(\frac{\pi }{2}-\mbox{Si}( \frac{W}{a}) \) |
104 | \(x\arctan ( \frac{x}{a})\) | \(a [ \cos ( \frac{x}{a}) -1 ] +w\mbox{Si}( \frac{W}{a}) \) |
105 | \(x\cot^{-1}( \frac{x}{a})\) | \(a [ 1-\cos ( \frac{x}{a}) ] -w\mbox{Si}( \frac{W}{a}) +\frac{\pi w}{2} \) |
106 | sinh(ax) | \(\frac{1}{2} [ \mbox{I}_{0}( 2\sqrt{aw}) -\mbox{J}_{0} ( 2\sqrt{aw}) ] \) |
107 | cosh(ax) | \(\frac{1}{2} [ \mbox{I}_{0}( 2\sqrt{aw}) +\mbox{J}_{0} ( 2\sqrt{aw}) ] \) |
108 | sinh^{2}(ax) | \(\frac{(aw)^{2}}{2}{}_{1}F_{4}( 1;\frac{3}{2},\frac{3}{2},2,2; \frac{(aw)^{2}}{4}) \) |
109 | cosh^{2}(ax) | \(1+\frac{(aw)^{2}}{2}{}_{1}F_{4}( 1;\frac{3}{2},\frac{3}{2},2,2; \frac{(aw)^{2}}{4}) \) |
110 | \(\frac{2\sinh (ax)}{x}\) | \(\sqrt{\frac{a}{w}} [ \mbox{I}_{1}( 2\sqrt{aw}) + \mbox{J}_{1}( 2\sqrt{aw}) ] \) |
111 | \(\frac{2\cosh (ax)}{x}\) | \(\sqrt{\frac{a}{w}} [ \mbox{I}_{1}( 2\sqrt{aw}) - \mbox{J}_{1}( 2\sqrt{aw}) ] \) |
112 | \(x^{v-1}\sinh (ax)\) | \(\frac{w^{\frac{v}{2}-1}}{2a^{\frac{v}{2}}} [v\mbox{I}_{v}( 2 \sqrt{aw}) +\sqrt{aw}\mbox{I}_{v+1}( 2\sqrt{aw}) - v\mbox{J}_{v}( 2\sqrt{aw}) +\sqrt{aw}\mbox{J}_{v+1} ( 2\sqrt{aw}) ] \) |
113 | \(x^{v-1}\cosh (ax)\) | \(\frac{w^{\frac{v-3}{2}}}{2a^{\frac{v+1}{2}}} [ v\sqrt{aw} \mbox{I}_{v}( 2\sqrt{aw}) +aw\mbox{I}_{v+1}( 2 \sqrt{aw}) - v\sqrt{aw}\mbox{J}_{v}( 2\sqrt{aw}) -aw \mbox{J}_{v+1}( 2\sqrt{aw}) ] \) |
114 | sin(ax)sinh(ax) | \(\frac{(aw)^{2}}{2}{}_{0}F_{7}( ;\frac{3}{4},\frac{3}{4},1, \frac{5}{4},\frac{5}{4}, \frac{3}{2},\frac{3}{2};- \frac{(aw)^{2}}{16\text{,}384}) \) |
115 | cos(ax)sinh(ax)aw | \({}_{0}F_{7}( ;\frac{1}{2},\frac{1}{2},\frac{3}{4},\frac{3}{4},1, \frac{5}{4},\frac{5}{4};-\frac{(aw)^{2}}{16\text{,}384}) - \frac{(aw)^{3}}{18}{}_{0}F_{7}( ;1,\frac{5}{4},\frac{5}{4}, \frac{3}{2},\frac{3}{2}, \frac{7}{4},\frac{7}{4};- \frac{(aw)^{2}}{16\text{,}384}) \) |
116 | sin(ax)cosh(ax) | \(aw{}_{0}F_{7}( ;\frac{1}{2},\frac{1}{2},\frac{3}{4},\frac{3}{4},1, \frac{5}{4},\frac{5}{4};-\frac{(aw)^{2}}{16\text{,}384}) + \frac{(aw)^{3}}{18}{}_{0}F_{7}( ;1,\frac{5}{4},\frac{5}{4}, \frac{3}{2},\frac{3}{2}, \frac{7}{4},\frac{7}{4};- \frac{(aw)^{2}}{16\text{,}384}) \) |
117 | cos(ax)cosh(ax) | \({}_{0}F_{7}( ;\frac{1}{4},\frac{1}{4},\frac{1}{2},\frac{1}{2}, \frac{3}{4},\frac{3}{4},1;-\frac{(aw)^{2}}{16\text{,}384}) \) |
118 | \(\sinh ( 2\sqrt{a}\sqrt{x})\) | \(\frac{4\sqrt{aw}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{3}{2},\frac{3}{2};aw)\) |
119 | \(\cosh ( 2\sqrt{a}\sqrt{x})\) | \({}_{0}F_{2}( ;\frac{1}{2},1;aw)\) |
120 | \(\sqrt{x}\sinh ( 2\sqrt{a}\sqrt{x})\) | \(2\sqrt{a}w{}_{0}F_{2}( ;\frac{3}{2},2;aw) \) |
121 | \(\sqrt{x}\cosh ( 2\sqrt{a}\sqrt{x})\) | \(\frac{2\sqrt{w}}{\sqrt{\pi }}{}_{0}F_{2}( ;\frac{1}{2}, \frac{3}{2};aw) \) |
122 | \(\frac{\sinh ( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\) | \(2\sqrt{a}{}_{0}F_{2}( ;1,\frac{3}{2};aw) \) |
123 | \(\frac{\cosh ( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\) | \(\frac{1}{\sqrt{w\pi }}{}_{0}F_{2}( ;\frac{1}{2},\frac{1}{2};aw) \) |
124 | \(\frac{\sinh^{2}( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\) | \(\frac{2a\sqrt{w}}{\sqrt{\pi }}{}_{1}F_{3}( 1;\frac{3}{2}, \frac{3}{2},2;aw) \) |
125 | \(\frac{\cosh^{2}( 2\sqrt{a}\sqrt{x}) }{\sqrt{x}}\) | \(\frac{1}{\sqrt{\pi w}} [ 1+2aw {}_{1}F_{3}( 1;\frac{3}{2}, \frac{3}{2},2;aw) ] \) |
126 | \(\frac{\sinh ( 2\sqrt{a}\sqrt{x}) }{x^{\frac{3}{4}}}\) | \(\frac{\sqrt{8a}}{w^{\frac{1}{4}}( -\frac{1}{4}) !} {}_{0}F _{2}( ;\frac{3}{4},\frac{3}{2};2aw) \) |
127 | \(\frac{\cosh ( 2\sqrt{a}\sqrt{x}) }{x^{\frac{3}{4}}}\) | \(\frac{1}{w^{\frac{3}{4}}( -\frac{1}{4}) !} {}_{0}F_{2}( ; \frac{1}{2},\frac{3}{4};2aw) \) |
128 | \(x^{v-1}\sinh ( 2\sqrt{a}\sqrt{x})\) | \(\frac{\sqrt{2a}w^{v-\frac{1}{2}}}{( v-\frac{1}{2}) !} {}_{0}F_{2}( ;\frac{3}{2},v+\frac{1}{2};\frac{aw}{2}) \) |
129 | \(x^{v-1}\cosh ( 2\sqrt{a}\sqrt{x})\) | \(\frac{w^{v-1}}{(v-1)!} {}_{0}F_{2}( ;\frac{1}{2},v;\frac{aw}{2}) \) |
130 | sinh^{−1}(x) | \(w{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1,\frac{3}{2}, \frac{3}{2};-\frac{w ^{2}}{4}) \) |
131 | cosh^{−1}(x) | \(-iw{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};1,\frac{3}{2}, \frac{3}{2};\frac{w^{2}}{4}) +\frac{i\pi }{2} \) |
132 | \(\cosh^{-1}( 1+\frac{x}{a})\) | \(\sqrt{\frac{8w}{a\pi }}{}_{2}F_{2}( \frac{1}{2},\frac{1}{2}; \frac{3}{2}, \frac{3}{2};-\frac{w}{2a}) \) |
133 | xsinh^{−1}(x) | \(\frac{w^{2}}{2}{}_{2}F_{3}( \frac{1}{2},\frac{1}{2};\frac{3}{2}, \frac{3}{2},2;-\frac{w^{2}}{4}) \) |
134 | sinh((2n + 1)sinh^{−1}(x)) | \((2n+1)w{}_{2}F_{3}( -n,n+1;1,\frac{3}{2}, \frac{3}{2};-\frac{w ^{2}}{4}) \) |
135 | cosh(2nsinh^{−1}(x)) | \({}_{2}F_{3}( n,-n;\frac{1}{2}, \frac{1}{2},1;-\frac{w^{2}}{4}) \) |
136 | sinh(vsinh^{−1}(x)) | \(vw{}_{2}F_{3}( \frac{1+v}{2},\frac{1-v}{2};1,\frac{3}{2}, \frac{3}{2};-\frac{w^{2}}{4}) \) |
137 | cosh(vsinh^{−1}(x)) | \({}_{2}F_{3}( \frac{v}{2},-\frac{v}{2};\frac{1}{2}, \frac{1}{2},1;-\frac{w ^{2}}{4}) \) |
138 | \(\sinh ( v\cosh^{-1}( 1+\frac{x}{a}) )\) | \(v\sqrt{\frac{8w}{a\pi }}{}_{2}F_{2}( \frac{1}{2}+v,\frac{1}{2}-v; \frac{3}{2}, \frac{3}{2};-\frac{w}{2a}) \) |
139 | \(\frac{\exp (n\sinh^{-1}(x))}{\sqrt{1+x^{2}}}\) | \(nw {}_{2}F_{3}( 1+\frac{n}{2},1-\frac{n}{2};1,\frac{3}{2}, \frac{3}{2}; -\frac{w^{2}}{4}) + {}_{2}F_{3}( \frac{1-n}{2}, \frac{1+n}{2};\frac{1}{2},\frac{1}{2},1; -\frac{w^{2}}{4}) \) |
140 | \(\frac{\exp (-n\sinh^{-1}(x))}{\sqrt{1+x^{2}}}\) | \(-nw {}_{2}F_{3}( 1-\frac{n}{2},1+\frac{n}{2};1,\frac{3}{2}, \frac{3}{2}; -\frac{w^{2}}{4}) + {}_{2}F_{3}( \frac{1-n}{2}, \frac{1+n}{2};\frac{1}{2},\frac{1}{2},1; -\frac{w^{2}}{4}) \) |
141 | \(\frac{\sinh (v\sinh^{-1}(x))}{\sqrt{x^{2}+1}}\) | \(vw {}_{2}F_{3}( 1-\frac{v}{2},1+\frac{v}{2};1,\frac{3}{2}, \frac{3}{2}; -\frac{w^{2}}{4}) \) |
142 | \(\frac{\cosh (n\sinh^{-1}(x))}{\sqrt{x^{2}+1}}\) | \({}_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2},\frac{1}{2},1; -\frac{w^{2}}{4}) \) |
143 | \(\frac{\cosh (n\cosh^{-1}(x))}{\sqrt{x^{2}-1}}\) | \(-\frac{1}{2e^{\frac{in\pi }{2}}} [ i( 1+e^{in\pi }) {}_{2}F_{3}( \frac{1-n}{2},\frac{1+n}{2};\frac{1}{2},\frac{1}{2},1;\frac{w ^{2}}{4}) + nw( 1+e^{in\pi }) {}_{2}F_{3}( 1- \frac{n}{2},1+\frac{n}{2};1,\frac{3}{2},\frac{3}{2};\frac{w^{2}}{4}) ] \) |
144 | \(\frac{\exp (2v\sinh^{-1}( \frac{x}{2a}) )}{\sqrt{x^{3}+4a ^{2}x}}\) | \(\frac{\operatorname{csgn}(a)}{2a^{2}\sqrt{\pi w}} [ a{}_{2}F_{3}( \frac{1}{2}+v, \frac{1}{2}-v;\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{w^{2}}{16a ^{2}}) + 2vw{}_{2}F_{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4}, \frac{3}{2};-\frac{w^{2}}{16a^{2}}) ] \) |
145 | \(\frac{\exp (-2v\sinh^{-1}( \frac{x}{2a}) )}{\sqrt{x^{3}+4a ^{2}x}}\) | \(\frac{\operatorname{csgn}(a)}{2a^{2}\sqrt{\pi w}} [ a{}_{2}F_{3}( \frac{1}{2}+v, \frac{1}{2}-v;\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{w^{2}}{16a ^{2}}) - 2vw{}_{2}F_{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4}, \frac{3}{2};-\frac{w^{2}}{16a^{2}}) ] \) |
146 | \(\frac{1}{\sqrt{x^{3}+4a^{2}x}} (\cos ( ( v+\frac{1}{4})\pi ) \exp ( -2v\sinh^{-1}( \frac{x}{2a}) )+\sin ( ( v+\frac{1}{4}) \pi ) \exp ( 2v\sinh^{-1}( \frac{x}{2a}) ) )\) | \(\frac{\operatorname{csgn}(a)}{a^{2}\sqrt{2\pi w}} [a\cos (\pi v){}_{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{2}, \frac{3}{4};-\frac{w^{2}}{16a^{2}}) + 2vw\sin (\pi v){}_{2}F_{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{16a ^{2}}) ] \) |
147 | \(\frac{1}{\sqrt{x^{3}+4a^{2}x}} (\sin ( ( v+\frac{1}{4})\pi ) \exp ( -2v\sinh^{-1}( \frac{x}{2a}) )-\cos ( ( v+\frac{1}{4}) \pi ) \exp ( 2v\sinh^{-1}( \frac{x}{2a}) ) ) \) | \(\frac{\operatorname{csgn}(a)}{a^{2}\sqrt{2\pi w}} [ a\sin (\pi v){} _{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{2}, \frac{3}{4};-\frac{w^{2}}{16a^{2}}) - 2vw\cos (\pi v){}_{2}F _{3}( 1-v,1+v;\frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{16a ^{2}}) ] \) |
148 | \(\frac{\sinh ( 2v\sinh^{-1}( \frac{x}{2a}) ) }{\sqrt{x^{3}+4a^{2}x}}\) | \(\frac{v\sqrt{w}\operatorname{csgn}(a)}{a^{2}\sqrt{\pi }}{}_{2}F_{3}( 1-v,1+v; \frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{w^{2}}{16a^{2}}) \) |
149 | \(\frac{\cosh ( 2v\sinh^{-1}( \frac{x}{2a}) ) }{\sqrt{x^{3}+4a^{2}x}}\) | \(\frac{\operatorname{csgn}(a)}{2a\sqrt{\pi w}}{}_{2}F_{3}( \frac{1}{2}+v,\frac{1}{2}-v;\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{w^{2}}{16a ^{2}})\) |
150 | \(\frac{\sinh^{-1}(x)}{x}\) | \(\mbox{J}_{0}(w)-\frac{\pi }{2} [ \mbox{J}_{0}(w)\textbf{H} _{1}(w) -\mbox{J}_{1}(w)\textbf{H}_{0}(w) ]\) |
Special functions definition
S. No | Function | Definition |
---|---|---|
1 | First kind Bessel function | \(\mbox{J}_{n}(x)=\sum_{k=0}^{\infty }\frac{(-1)^{k}( \frac{x}{2})^{2k+n}}{k!(n+k)!}\) |
2 | Modified first kind Bessel function | \(\mbox{I}_{n}(x)=\sum_{k=0}^{\infty }\frac{( \frac{x}{2}) ^{2k+n}}{k!(n+k)!} \) |
3 | Kelvin real function | \(\mbox{ber}_{n}(x)=\operatorname{Re} J_{n}(i^{\frac{3}{2}}x) \) |
4 | Kelvin imaginary function | \(\operatorname{bei}_{n}(x)=\operatorname{Im} J_{n}(i^{\frac{3}{2}}x) \) |
5 | Error function | \(\mbox{erf}(x)=\frac{2}{\sqrt{\pi }}\int_{0}^{x}e^{-z^{2}}\,dz \) |
6 | Complementary error function | \(\mbox{erfc}(x)=\frac{2}{\sqrt{\pi }}\int_{x}^{\infty }e^{-z^{2}}\,dz \) |
7 | Struve function | \(\textbf{H}_{v}(x)=( \frac{x}{2}) ^{v+1} \sum_{k=0} ^{\infty }\frac{(-1)^{k}( \frac{x}{2}) ^{2k}}{\Gamma ( k+ \frac{3}{2}) \Gamma ( k+v+\frac{3}{2}) } \) |
8 | Modified Struve function | \(\textbf{L}_{v}(x)=( \frac{x}{2}) ^{v+1} \sum_{k=0} ^{\infty }\frac{( \frac{x}{2}) ^{2k}}{\Gamma ( k+ \frac{3}{2}) \Gamma ( k+v+\frac{3}{2}) } \) |
9 | Generalized hypergeometric function | \({}_{p}F_{q}( (a_{p});(b_{q});x) =\sum_{k=0}^{\infty }\frac{(a_{1})_{k}\cdot(a_{2})_{k}\cdots (a_{p})_{k}x^{k}}{(b_{1})_{k}\cdot(b_{2})_{k} \cdots (b_{q})_{k}k!} \) |
10 | Lommel S1 function | \(\textbf{S}^{(1)}_{\mu ,v}(x)=\frac{x^{\mu +1}{}_{1}F_{2} ( 1;\frac{ \mu -v+3}{2},\frac{\mu +v+3}{2};-\frac{x^{2}}{4}) }{(\mu +1)^{2}-v ^{2}} \) |
11 | Whittaker M function | \(\mbox{M}_{\kappa ,\mu }(x)=e^{-\frac{x}{2}}x^{\mu +\frac{1}{2}} \mbox{M} ( \mu -\kappa +\frac{1}{2},1+2\mu ;x) \) |
12 | Kummer function | \(\mbox{M}(a,b,c)=\sum_{n=0}^{\infty }\frac{a^{(n)}x^{n}}{b^{(n)}n!}= {}_{1}F_{1}(a;b;x) \) |
13 | Sign function | \(\operatorname{csgn}(x)= \begin{cases} 1 ; x<\mathscr{R}(x), \\ -1 ; x>\mathscr{R}(x) \end{cases} \) |
14 | Sine integral | \(\mbox{Si}(x)=\int_{0}^{x}\frac{\sin (z)\,dz}{z} \) |
15 | Cosine integral | \(\mbox{Ci}(x)=-\int_{x}^{\infty }\frac{\cos (z)\,dz}{z} \) |
16 | Hyperbolic cosine integral | \(\mbox{Chi}(x)=\gamma +\ln (x)+\int_{x}^{\infty }\frac{(\cos (z)-1)\,dz}{z} \) |
17 | Exponential integral | \(\mbox{Ei}(x)=\int_{x}^{\infty }\frac{e^{-z}\,dz}{z} \) |
18 | Laguerre polynomials | \(\mbox{L}_{n}(x)=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}( x^{n}e^{-x}) \) |
19 | Euler’s constant | γ = 0.5772156 |
Theorem 1
Proof
The proof is straightforward. □
The following steps of Algorithm 1 solve the ordinary differential equations [14].
3 New exact solutions of AGE-F1
Example 1
Example 2
Example 3
Example 4
Example 5
Remark 1
4 New exact complex solutions of AGE-F1
By computing another power series solution of Step 3 of Algorithm 1, the second set of solutions of (9) is studied in this section which are new exact solutions. All the computations are worked through the Maple computer algebra system.
Example 6
Example 7
Example 8
Example 9
Example 10
Remark 2
Next, when solving for general l, (9), DIST method in both the above methods gives the new approximate analytical solution in terms of Lommel S1 function which will be studied in a separate work numerically.
5 Conclusion
Through this research communication an algorithm based on the discrete inverse Sumudu Transform (DIST) was described to solve ordinary differential equations for their new exact analytical and complex solutions. An algebro-geometric equation for different integer value coefficients was studied with the algorithm and the method was proven by deriving their new solutions. Efficiency of the DIST method was shown via the comparative study in Remarks 1 and 2, Maple plots were shown graphically in Figs. 3 and 6. The enlarged list of functions and their inverse Sumudu transforms in Table 1 shows that inverting the elementary functions upon Sumudu discrete-wise gives the special functions in Table 2 and will help future research.
Declarations
Acknowledgements
The authors are very grateful to Universiti Putra Malaysia for the partial support under the research grant having vot number UPM-GPB/2017/9543000.
Funding
Universiti Putra Malaysia.
Authors’ contributions
All authors contributed equally. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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