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<title>Time-scale calculus</title>

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Time-scale calculus

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In <a href="Mathematics" title="Mathematics">mathematics</a>, time-scale calculus is a unification of the theory of <a href="Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a> with that of <a href="Differential_equation" title="Differential equation">differential equations</a>, unifying <a href="Integral" title="Integral">integral</a> and <a href="Differential_calculus" title="Differential calculus">differential calculus</a> with the <a href="Calculus_of_finite_differences" class="mw-redirect" title="Calculus of finite differences">calculus of finite differences</a>, offering a formalism for studying <a href="Hybrid_system" title="Hybrid system">hybrid systems</a>. It has applications in any field that requires simultaneous modelling of <a href="Discrete_time_and_continuous_time" title="Discrete time and continuous time">discrete and continuous</a> data. It gives a new definition of a <a href="Derivative" title="Derivative">derivative</a> such that if one differentiates a function defined on the <a href="Real_number" title="Real number">real numbers</a> then the definition is equivalent to standard differentiation, but if one uses a function defined on the <a href="Integer" title="Integer">integers</a> then it is equivalent to the <a href="Forward_difference" class="mw-redirect" title="Forward difference">forward difference</a> operator.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger.<a href="#cite_note-hilger-1">[1]</a> However, similar ideas have been used before and go back at least to the introduction of the <a href="Riemann%E2%80%93Stieltjes_integral" title="Riemann–Stieltjes integral">Riemann–Stieltjes integral</a>, which unifies sums and integrals.

<div class="mw-heading mw-heading2"><h2 id="Dynamic_equations">Dynamic equations</h2></div>

Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their <a href="Continuous_function" title="Continuous function">continuous</a> counterparts.<a href="#cite_note-bp-2">[2]</a> The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown <a href="Function_(mathematics)" title="Function (mathematics)">function</a> is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="Real_number" title="Real number">real numbers</a> or set of <a href="Integer" title="Integer">integers</a> but to more general time scales such as a <a href="Cantor_set" title="Cantor set">Cantor set</a>.

The three most popular examples of <a href="Calculus" title="Calculus">calculus</a> on time scales are <a href="Differential_calculus" title="Differential calculus">differential calculus</a>, <a href="Finite_differences" class="mw-redirect" title="Finite differences">difference calculus</a>, and <a href="Quantum_calculus" title="Quantum calculus">quantum calculus</a>. Dynamic equations on a time scale have a potential for applications such as in <a href="Population_dynamics" title="Population dynamics">population dynamics</a>. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population.

<div class="mw-heading mw-heading2"><h2 id="Formal_definitions">Formal definitions</h2></div>

A time scale (or measure chain) is a <a href="Closed_subset" class="mw-redirect" title="Closed subset">closed subset</a> of the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. The common notation for a general time scale is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation>

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</math><img src="./9c039979935c00b3b216cbb065999207872677f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }" loading="lazy">.

The two most commonly encountered examples of time scales are the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> and the <a href="Discrete_time" class="mw-redirect" title="Discrete time">discrete time</a> scale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle h\mathbb {Z} }</annotation>

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</math><img src="./524b90bece34462fbc03e94939f0175999d8172c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.889ex; height:2.176ex;" alt="{\displaystyle h\mathbb {Z} }" loading="lazy">.

A single point in a time scale is defined as:

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<annotation encoding="application/x-tex">{\displaystyle t:t\in \mathbb {T} }</annotation>

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</math><img src="./5d6433ab1f77088f03370484c41e086f288e8ac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.007ex; height:2.176ex;" alt="{\displaystyle t:t\in \mathbb {T} }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Operations_on_time_scales">Operations on time scales</h3></div>

The forward jump and backward jump operators represent the closest point in the time scale on the right and left of a given point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, respectively. Formally:

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<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=\inf\{s\in \mathbb {T} :s>t\}}</annotation>

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</math><img src="./94b32e70a3f79325e389300ac6d5a8a38a8a7883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.5ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=\inf\{s\in \mathbb {T} :s>t\}}" loading="lazy"> (forward shift/jump operator)</dd>

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<annotation encoding="application/x-tex">{\displaystyle \rho (t)=\sup\{s\in \mathbb {T} :s<t\}}</annotation>

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</math><img src="./ca7eab21011c389094e677e84b1ec1e2b72d2e80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.223ex; height:2.843ex;" alt="{\displaystyle \rho (t)=\sup\{s\in \mathbb {T} :s<t\}}" loading="lazy"> (backward shift/jump operator)</dd></dl>

The graininess <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is the distance from a point to the closest point on the right and is given by:

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<annotation encoding="application/x-tex">{\displaystyle \mu (t)=\sigma (t)-t.}</annotation>

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</math><img src="./e9ee014c10ade4970a5e6e63c819ae053ee3880d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.455ex; height:2.843ex;" alt="{\displaystyle \mu (t)=\sigma (t)-t.}" loading="lazy"></dd></dl>

For a right-dense <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (t)=t}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=t}</annotation>

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</math><img src="./fe62d7d5876511a1a8890aa7fb044dfde8824a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.917ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=t}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (t)=0}">

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<annotation encoding="application/x-tex">{\displaystyle \mu (t)=0}</annotation>

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</math><img src="./052a6ef207129be9fa0f107cca8d1839c54279c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.312ex; height:2.843ex;" alt="{\displaystyle \mu (t)=0}" loading="lazy">.

For a left-dense <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (t)=t.}">

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<annotation encoding="application/x-tex">{\displaystyle \rho (t)=t.}</annotation>

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</math><img src="./d69c1825aeb9fde2caabae53a0d227a0fc8cbb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.436ex; height:2.843ex;" alt="{\displaystyle \rho (t)=t.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Classification_of_points">Classification of points</h3></div>

For any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {T} }">

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<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {T} }</annotation>

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</math><img src="./f9d5c857bec85da0d226e496492aa2cf78299f78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {T} }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is:

<ul><li>left dense if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (t)=t}">

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<annotation encoding="application/x-tex">{\displaystyle \rho (t)=t}</annotation>

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</math><img src="./a257bb02e4000e70acd75991dc4651d0a679824d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.789ex; height:2.843ex;" alt="{\displaystyle \rho (t)=t}" loading="lazy"></li>

<li>right dense if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (t)=t}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=t}</annotation>

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</math><img src="./fe62d7d5876511a1a8890aa7fb044dfde8824a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.917ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=t}" loading="lazy"></li>

<li>left scattered if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (t)<t}">

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<annotation encoding="application/x-tex">{\displaystyle \rho (t)<t}</annotation>

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</math><img src="./e90d78d1ea507baf5dc973eb1254aed013a83bef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.789ex; height:2.843ex;" alt="{\displaystyle \rho (t)<t}" loading="lazy"></li>

<li>right scattered if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (t)>t}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (t)>t}</annotation>

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</math><img src="./f786f2965767c785892a75b9d20327a0513eec99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.917ex; height:2.843ex;" alt="{\displaystyle \sigma (t)>t}" loading="lazy"></li>

<li>dense if both left dense and right dense</li>

<li>isolated if both left scattered and right scattered</li></ul>

As illustrated by the figure at right:

<ul><li>Point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation>

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</math><img src="./cb0768c0bd659f2f84fb5ef9f4b74f336123d915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}" loading="lazy"> is dense</li>

<li>Point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle t_{2}}</annotation>

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</math><img src="./749fee708b41e7079eabd50d61c8bf3e965db16f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{2}}" loading="lazy"> is left dense and right scattered</li>

<li>Point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{3}}">

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{3}}</annotation>

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</math><img src="./c62226a0dd66a3f9ad5706a365227331d9acb3a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{3}}" loading="lazy"> is isolated</li>

<li>Point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{4}}">

<msub>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle t_{4}}</annotation>

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</math><img src="./733c1c9afebab516de6c89491b05d5478767b820.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{4}}" loading="lazy"> is left scattered and right dense</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Continuity">Continuity</h3></div>

<a href="Continuous_function" title="Continuous function">Continuity</a> of a time scale is redefined as equivalent to density. A time scale is said to be right-continuous at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> if it is right dense at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">. Similarly, a time scale is said to be left-continuous at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> if it is left dense at point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Derivative">Derivative</h2></div>

Take a function:

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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {T} \to \mathbb {R} ,}</annotation>

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</math><img src="./622d3d511ce44c5ac9d9df38e14fef7838f832ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.705ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {T} \to \mathbb {R} ,}" loading="lazy"></dd></dl>

(where R could be any <a href="Banach_space" title="Banach space">Banach space</a>, but is set to the real line for simplicity).

Definition: The delta derivative (also Hilger derivative) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\Delta }(t)}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle f^{\Delta }(t)}</annotation>

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</math><img src="./80a5f1ac760d00b8a2278a90fb9f5849219df9b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.571ex; height:3.176ex;" alt="{\displaystyle f^{\Delta }(t)}" loading="lazy"> exists if and only if:

For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

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</math><img src="./e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> there exists a neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> such that:

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|f(\sigma (t))-f(s)-f^{\Delta }(t)(\sigma (t)-s)\right|\leq \varepsilon \left|\sigma (t)-s\right|}</annotation>

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</math><img src="./3d79625b6caa0dd949902392ff58d86e86ad4689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:47.281ex; height:3.509ex;" alt="{\displaystyle \left|f(\sigma (t))-f(s)-f^{\Delta }(t)(\sigma (t)-s)\right|\leq \varepsilon \left|\sigma (t)-s\right|}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">.

Take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} =\mathbb {R} .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} =\mathbb {R} .}</annotation>

</semantics>

</math><img src="./2651494774b761e4d6dcea7f37a6749162757412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.974ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} =\mathbb {R} .}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (t)=t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=t}</annotation>

</semantics>

</math><img src="./fe62d7d5876511a1a8890aa7fb044dfde8824a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.917ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=t}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (t)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (t)=0}</annotation>

</semantics>

</math><img src="./052a6ef207129be9fa0f107cca8d1839c54279c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.312ex; height:2.843ex;" alt="{\displaystyle \mu (t)=0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\Delta }=f'}">

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\Delta }=f'}</annotation>

</semantics>

</math><img src="./84f68b38f580769301258333117980cd7be63903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.025ex; height:3.009ex;" alt="{\displaystyle f^{\Delta }=f'}" loading="lazy">; is the derivative used in standard <a href="Calculus" title="Calculus">calculus</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} =\mathbb {Z} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} =\mathbb {Z} }</annotation>

</semantics>

</math><img src="./4d68b1a4291295e53df9b73825f72fab01111cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.199ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} =\mathbb {Z} }" loading="lazy"> (the <a href="Integer" title="Integer">integers</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (t)=t+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (t)=t+1}</annotation>

</semantics>

</math><img src="./667077766543627c13cc4a3a6e6ffaebb1806eae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle \sigma (t)=t+1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (t)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (t)=1}</annotation>

</semantics>

</math><img src="./e57a714259c2070d5ea331a06d732457a94a187a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.312ex; height:2.843ex;" alt="{\displaystyle \mu (t)=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\Delta }=\Delta f}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\Delta }=\Delta f}</annotation>

</semantics>

</math><img src="./ada9ada97408f4ba340b6b08c39eabf65311115f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.235ex; height:3.009ex;" alt="{\displaystyle f^{\Delta }=\Delta f}" loading="lazy"> is the <a href="Forward_difference_operator" class="mw-redirect" title="Forward difference operator">forward difference operator</a> used in difference equations.

<div class="mw-heading mw-heading2"><h2 id="Integration">Integration</h2></div>

The delta integral is defined as the <a href="Antiderivative" title="Antiderivative">antiderivative</a> with respect to the delta derivative. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(t)}</annotation>

</semantics>

</math><img src="./5b57ed3bbf501fb3c7f4bc5c4eafa96bb9e32165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle F(t)}" loading="lazy"> has a continuous derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=F^{\Delta }(t)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)=F^{\Delta }(t)}</annotation>

</semantics>

</math><img src="./43a0a4f94e1cd57d2ea24b1a55b0132f084546e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.091ex; height:3.176ex;" alt="{\displaystyle f(t)=F^{\Delta }(t)}" loading="lazy"> one sets

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{r}^{s}f(t)\Delta (t)=F(s)-F(r).}</annotation>

</semantics>

</math><img src="./7323a3afa496f50370922017e94ca7ef48ad1cc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.192ex; height:5.843ex;" alt="{\displaystyle \int _{r}^{s}f(t)\Delta (t)=F(s)-F(r).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Laplace_transform_and_z-transform">Laplace transform and z-transform</h2></div>

A <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal<a href="#cite_note-bp-2">[2]</a> to a modified <a href="Z-transform" title="Z-transform">Z-transform</a>:

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}'\{x[z]\}={\frac {{\mathcal {Z}}\{x[z+1]\}}{z+1}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Partial_differentiation">Partial differentiation</h2></div>

<a href="Partial_differential_equation" title="Partial differential equation">Partial differential equations</a> and <a href="Partial_difference_equation" class="mw-redirect" title="Partial difference equation">partial difference equations</a> are unified as partial dynamic equations on time scales.<a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="Multiple_integration">Multiple integration</h2></div>

<a href="Multiple_integration" class="mw-redirect" title="Multiple integration">Multiple integration</a> on time scales is treated in Bohner (2005).<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Stochastic_dynamic_equations_on_time_scales">Stochastic dynamic equations on time scales</h2></div>

<a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equations</a> and stochastic difference equations can be generalized to stochastic dynamic equations on time scales.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Measure_theory_on_time_scales">Measure theory on time scales</h2></div>

Associated with every time scale is a natural <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a><a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a> defined via

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu ^{\Delta }(A)=\lambda (\rho ^{-1}(A)),}</annotation>

</semantics>

</math><img src="./ffd99670b0bbec9f9bd6d3d5240f0a9fac0aa699.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.552ex; height:3.176ex;" alt="{\displaystyle \mu ^{\Delta }(A)=\lambda (\rho ^{-1}(A)),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> denotes <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is the backward <a href="Shift_operator" title="Shift operator">shift operator</a> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. The delta integral turns out to be the usual <a href="Lebesgue%E2%80%93Stieltjes_integral" class="mw-redirect" title="Lebesgue–Stieltjes integral">Lebesgue–Stieltjes integral</a> with respect to this measure

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{r}^{s}f(t)\Delta t=\int _{[r,s)}f(t)d\mu ^{\Delta }(t)}</annotation>

</semantics>

</math><img src="./0ec4f6eb5d93c1c8188388a25738fb65fc560243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.43ex; height:6.176ex;" alt="{\displaystyle \int _{r}^{s}f(t)\Delta t=\int _{[r,s)}f(t)d\mu ^{\Delta }(t)}" loading="lazy"></dd></dl>

and the delta derivative turns out to be the <a href="Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a> with respect to this measure<a href="#cite_note-10">[10]</a>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{\Delta }(t)={\frac {df}{d\mu ^{\Delta }}}(t).}</annotation>

</semantics>

</math><img src="./1bbe3f13d72980649b58683dafb5778b411d8187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.02ex; height:6.176ex;" alt="{\displaystyle f^{\Delta }(t)={\frac {df}{d\mu ^{\Delta }}}(t).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Distributions_on_time_scales">Distributions on time scales</h2></div>

The <a href="Dirac_delta" class="mw-redirect" title="Dirac delta">Dirac delta</a> and <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a> are unified on time scales as the Hilger delta:<a href="#cite_note-11">[11]</a><a href="#cite_note-12">[12]</a>

</mrow>

</mrow>

</msubsup>

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{a}^{\mathbb {H} }(t)={\begin{cases}{\dfrac {1}{\mu (a)}},&t=a\\0,&t\neq a\end{cases}}}</annotation>

</semantics>

</math><img src="./2f80d25e9ff9858e87a893c77c33cb3575febbaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.578ex; margin-bottom: -0.26ex; width:24.543ex; height:8.843ex;" alt="{\displaystyle \delta _{a}^{\mathbb {H} }(t)={\begin{cases}{\dfrac {1}{\mu (a)}},&t=a\\0,&t\neq a\end{cases}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Fractional_calculus_on_time_scales">Fractional calculus on time scales</h2></div>

<a href="Fractional_calculus" title="Fractional calculus">Fractional calculus</a> on time scales is treated in Bastos, Mozyrska, and Torres.<a href="#cite_note-13">[13]</a>

<ul><li><a href="Analysis_on_fractals" title="Analysis on fractals">Analysis on fractals</a> for dynamic equations on a <a href="Cantor_set" title="Cantor set">Cantor set</a>.</li>

<li><a href="Multiple-scale_analysis" title="Multiple-scale analysis">Multiple-scale analysis</a></li>

<li><a href="Method_of_averaging" title="Method of averaging">Method of averaging</a></li>

<li><a href="Krylov%E2%80%93Bogoliubov_averaging_method" title="Krylov–Bogoliubov averaging method">Krylov–Bogoliubov averaging method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHilger1989" class="citation thesis cs1">Hilger, Stefan (1989). Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (PhD thesis). Universität Würzburg. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/246538565">246538565</a>.</cite>

</li>

<li id="cite_note-bp-2">^ <a href="#cite_ref-bp_2-0">a</a> <a href="#cite_ref-bp_2-1">b</a> <cite id="CITEREFMartin_Bohner_&_Allan_Peterson2001" class="citation book cs1">Martin Bohner & Allan Peterson (2001). Dynamic Equations on Time Scales. Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4225-9</bdi>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFAhlbrandtMorian2002" class="citation journal cs1">Ahlbrandt, Calvin D.; Morian, Christina (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2801%2900434-4">"Partial differential equations on time scales"</a>. Journal of Computational and Applied Mathematics. 141 (1–2): 35–55. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JCoAM.141...35A">2002JCoAM.141...35A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2801%2900434-4">10.1016/S0377-0427(01)00434-4</a>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFJackson2006" class="citation journal cs1">Jackson, B. (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2005.02.011">"Partial dynamic equations on time scales"</a>. Journal of Computational and Applied Mathematics. 186 (2): 391–415. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JCoAM.186..391J">2006JCoAM.186..391J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2005.02.011">10.1016/j.cam.2005.02.011</a>.</cite>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFBohnerGuseinov2004" class="citation journal cs1">Bohner, M.; Guseinov, G. S. (2004). <a rel="nofollow" class="external text" href="https://web.mst.edu/~bohner/papers/pdots.pdf">"Partial differentiation on time scales"</a> (PDF). Dynamic Systems and Applications. 13: 351–379.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFBohnerGuseinov2005" class="citation journal cs1">Bohner, M; Guseinov, GS (2005). "Multiple integration on time scales". Dynamic Systems and Applications. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.79.8824">10.1.1.79.8824</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFAgarwalBohnerO’ReganPeterson2002" class="citation journal cs1">Agarwal, Ravi; Bohner, Martin; O’Regan, Donal; Peterson, Allan (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2801%2900432-0">"Dynamic equations on time scales: a survey"</a>. Journal of Computational and Applied Mathematics. 141 (1–2): 1–26. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JCoAM.141....1A">2002JCoAM.141....1A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2801%2900432-0">10.1016/S0377-0427(01)00432-0</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://web.mst.edu/~bohner/tisc.html">Dynamic Equations on Time Scales</a> Special issue of Journal of Computational and Applied Mathematics (2002)</li>

<li><a rel="nofollow" class="external text" href="http://www.hindawi.com/journals/ade/volume-2006/si.1.html">Dynamic Equations And Applications</a> Special Issue of Advances in Difference Equations (2006)</li>

<li><a rel="nofollow" class="external text" href="http://www.e-ndst.kiev.ua/v9n1.htm">Dynamic Equations on Time Scales: Qualitative Analysis and Applications</a> Special issue of Nonlinear Dynamics And Systems Theory (2009)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.timescales.org">The Baylor University Time Scales Group</a></li>

<li><a rel="nofollow" class="external text" href="http://timescalewiki.org/index.php/Main_Page">Timescalewiki.org</a></li></ul></div><div><div class="zim-footer">

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